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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 4 Issue 2

On discrete three parameter burr type XII and discrete lomax distributions and their applications to model count data from medical science

Para BA, Jan TR

Department of statistics, University of Kashmir, India

Correspondence: Bilal Ahmad Para, Department of statistics, University of Kashmir, Srinagar, J&K(India)-192301, Srinagar, Jammu and Kashmir, India

Received: May 19, 2016 | Published: July 23, 2016

Citation: Para BA, Jan TR. On discrete three parameter burr type xii and discrete lomax distributions and their applications to model count data from medical science. Biom Biostat Int J. 2016;4(2):70-82. DOI: 10.15406/bbij.2016.04.00092

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Abstract

In this paper we propose a discrete analogue of three parameter Burr type XII distribution and discrete Lomax distribution as new discrete models using the general approach of discretization of continuous distribution. The models are plausible in modeling discrete data and exhibit both increasing and decreasing hazard rates. We shall first study some basic distributional and moment properties of these new distributions. Then, certain structural properties of the distributions such as their unimodality, hazard rate behaviors and the second rate of failure functions are discussed. Developing a discrete versions of three parameter Burr type XII and Lomax distributions would be helpful in modeling a discrete data which exhibits heavy tails and can be useful in medical science and other fields. The equivalence of discrete three parameter Burr type XII (DBD-XII) and continuous Burr type XII (BD-XII) distributions has been established and similarly characterization results have also been made to establish a direct link between the discrete Lomax distribution and its continuous counterpart. Various theorems relating a three parameter discrete Burr type XII distribution and discrete Lomax distribution with other statistical distributions have also been proved. Finally, the models are examined with an example data set originated from a study,1,2 data set of counts of cysts of kidneys using steroids and compared with the classical models.

Keywords: discrete lomax distribution, AIC, ML estimate, failure rate, medical sciences, index of dispersion

Introduction

Statistical models describe a phenomenon in the form of mathematical equations. Plethora of continuous lifetime models in reliability theory is now available in the subject to portray the survival behavior of a component or a system. Most of the lifetimes are continuous in nature and hence many continuous life distributions have been studied in literature Kapur & Lamberson,3 Lawless4 and Sinha.5 However, it is sometimes impossible or inconvenient in life testing experiments to measure the life length of a device on a continuous scale. Equipment or a piece of equipment operates in cycles and experimenter observes the number of cycles successfully completed prior to failure. A frequently referred example is copier whose life length would be the total number of copies it produces. Another example is the lifetime of an on/off switching device is a discrete random variable, or life length of a device receiving a number of shocks it sustain before it fails. Or in case of survival analysis, we may record the number of days of survival for lung cancer patients since therapy, or the times from remission to relapse are also usually recorded in number of days. In the recent past special roles of discrete distribution is getting recognition from the analysts in the field of reliability theory. In this context, the well known distributions namely geometric and negative binomial are known discrete alternatives for the exponential and gamma distributions, respectively. It is also well known that these discrete distributions have monotonic hazard rate functions and thus they are unsuitable for some situations. Fortunately, many continuous distributions can be discretized. As mentioned earlier, the discrete versions of exponential and gamma are geometric and negative binomial. There are three discrete versions of the continuous Weibull distribution.14 The discrete versions of the normal and rayleigh distributions were also proposed by Roy.6,7 Discrete analogues of two parameter Burr XII and Pareto distributions were also proposed by Krishna & Punder.8 Recently discrete inverse Weibull distribution was studied,9 which is a discrete version of the continuous inverse Weibull variable, defined as X1 where X  denotes the continuous Weibull random variable. Para & Jan10 proposed a discrete version of two parameter Burr type III distribution as a reliability model to fit a range of discrete life time data. Deniz & Ojeda11 introduced a discrete version of Lindley distribution by discretizing the continuous failure model of the Lindley distribution. Also, a compound discrete Lindley distribution in closed form is obtained after revising some of its properties. Nekoukhou et al.12 presented a discrete analog of the generalized exponential distribution, which can be viewed as another generalization of the geometric distribution, and some of its distributional and moment properties were discussed.

 In the present paper we propose a three parameter discrete Burr type XII (DBD-XII) model and a two parameter discrete Lomax model as there is a need to find more plausible discrete life time distributions or survival models in medical science and other fields, to fit to various life time data. The model has a flexible index of dispersion which broaden its range to fit a data sets arising in medical science/biological science, engineering, finance etc.

Burr13 introduced a family of distributions includes twelve types of cumulative distribution functions, which yield a variety of density shapes. The two important members of the family are Burr type III and Burr type XII distributions. Types III and XII are the simplest functionally and therefore, the two distributions are the most desirable for statistical modeling.

 A continuous random variable X is said to follow a three parameter Burr type XII distribution if its pdf is given by

f(x)={ckγ(xy)c10(1+(xy)c)(k+1),x>0,c>0,k>o,γ>0elsewhere

and its cumulative distribution function is given by

F(x) =1(1+(xγ)c)k

x>0,k>0,c>0,γ>0

When c=1, the three parameter Burr type XII distribution becomes Lomax distribution with pdf given

f(x)={kγ (1+(xγ))(k+1)  ,x>0 , k>0,γ>00                                                     elsewhere

and its cumulative distribution function is given by

F(x) =1(1+(xγ))k

x>0,k>0,γ>0

Figures 1-4 gives the pdf plot for three parameter Burr type XII distribution and Lomax distribution for different values of parameters. Figure 3 & Figure 4 are especially for Lomax distribution. It is evident that the distribution of the rv X exhibit a right skewed nature.

Figure 1 pdf plot for BD-XII (c,k,γ)

Figure 2 pdf plot for BD-XII (c,k,γ)

Figure 3 pdf plot for BD-XII (c,k, γ).

Figure 4 PDF plot for BD-XII (c, k, γ).

The various reliability measures of three parameter Burr type XII random variable X are given by

  1. Survival function

                s(x)=1x0f(x)dx

=1x0ckγ (xγ)c1(1+(xγ)c)(k+1)dx

=(1+(xγ)c)γk

                               > 0; c > 0; k > 0; γ>0        

  1. The failure rate is given by

          r(x)= ckγ (xγ)c1/1+(xγ)c

> 0; c > 0; k > 0; γ>0

  1. The second rate of failure is given by

         SRF(x)=log(s(x)s(x+1))=klog(1+(x/γ)c1+((x+1)/γ)c)

> 0; c > 0; k > 0; γ>0

  1. The rth moment is
  2.     E(xr)=0xrf(x)dx

    =rβ(rc+1, krc)
 Where

β(a,b)=0xa1(1+x)a+bdx  ,

> 0; c > 0; k > 0; σ>0;ck>r

The convergence of the rth moment is only possible if ck>r

Three parameter discrete Burr type XII and discrete lomax model

Roy14 pointed out that the univariate geometric distribution can be viewed as a discrete concentration of a corresponding exponential distribution in the following manner:

[X=x]=s(x) s (x+1)

 When x = 0, 1, 2,…..

Where X is discrete random variable following geometric distribution with probability mass functions as

(x)= θx(1θ)   x = 0,1,2,…….

Where s(x) represents the survival function of an exponential distribution of the form s(x) = exp(λx) clearly θ= exp(λ), 0 < θ< 1 .

Thus, one to one correspondence between the geometric distribution and the exponential distribution can be established, the survival functions being of the same form.

The general approach of dicretising a continuous variable is to introduce a greatest integer function of X i.e., [X] (the greatest integer less than or equal to X till it reaches the integer), in order to introduce grouping on a time axis.

A discrete Burr type XII variable, dX can be viewed as the discrete concentration of the continuous Burr type XII variable X, where the corresponding probability mass function of dX can be written as:

P(dX=x)=p(x)=s(x)s(x+1)

The probability mass function takes the form

P(x)=βlog(1+(x/γ)c)βlog(1+((x+1)/γ)c)   x=0,1,2,3.                                                 (3.1) 

Where β=ek; 0<β<1;γ>0;c>0

And the cumulative distribution function is given by

F(x)=1βlog(1+((x+1)/γ)c)    Where β=e(k);0<β<1;γ>0;c>0                          (3.2)

When c=1, the three parameter discrete Burr type XII distribution becomes discrete Lomax distribution with pdf and cdf given by

P(x)=βlog(1+(x/γ))βlog(1+((x+1)/γ))   x=0,1,2,3.                                                  (3.3) 

Where β=ek; 0 < β<1;γ>0

F(x)= 1βlog(1+(x/y))

   Where β=ek;0<β<1;σ>0                              (3.4)

The quantile functions for three parameter discrete Burr type XII and discrete Lomax distributions can be obtained by inverting (3.2) and (3.4) respectively.

xφ=[γ(ef(φ,β)1)1c1]   for DBD-XII and xφ=[γ(ef(φ,β)1)1]     for DLomax distribution.

Where f(φ,β)=log(1φ)logβ ;c>0, γ>0, β>0 

Where [ ] denotes the greatest integer function (the largest integer less than or equal). In particular, the median can be written as x0.5=[γ(ef(β)1)1c1]    for three parameter discrete Burr type XII distribution and for discrete Lomax distribution the median is x0.5=[γ(ef(β)1)1]      Where f(β)=log(2)logβ ;c>0, γ>0, β>0

The parameter β  completely determines the pmf (3.1) at x = 0 and = 1. It should be also noted that in this case the p(x) is always monotonic decreasing for x = 1,2,3,4,….

    When c<log(e(β)1)log2  Where (β)=log(2βlog(2)1)logβ

Where β=ek;0<β<1;c>0  otherwise it is no longer monotonic decreasing but is unimodal, having a mode at x=1 i.e., it takes a jump at x=1 and then decreases for all x1 Figures 5-10 exhibit a graphical overview of the pmf plot for both three parameter discrete Burr type XII and discrete Lomax models for different values of parameters.

Figure 5 pmf plot for DBD-XII (β, c, γ).

Figure 6 pmf plot for DBD-XII (β, c, γ).

Figure 7 pmf plot for DBD-XII (β, c, γ).

Figure 8 pmf plot for DBD-XII (β, c, γ).

Figure 9 pmf plot for DLomax (β, γ).

Figure 10 pmf plot for DLomax (β, γ).

In addition, the modal value of three parameter discrete Burr type XII distribution, xm  is given by xm=[γ(c1clog(β)+1)1c] , in case when c>1 [if c1 , then the distribution is monotonic decreasing for all x=0,1,2,…..] , the value of c plays a very important role in determining the shape of the cdf curve , the lower the value of c , the sharper the fall of cdf curve, while lower the value of k parameter, the sharper the initial rise of the cdf curve.

When γ1 and c>1 , the distribution of three parameter discrete Burr type XII model can attain model value other than at x=1 and x=0 also. Figures 11-13 provides display of pmf plot when the model value of the distribution is other than at x=1 also.

Figure 11 pmf plot for DBD-XII (β, c, γ).

Figure 12 pmf plot for DBD-XII (β, c, γ).

Figure 13 pmf plot for DBD-XII (β, c, γ).

Reliability measures of three parameter discrete Burr type XII random variable Dx are given by

  1. Survival function

s(x)=p(dXx)=βlog(1+(x/γ)c) where β=ek; 0 <β<1;γ>0;c>0 x=0,1,2,

s(x) is same for continuous Burr type XII distribution and discrete Burr type XII distribution at the integer points of x.

  1. Rate of Failure, r(x) is given by

r(x)=p(x)s(x)=βlog(1+(x/γ))βlog(1+((x+1)/γ))βlog(1+(x/γ)c)

where β=ek; 0 <β<1;γ>0;c>0     x=0,1,2,

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  1. Second Rate of Failure is given by

SRF(x)=log(βlog(1+(x/γ)c)βlog(1+((x+1)/γ)c)) where β=ek; 0 <β<1;γ>0;c>0     x=0,1,2,

The reliability measures for discrete Lomax distribution can be directly obtained from reliability measures of three parameter discrete Burr type XII distribution by taking c=1.

It could be seen that r(x) and SRF(x) are always monotonic decreasing functions if

γ=1

and c<log[eϕ(β)1]/log2=α(say) Where (β)=log(β2log2)logβ

β=e(k);0<β<1;σ>0;c>0

Figures 14-19 illustrates the second rate of failure plot for DBD-XII and discrete Lomax models for different values of parameters. For c >α; r(0)< r(1) and SRF(0)< SRF(1) and for all other values of x ≥ 1, r(x) and SRF(x) decreases, clearly the hazard rates of continuous model and the discrete modal shows the same monotonocity. In case γ1 the hazard rate function for three parameter Burr type XII can attain maximum at other than x=0 and x=1 also as illustrated in Figure 18.

Moments of three parameter discrete burr type XII distribution and discrete lomax distribution

Figure 14 SRF(x) plot for DBD-XII (β, c, γ).

Figure 15 SRF(x) plot for DBD-XII (β, c, γ).

Figure 16 SRF(x) plot for DBD-XII (β, c, γ).

Figure 17 SRF(x) plot for DBD-XII (β, c, γ).

Figure 18 SRF(x) plot for DBD-XII (β, c, γ).

Figure 19 SRF(x) plot for DBD-XII (β, c, γ).

E(xr)=x=0xrp(x)

=x=1[xr(x1)r]s(x)

Now, E(x)=1s(x) =1βlog(1+(x/γ)c)

E(x2)=1(2x1)s(x) =1(2x1)βlog(1+(x/γ)c)

V(x)=1(2x1)βlog(1+(x/γ)c){1(2x1)βlog(1+(x/γ)c)}2

For checking purpose of moments convergence or divergence, we have

E(xr)=x=1[xr(x1)r]s(x)     rx=1xr1βlog(1+(x/γ)c)ckx=11xckr+1

Where β=ek; 0 <β<1;γ>0;c>0;k=logeβ which is convergent if ck-r+1>1 or ck>r

In case of discrete Lomax distribution, for the convergence of moments k should be greater than r. Hence, E(xr) for three parameter Burr type XII distribution and discrete Lomax distribution exists if and only if ck>r and k>r respectively. Or in other words when β<er/c moments of three parameter Burr type XII distribution exists. There is a one to one correspondence between three parameter continuous Burr type XII distribution and three parameter discrete Burr type XII distribution, as the expressions for survival function, failure rate function, second rate of failure function for DBD-XII (β,c,γ) can be directly obtained from continuous Burr type XII distribution by replacing k=logeβ .

Table 1 and Table 2 exhibits the index of dispersion D = [E(X2) − (E(X))2]/E(X), for different values of the parameters c, β and γ for three parameter discrete Burr type XII distribution and discrete Lomax distribution. It can be seen that this variance to mean ratio goes on increasing in case of discrete Lomax distribution as the parameters goes on increasing, and therefore in this case the discrete Lomax distribution seems over dispersed. In case of discrete Burr type XII as β and c goes on increasing the distribution shows under dispersion.

Different Values of γ

Different Values of β

Parameters

0.0001

0.0003

0.0009

0.0060

0.0200

0.0300

0.0400

0.0500

0.10

1.0060

1.0120

1.0250

1.1030

1.3000

1.4610

1.6510

1.8850

0.11

1.0060

1.0120

1.0260

1.1060

1.3060

1.4690

1.6630

1.8990

0.12

1.0060

1.0120

1.0260

1.1080

1.3120

1.4780

1.6740

1.9140

0.14

1.0060

1.0130

1.0280

1.1130

1.3240

1.4950

1.6960

1.9430

0.17

1.0070

1.0150

1.0310

1.1210

1.3420

1.5210

1.7310

1.9860

0.20

1.0080

1.0160

1.0330

1.1300

1.3610

1.5470

1.7650

2.0300

0.25

1.0100

1.0190

1.0380

1.1440

1.3930

1.5900

1.8220

2.1030

0.33

1.0120

1.0240

1.0470

1.1680

1.4440

1.6610

1.9140

2.2200

0.50

1.0200

1.0360

1.0680

1.2220

1.5550

1.8120

2.1090

2.4680

1.00

1.0530

1.0870

1.1450

1.3940

1.8850

2.2550

2.6800

3.1940

1.11

1.0620

1.1000

1.1640

1.4330

1.9580

2.3520

2.8060

3.3530

2.00

1.1530

1.2220

1.3310

1.7550

2.5520

3.1450

3.8280

4.6530

2.50

1.2110

1.2970

1.4310

1.9400

2.8890

3.5940

4.4080

5.3900

3.33

1.3140

1.4270

1.6020

2.2520

3.4540

4.3460

5.3760

6.6210

5.00

1.5380

1.7060

1.9600

2.8920

4.6040

5.8730

7.3390

9.1120

10.00

2.2650

2.5920

3.0810

4.8530

8.0870

10.4860

13.2590

16.6180

Table 1 Index of dispersion for DLomax for different values of β and γ

Different values of c

Different values of  β

 

 

 

 

 

 

 

Parameters

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

2

0.99

0.9945

1.0035

1.0156

1.0307

1.0484

1.0688

1.092

1.1181

3

0.9609

0.9391

0.9222

0.9083

0.8967

0.887

0.8789

0.8722

0.8668

4

0.959

0.934

0.9131

0.8945

0.8778

0.8624

0.8481

0.8349

0.8225

5

0.9589

0.9336

0.9121

0.8928

0.8751

0.8584

0.8428

0.8279

0.8137

6

0.9589

0.9336

0.912

0.8926

0.8747

0.8578

0.8419

0.8266

0.812

7

0.9589

0.9336

0.912

0.8926

0.8746

0.8578

0.8417

0.8264

0.8117

Different values of c

Different values of β

Parameters

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

2

1.1803

1.2169

1.2579

1.3036

1.3548

1.4122

1.4768

1.5497

1.6324

3

0.8598

0.8582

0.8577

0.8584

0.8604

0.8636

0.8682

0.8742

0.8816

4

0.8003

0.7904

0.7811

0.7726

0.7647

0.7576

0.7511

0.7453

0.7403

5

0.7872

0.7748

0.7628

0.7514

0.7404

0.7299

0.7198

0.7102

0.701

6

0.7843

0.7711

0.7583

0.746

0.7339

0.7222

0.7109

0.6998

0.6891

7

0.7836

0.7703

0.7572

0.7445

0.7322

0.7201

0.7083

0.6967

0.6854

Different values of c

Different values of β

Parameters

0.41

0.42

0.43

0.5

0.51

0.52

0.53

0.54

0.6

5

0.654

0.6636

0.6756

0.848

0.8929

0.9461

1.0094

1.0854

1.2897

6

0.55

0.5504

0.5518

0.6015

0.617

0.6356

0.6578

0.6844

0.7539

7

0.5046

0.5007

0.4973

0.4969

0.5013

0.5074

0.5153

0.5253

0.5531

8

0.4831

0.4769

0.4711

0.4454

0.4446

0.4447

0.4459

0.4483

0.4577

9

0.4724

0.465

0.4579

0.4181

0.4143

0.4112

0.4088

0.4071

0.4068

 

10

0.467

0.4589

0.451

0.4029

0.3974

0.3923

0.3878

0.3838

0.3777

Table 2 Index of dispersion for DBD-XII for different values of β and c when γ=1

Estimation of the parameters of three parameter discrete Burr type XII distribution and discrete Lomax distribution

Estimation of the parameters based on the ML method: Let X1,X2X3,Xn be a random sample of size n. If these Xi.'s are assumed to be iid random variables following three parameter discrete Burr type XII distribution i.e., DBDXII(β,c,γ) their likelihood function is given by L(β,c,γ;x)=ni=1p(xi)

=ni=1(βlog(1+(x/γ)c)βlog(1+((x+1)/γ)c)) (5.1)

And (5.1) can be rewritten as follows L(β,c,γ;x)=ni=1βlog(1+(x/γ)c)(1β(xi,c,γ)) (5.2)

where (xi,c,γ)=log[(1+((xi+1)/γ)c)(1+(xi/γ)c)]

logL=[log(1+(x/γ)c)logβ+log(1β(xi,c,γ))] (5.3)

Taking partial derivatives with respect to β,c and γ and equating them to zero, we obtain the normal equations. Which can be solved to obtain the maximum likelihood estimators. logLβ=ni=1[log(1+(xi/γ)c)ˆβ(xi,c,γ)ˆβ(xi,c,γ)1)1ˆβ(xi,c,γ))]=0 (5.4)

logLc=ni=1[((xi/γ)ˆc)logβlogxi1+(xγ)ˆclogβ'(xi,ˆc,γ)β(xi,ˆc,γ)1β(xi,ˆc,γ)]   =0 (5.5)

Where '(xi,ˆc,γ)=(xi,c,γ)c (5.6)

logLγ=ni=1[(cxicˆγ(c+1))logβ1+(xˆγ)clogβ'(xi,c,ˆγ)β(xi,c,ˆγ)β(xi,c,ˆγ)1]   =0 (5.6)

'(xi,c,ˆγ)=(xi,c,γ)γ

The solution of this system is not possible in a closed form, so by using numerical computation, the solution of the three log-likelihood equations (5.4), (5.5) and (5.6) will provide the MLE of (β,c,γ) .

In this study, maximum likelihood estimates of were computed by numerical methods, using the R studio statistical software with the help of “MASS” package. For solving the equations analytically Nelder_Mead optimization method15 is employed.

We here now consider the four possible cases for estimating the parameters.

Case I: known parameters c and γ and unknown parameter β.

logLβc=ˆc,γ=ˆγ, β=ˆβ=0 yields

ni=1[log(1+(xi/γ)c)ˆβϕ(xi,c,γ)ˆβϕ(xi,c,γ)1)1ˆβϕ(xi,c,γ))]=0

Solving the Equation (5.7) analytically gives the maximum likelihood estimator ˆβ of the parameter β .

Case II: known parameter c and unknown parameters β and γ.

ni=1[(cXciˆγ(c+1))logβ1+(Xˆγ)clogβϕ' (5.8)

Solving the Equations (5.7) and (5.8) analytically gives the maximum likelihood estimators β ̂ and γ ̂ of the parameters β and γ.

Case III: known parameter γ and unknown parameters β and c.

logL c c= c ̂ ,γ= γ ̂ , β= β ̂ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaae4yaaaacaqG8bGaaeiOa8aadaWgaaqaa8 qacaqGJbGaeyypa0ZdamaaxacabaWdbiaabogaa8aabeqaa8qacqWI cmajaaGaaiilaiaabo7acqGH9aqppaWaaCbiaeaapeGaae4SdaWdae qabaWdbiablkWaKaaacaGGSaGaaeiOaiaabk7acqGH9aqppaWaaCbi aeaapeGaaeOSdaWdaeqabaWdbiablkWaKaaaa8aabeaacqGH9aqpca aIWaaaaa@5255@ yields

i=1 n [ (( x i /γ ) c ̂ ) logβlogx i 1+ ( x γ ) c ̂ logβ'( x i , c, ̂ γ ) β ( x i , c ̂ ,γ ) 1 β ( x i , c, ̂ γ ) ]   =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqabKqbG8aabaWdbiaabMgacqGH9aqpcaaIXaaapaqa a8qacaqGUbaajuaGpaqaa8qacqGHris5aaWaamWaa8aabaWdbmaala aapaqaa8qacaGGOaWaaeWaa8aabaWdbiaabIhapaWaaSbaaKqbGeaa peGaaeyAaaqcfa4daeqaa8qacaGGVaGaae4SdiaacMcapaWaaWbaae qabaWaaCbiaKqbGeaapeGaae4yaaqcfa4daeqabaWdbiablkWaKaaa aaaacaGLOaGaayzkaaGaaeiBaiaab+gacaqGNbGaaeOSdiaabYgaca qGVbGaae4zaiaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqa aaqaa8qacaaIXaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qaca qG4baapaqaa8qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabeaa daWfGaqcfasaa8qacaqGJbaajuaGpaqabeaapeGaeSOadqcaaaaaaa GaeyOeI0YaaSaaa8aabaWdbiaabYgacaqGVbGaae4zaiaabk7acqaH fiIXcaqGNaWaaeWaa8aabaWdbiaabIhapaWaaSbaaeaapeGaaeyAaa Wdaeqaa8qacaGGSaWdamaaxacabaWdbiaabogacaGGSaaapaqabeaa peGaeSOadqcaaiaabo7aaiaawIcacaGLPaaacaqGYoWdamaaCaaabe qcfasaa8qacqaHfiIXjuaGdaqadaqcfaYdaeaapeGaaeiEaKqba+aa daWgaaqcfasaa8qacaqGPbaapaqabaWdbiaacYcajuaGpaWaaCbiaK qbGeaapeGaae4yaaWdaeqabaWdbiablkWaKaaacaGGSaGaae4SdaGa ayjkaiaawMcaaaaaaKqba+aabaWdbiaaigdacqGHsislcaqGYoWdam aaCaaabeqcfasaa8qacqaHfiIXjuaGdaqadaqcfaYdaeaapeGaaeiE aKqba+aadaWgaaqcfasaa8qacaqGPbaapaqabaWdbiaacYcajuaGpa WaaCbiaKqbGeaapeGaae4yaiaacYcaa8aabeqaa8qacqWIcmajaaGa ae4SdaGaayjkaiaawMcaaaaaaaaajuaGcaGLBbGaayzxaaGaaeiOai aabckacaqGGcGaeyypa0JaaGimaaaa@9121@ (5.9)

Solving the Equations (5.7) and (5.9) analytically gives the maximum likelihood estimators β ̂ and c ̂ of the parameters and .

Case IV: Unknown parameters β , c and γ . Solving the Equations (5.7), (5.8) and (5.9) analytically gives the maximum likelihood estimators c ̂  ,  γ ̂   and  β ̂ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbiae aaqaaaaaaaaaWdbiaabogaa8aabeqaa8qacqWIcmajaaGaaeiOaiaa cYcapaWaaCbiaeaapeGaaeiOaiaabo7aa8aabeqaa8qacqWIcmajaa GaaeiOaiaabckacaqGHbGaaeOBaiaabsgacaqGGcWdamaaxacabaWd biaabk7aa8aabeqaa8qacqWIcmajaaaaaa@4699@ of the parameters c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbaaaa@378D@ , γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzaaa@384C@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ respectively.

Estimation of the parameters based on the proportion method: Khan et al.16 proposed and provided a motivation for the method of proportions to estimate the parameters for discrete Weibull distribution. Now, we present a similar method for the three parameter discrete Burr type XII distribution and discrete Lomax distribution for the same reasons as outlined.16 Let x 1, x 2 , x 3, , x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaaigdacaGGSaaajuaGpaqa baWdbiaabIhapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaa8qaca GGSaGaaeiEa8aadaWgaaqcfasaa8qacaaIZaGaaiilaaqcfa4daeqa a8qacqGHMacVcaGGSaGaaeiEa8aadaWgaaqcfasaa8qacaqGUbaaju aGpaqabaaaaa@4660@ be a random sample from the distribution with pmf (3.1). Define the indicator function by I u ( x i )={ 1            if  x i =u  0            if  x i u  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbWdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeWa aeWaa8aabaWdbiaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4dae qaaaWdbiaawIcacaGLPaaacqGH9aqpdaGabaWdaeaafaqabeGabaaa baWdbiaaigdacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaWGPbGaamOzaiaa cckacaqG4bWdamaaBaaajuaibaWdbiaabMgaaKqba+aabeaapeGaey ypa0JaamyDaiaacckaa8aabaWdbiaaicdacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaWGPbGaamOzaiaacckacaqG4bWdamaaBaaajuaibaWdbiaa bMgaaKqba+aabeaapeGaeyiyIKRaamyDaiaacckaaaaacaGL7baaaa a@7079@

Denote f u = I u ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaabwhaaKqba+aabeaapeGa eyypa0ZdamaavacabeqabeaacaaMb8oabaWdbiabggHiLdaacaWGjb WdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeWaaeWaa8aabaWd biaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaaaWdbiaawI cacaGLPaaaaaa@45C1@ by the frequency of the value u in the observed sample.

Therefore, the proportion (relative frequency) R u = f u n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbWdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeGa eyypa0ZaaSaaa8aabaWdbiaadAgapaWaaSbaaKqbGeaapeGaamyDaa qcfa4daeqaaaqaa8qacaWGUbaaaaaa@3EB9@ can be used to estimate the probability P( u;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWaaeWaa8aabaWdbiaadwhacaGG7aGaeqOSdiMaaiil aiaadogacaGGSaGaae4SdaGaayjkaiaawMcaaaaa@3FFD@ . Now we consider the following cases for the purpose of parameter estimation.

Case I: known parameters c and γ and unknown parameter β.

This is the simplest case. The unknown parameter β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ has a proportion estimator in exact solution, where

P( 0;β,c,γ )= 1 β log( 1+ 1 γ c )  = f 0 n      Where 0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWaaeWaa8aabaWdbiaaicdacaGG7aGaeqOSdiMaaiil aiaadogacaGGSaGaae4SdaGaayjkaiaawMcaaiabg2da9iaabckaca aIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaapeGaciiBaiaac+ga caGGNbqcfa4aaeWaaKqbG8aabaWdbiaaigdacqGHRaWkjuaGdaWcaa qcfaYdaeaapeGaaGymaaWdaeaapeGaae4SdKqba+aadaahaaqcfasa beaapeGaae4yaaaaaaaacaGLOaGaayzkaaaaaKqbakaabckacqGH9a qpdaWcaaWdaeaapeGaamOza8aadaWgaaqcfasaa8qacaaIWaaajuaG paqabaaabaWdbiaad6gaaaGaaeiOaiaabckacaqGGcGaaeiOaiaabc kacaqGxbGaaeiAaiaabwgacaqGYbGaaeyzaiaabckacaaIWaGaeyip aWJaeqOSdiMaeyipaWJaaGymaiaacUdacqaHZoWzcqGH+aGpcaaIWa Gaai4oaiaadogacqGH+aGpcaaIWaaaaa@6F8B@ (5.10)

β * = e log( ( 1 f 0 n ) ( 1 1 γ c ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaGaeyypa0Jaamyz a8aadaahaaqabKqbGeaapeGaamiBaiaad+gacaWGNbqcfa4aaeWaaK qbG8aabaqcfa4dbmaalaaajuaipaqaaKqba+qadaqadaqcfaYdaeaa peGaaGymaiabgkHiTKqbaoaalaaajuaipaqaa8qacaWGMbqcfa4dam aaBaaajuaibaWdbiaaicdaa8aabeaaaeaapeGaamOBaaaaaiaawIca caGLPaaaa8aabaqcfa4dbmaabmaajuaipaqaa8qacaaIXaGaeyOeI0 scfa4aaSaaaKqbG8aabaWdbiaaigdaa8aabaWdbiaabo7ajuaGpaWa aWbaaKqbGeqabaWdbiaadogaaaaaaaGaayjkaiaawMcaaaaaaiaawI cacaGLPaaaaaaaaa@536C@

f 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@3955@ denotes the number of zero’s in a sample of size n.

Case II: known parameter c and unknown parameters β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ and γ.

Let f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@3956@ denote the number of one’s in the sample of size n.

β log( 1+ 1 γ c ) β log( 1+ ( 2 γ ) c ) = f 1 n    Where 0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEga juaGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaalaaajuaipa qaa8qacaaIXaaapaqaa8qacaqGZoqcfa4damaaCaaajuaibeqaa8qa caqGJbaaaaaaaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaeOSd8aada ahaaqabKqbGeaapeGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbG8aa baWdbiaaigdacqGHRaWkjuaGdaqadaqcfaYdaeaajuaGpeWaaSaaaK qbG8aabaWdbiaaikdaa8aabaWdbiaabo7aaaaacaGLOaGaayzkaaqc fa4damaaCaaajuaibeqaa8qacaqGJbaaaaGaayjkaiaawMcaaaaaju aGcqGH9aqpdaWcaaWdaeaapeGaamOza8aadaWgaaqcfasaa8qacaaI XaaajuaGpaqabaaabaWdbiaad6gaaaGaaeiOaiaabckacaqGGcGaae 4vaiaabIgacaqGLbGaaeOCaiaabwgacaqGGcGaaGimaiabgYda8iab ek7aIjabgYda8iaaigdacaGG7aGaeq4SdCMaeyOpa4JaaGimaiaacU dacaWGJbGaeyOpa4JaaGimaaaa@7073@ (5.11)

Solving equations (5.10) and (5.11) numerically using Newton_Raphson (N_R) method, gives the proportion estimators β * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaaaaa@3935@ and γ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoWdamaaCaaabeqaa8qacaqGQaaaaaaa@38CC@ of the parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382C@ .

Case III: known parameter γ and unknown parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and c. Solving equations (5.10) and (5.11) numerically using Newton_Raphson (N_R) method, gives the proportion estimators β* MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycaGGQaaaaa@38F4@ and c* MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yai aacQcaaaa@381A@ of the parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoaaaa@37DE@ .

Let f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaaaaa@3957@ denote the number of two’s in the sample of size n.

β log( 1+ ( 2 γ ) c ) β log( 1+ ( 3 γ ) c ) = f 2 n    Where 0<β<1;γ>0;c>0  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEga juaGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaabmaajuaipa qaaKqba+qadaWcaaqcfaYdaeaapeGaaGOmaaWdaeaapeGaae4Sdaaa aiaawIcacaGLPaaajuaGpaWaaWbaaKqbGeqabaWdbiaabogaaaaaca GLOaGaayzkaaaaaKqbakabgkHiTiaabk7apaWaaWbaaeqajuaibaWd biGacYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaIXaGaey 4kaSscfa4aaeWaaKqbG8aabaqcfa4dbmaalaaajuaipaqaa8qacaaI Zaaapaqaa8qacaqGZoaaaaGaayjkaiaawMcaaKqba+aadaahaaqcfa sabeaapeGaae4yaaaaaiaawIcacaGLPaaaaaqcfaOaeyypa0ZaaSaa a8aabaWdbiaadAgapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaaa qaa8qacaWGUbaaaiaabckacaqGGcGaaeiOaiaabEfacaqGObGaaeyz aiaabkhacaqGLbGaaeiOaiaaicdacqGH8aapcqaHYoGycqGH8aapca aIXaGaai4oaiabeo7aNjabg6da+iaaicdacaGG7aGaam4yaiabg6da +iaaicdacaGGGcaaaa@73FE@ (5.12)

Solving the Equations (5.10), (5.11) and (5.12) analytically using Newton_Raphson (N_R) method, gives the proportion estimators β * ,  c * and  γ *   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaGaaiilaiaabcka caqGJbWdamaaCaaabeqaa8qacaqGQaaaaiaabggacaqGUbGaaeizai aabckacaqGGcGaae4Sd8aadaahaaqabeaapeGaaeOkaaaacaqGGcaa aa@4528@ of the parameters β,c and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi MaaiilaiaadogaqaaaaaaaaaWdbiaacckapaGaamyyaiaad6gacaWG KbWdbiaacckapaGaeq4SdCgaaa@40BC@ respectively.

Some theorems related to three parameter discrete Burr type XII distribution and discrete Lomax distribution

In this section we discuss some important theorems which relate three parameter discrete Burr type XII Distribution and discrete Lomax distribution with some important discrete class of continuous distributions already in the literature.

Theorem 1: Let X be random variable following three parameter continuous Burr XII distribution with E( X r )<      r=1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIfapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaGaeyipaWJaeyOhIuQaaiiOai aacckacaGGGcGaaiiOaiaacckacqGHaiIicaGGGcGaamOCaiabg2da 9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaeyOjGWRaaiOlaaaa@4DD0@ Then E( Y r )<  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabMfapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaGaeyipaWJaaiiOaiabg6HiLc aa@3F7C@ where Y=[ X ]~DBX( x;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbiaabIfaaiaawUfacaGL DbaacaGG+bGaaeiraiaabkeacaqGybWaaeWaa8aabaWdbiaadIhaca GG7aGaeqOSdiMaaiilaiaadogacaGGSaGaae4SdaGaayjkaiaawMca aaaa@4762@

Proof: Proof is straight forward, since 0[ X ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyizIm6aamWaa8aabaWdbiaabIfaaiaawUfacaGL DbaacqGHKjYOcaqGybGaaeiOaaaa@3FB3@ , so clearly if E( X r )<           r=1,2,3....... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaaiikaiaadIfapaWaaWbaaeqajuaibaWdbiaadkha aaqcfaOaaiykaiabgYda8iabg6HiLkaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaeyia IiIaamOCaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaai Olaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@55D9@ Then E( [X] r )< MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaacUfacaqGybGaaiyxa8aadaah aaqabKqbGeaapeGaaeOCaaaaaKqbakaawIcacaGLPaaacqGH8aapcq GHEisPaaa@4017@ .

Theorem 2: If X~DBDXII( x;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaaiOFaiaadseacaWGcbGaamiraiabgkHiTiaadIfa caWGjbGaamysamaabmaabaGaamiEaiaacUdacqaHYoGycaGGSaGaam 4yaiaacYcacqaHZoWzaiaawIcacaGLPaaaaaa@4717@ then Y=[ [ log( 1+ ( x γ ) c ) ] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaadmaapaqaa8qacaqG SbGaae4BaiaabEgadaqadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaay zkaaaacaGLBbGaayzxaaWdamaaCaaabeqcfasaa8qacqGHsislcaaI XaGaai4laiaabogaaaaajuaGcaGLBbGaayzxaaaaaa@4D04@ follows discrete inverse Weibull distribution i.e., DIW ( c,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaaiilaiabek7aIbaa@39DE@ )

β= e k  ;  0<β<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaqGGcGaai4oaiaabckacaqGGcGaaGimaiabgY da8iabek7aIjabgYda8iaaigdaaaa@45E9@

Proof:-

P[ Yy ]=P[ [ [ log( 1+ ( x γ ) c ) ] 1/c ]y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaamWaa8aabaWdbiaabMfacqGHLjYScaqG5baacaGL BbGaayzxaaGaeyypa0Jaaeiuamaadmaapaqaa8qadaWadaWdaeaape WaamWaa8aabaWdbiaabYgacaqGVbGaae4zamaabmaapaqaa8qacaaI XaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8 qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaae4y aaaaaKqbakaawIcacaGLPaaaaiaawUfacaGLDbaapaWaaWbaaeqaju aibaWdbiabgkHiTiaaigdacaGGVaGaae4yaaaaaKqbakaawUfacaGL DbaacqGHLjYScaqG5baacaGLBbGaayzxaaaaaa@5850@

=P[ [ log( 1+ ( x γ ) c ) ] 1/c y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbmaadmaapaqaa8qacaqG SbGaae4BaiaabEgadaqadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaay zkaaaacaGLBbGaayzxaaWdamaaCaaabeqcfasaa8qacqGHsislcaaI XaGaai4laiaabogaaaqcfaOaeyyzImRaaeyEaaGaay5waiaaw2faaa aa@4FBD@

=P[ X [ γ c ( e y c 1)] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbiaabIfacqGHLjYScaGG BbGaae4Sd8aadaahaaqabKqbGeaapeGaae4yaaaajuaGcaGGOaGaae yza8aadaahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqcfasabeaa peGaeyOeI0Iaae4yaaaaaaqcfaOaeyOeI0IaaGymaiaacMcacaGGDb WdamaaCaaabeqcfasaa8qacaaIXaGaai4laiaabogaaaaajuaGcaGL BbGaayzxaaaaaa@4E0F@

=1 β log[ 1+ [ [ γ c ( e y c 1)] 1/c γ ] c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaa peGaaeiBaiaab+gacaqGNbqcfa4aamWaaKqbG8aabaWdbiaaigdacq GHRaWkjuaGdaWadaqcfaYdaeaajuaGpeWaaSaaaKqbG8aabaWdbiaa cUfacaqGZoqcfa4damaaCaaajuaibeqaa8qacaqGJbaaaiaacIcaca qGLbqcfa4damaaCaaajuaibeqaa8qacaqG5bqcfa4damaaCaaajuai beqaa8qacqGHsislcaqGJbaaaaaacqGHsislcaaIXaGaaiykaiaac2 fajuaGpaWaaWbaaKqbGeqabaWdbiaaigdacaGGVaGaae4yaaaaa8aa baWdbiaabo7aaaaacaGLBbGaayzxaaqcfa4damaaCaaajuaibeqaa8 qacaqGJbaaaaGaay5waiaaw2faaaaaaaa@5A01@

=1 β log e y c =1 β y c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaa peGaciiBaiaac+gacaGGNbGaaeyzaKqba+aadaahaaqcfasabeaape GaaeyEaKqba+aadaahaaqcfasabeaapeGaeyOeI0Iaae4yaaaaaaaa aKqbakabg2da9iaaigdacqGHsislcaqGYoWdamaaCaaabeqcfasaa8 qacaqG5bqcfa4damaaCaaajuaibeqaa8qacqGHsislcaqGJbaaaaaa aaa@4C2A@

Which is the survival function of a discrete inverse Weibull distribution.

Hence Y~ DIW( c, β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGzbGaaiOFaiaacckacaWGebGaamysaiaadEfadaqadaWd aeaapeGaam4yaiaacYcacaGGGcGaeqOSdigacaGLOaGaayzkaaaaaa@4220@

Theorem3: If X DBDXII (x;β,c,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaaiiOaiablYJi6iaadseacaWGcbGaamiraiabgkHi TiaadIfacaWGjbGaamysaiaabccapaGaaiika8qacaqG4bGaai4oai aabk7acaGGSaGaae4yaiaacYcacaqGZoWdaiaacMcaaaa@4828@ then Y=[ log[ 1+ ( x γ ) c ] ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaakaaapaqaa8qacaqG SbGaae4BaiaabEgadaWadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLBbGaay zxaaaabeaaaiaawUfacaGLDbaaaaa@474D@ follows discrete Raleigh distribution i.e., DRel ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGOaGaaeOSdiaacMcaaaa@3935@

β= e k  ;  0<β<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaqGGcGaai4oaiaabckacaqGGcGaaGimaiabgY da8iabek7aIjabgYda8iaaigdaaaa@45E9@

Proof:-

P[ Yy ]=P[ [ log[ 1+ ( x γ ) c ] ]y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaamWaa8aabaWdbiaabMfacqGHLjYScaqG5baacaGL BbGaayzxaaGaeyypa0Jaaeiuamaadmaapaqaa8qadaWadaWdaeaape WaaOaaa8aabaWdbiaabYgacaqGVbGaae4zamaadmaapaqaa8qacaaI XaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8 qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaae4y aaaaaKqbakaawUfacaGLDbaaaeqaaaGaay5waiaaw2faaiabgwMiZk aabMhaaiaawUfacaGLDbaaaaa@5299@

=P[ X[ γ c ( e y 2 1 ) ] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbiaabIfacqGHLjYSdaqc saWdaeaapeGaae4Sd8aadaahaaqabKqbGeaapeGaae4yaaaajuaGca GGOaGaaeyza8aadaahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqc fasabeaapeGaaGOmaaaaaaqcfaOaeyOeI0IaaGymaaGaay5waiaawM caaiaac2fapaWaaWbaaeqajuaibaWdbiaaigdacaGGVaGaae4yaaaa aKqbakaawUfacaGLDbaaaaa@4D5E@

= β log[ 1+ [ [ γ c ( e y 2 1 ) ] 1/c γ ] c ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcqaHYoGydaahaaqabeaaciGGSbGaai4BaiaacEga daWadaqaaiaaigdacqGHRaWkdaWadaqaamaalaaabaWaamWaaeaacq aHZoWzdaahaaqabKqbGeaacaWGJbaaaKqbaoaabmaabaGaamyzamaa CaaabeqcfasaaiaadMhajuaGdaahaaqcfasabeaacaaIYaaaaaaaju aGcqGHsislcaaIXaaacaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWba aeqajuaibaGaaGymaiaac+cacaWGJbaaaaqcfayaaiabeo7aNbaaai aawUfacaGLDbaadaahaaqabKqbGeaacaWGJbaaaaqcfaOaay5waiaa w2faaaaaaaa@555F@

= β log e y 2 = β y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4B aiaacEgacaqGLbqcfa4damaaCaaajuaibeqaa8qacaqG5bqcfa4dam aaCaaajuaibeqaa8qacaaIYaaaaaaaaaqcfaOaeyypa0JaaeOSd8aa daahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqcfasabeaapeGaaG Omaaaaaaaaaa@46AC@

which is the survival function of a discrete Raleigh distribution. Hence Y~ DRaleigh(  β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGzbGaaiOFaiaacckacaWGebGaamOuaiaadggacaWGSbGa amyzaiaadMgacaWGNbGaamiAamaabmaapaqaa8qacaGGGcGaeqOSdi gacaGLOaGaayzkaaaaaa@453D@ . Corollary. If X DLomax (x;β,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaeSipIOJaaiiOaiaadseacaWGmbGaam4Baiaad2ga caWGHbGaamiEaiaabccapaGaaiikaiaacIhacaGG7aGaeqOSdiMaai ilaiabeo7aNjaacMcaaaa@46EF@ then Y=[ log[ 1+( x γ ) ] ]~DRaleigh( β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaakaaapaqaa8qacaqG SbGaae4BaiaabEgadaWadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaaaiaawUfacaGLDbaaaeqaaaGaay5waiaaw2faaiaac6haca WGebGaamOuaiaadggacaWGSbGaamyzaiaadMgacaWGNbGaamiAamaa bmaapaqaa8qacqaHYoGyaiaawIcacaGLPaaaaaa@50DD@

Application of discrete Lomax distribution and three parameter discrete Burr type XII distribution in medical science

Here we consider the data set of counts of cysts of kidneys using steroids as given in the Table 3. The example data set originated from a study1 investigating the effect of a corticosteroid on cyst formation in mice fetuses undertaken within the Department of Nephro-Urology at the Institute of Child Health of University College London. Embryonic mouse kidneys were cultured, and a random sample was subjected to steroids (110). Table 4 exhibits some descriptive measures of count data of cysts of kidneys using steroids based on 1000 bootstrap samples.

For the purpose of parameter estimation, we employ the fitdistr procedure in R studio statistical software to find out the estimates of the parameters. The ML estimates and their standard errors provided by the fitdistr procedure are given in the Table 5. In Figure 20 the empirical cdf of the number of cysts in a kidney using steroid has been shown.

We compute the expected frequencies for fitting discrete Lomax, three parameter discrete Burr type XII, Poisson, Geometric, Inflated Poisson and DRayleigh distributions with the help of R studio statistical software and Pearson’s chi-square test is applied to check the goodness of fit of the models discussed. The calculated figures are given in the Table 5.

The p-values of Pearson’s Chi-square statistic are 0.532, 0.352, 0.0008, 0.000, 0.000 and 0.0006 for three parameter discrete Burr type XII, discrete Lomax, Zero Inflated Poisson, Poisson, Discrete Raleigh and geometric distributions, respectively Table 6. This reveals that Zero Inflated Poisson, Poisson, Geometric and discrete Rayleigh distributions are not good fit at all, whereas three parameter Burr type XII distribution and two parameter discrete Lomax distributions are good fit distributions with three parameter discrete Burr type XII model being the best one. The null hypothesis that data come from three parameter Burr type XII and two parameter discrete Lomax distributions is accepted. Figure 21 exhibits the graphical overview of the fitted distributions.

We have compared three parameter discrete Burr type XII distribution and two parameter discrete Lomax distribution with discrete Raleigh, Poisson, Zero Inflated Poisson and Geometric distributions using the Akaike information criterion (AIC), given by Akaike17 and the Bayesian information criterion (BIC), given by Schwarz.18 Generic function calculating Akaike's ‘An Information Criterion’ for one or several fitted model objects for which a log-likelihood value can be obtained, according to the formula -2*log-likelihood + k*npar, where npar represents the number of parameters in the fitted model, and k = 2 for the usual AIC, or k = log(n) (n being the number of observations) for the so called BIC or SBC (Schwarz's Bayesian criterion).

From Table 7, it is obvious that AIC and BIC criterion favors three parameter discrete Burr type XII and two parameter Lomax distributions in comparison with the Poisson, Zero Inflated Poisson, discrete Raleigh and Geometric distributions, in the case of Counts of cysts of kidneys using steroids.

Figure 21 exhibits graphical overview of the AIC, BIC and negative loglikelihood values for fitted distributions.

Counts of cysts of kidneys using steroids

0

1

2

3

4

5

6

7

8

9

10

11

Total

Frequency

65

14

10

6

4

2

2

2

1

1

1

2

110

Table 3 Counts of cysts of kidneys using steroids

Descriptive Measures

Statistic

Standard Error

Bootstrapa

Bias

Standard Error

95% Confidence Interval

Lower

Upper

Sum

153

Mean

1.39

0.236

0.01

0.23

0.95

1.87

Standard Deviation

2.472

-0.018

0.309

1.812

3.053

Variance

6.112

0.009

1.511

3.283

9.324

Skewness

2.293

0.23

-0.04

0.308

1.685

2.908

Kurtosis

5.089

0.457

-0.069

1.911

1.963

9.531

Valid N (listwise)

N

110

0

0

110

110

a. Bootstrap results are based on 1000 bootstrap samples

Table 4 Descriptive statistics of Counts of cysts of kidneys using steroids 

Distribution

Parameter Estimates

Standard Error of the Estimates

Model Function

Discrete Lomax

β=0.15,γ=1.83 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGimaiaac6cacaaIXaGaaGynaiaacYcacqaHZoWzcqGH 9aqpcaaIXaGaaiOlaiaaiIdacaaIZaaaaa@425A@

[0.098, 0.953]

β log( 1+( x y ) ) β log( 1+( x+1 γ ) )       x=0,1,2,3.... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaWbaaeqajuaibaGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbGeaa caaIXaGaey4kaSscfa4aaeWaaKqbGeaajuaGdaWcaaqcfasaaiaadI haaeaacaWG5baaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaajuaG cqGHsislcqaHYoGydaahaaqabKqbGeaaciGGSbGaai4BaiaacEgaju aGdaqadaqcfasaaiaaigdacqGHRaWkjuaGdaqadaqcfasaaKqbaoaa laaajuaibaGaamiEaiabgUcaRiaaigdaaeaacqaHZoWzaaaacaGLOa GaayzkaaaacaGLOaGaayzkaaaaaKqbacbaaaaaaaaapeGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiEaiabg2da9iaaicdaca GGSaGaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGUaGaaiOlaiaa c6cacaGGUaaaaa@66F1@ where β= e k ;0<β<1;γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaamyzamaaCaaabeqcfasaaiabgkHiTiaadUgaaaqcfaOa ai4oaiaaicdacqGH8aapcqaHYoGycqGH8aapcaaIXaGaai4oaiabeo 7aNjabg6da+iaaicdaaaa@46D5@

Poisson

λ=1.39 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW Maeyypa0JaaGymaiaac6cacaaIZaGaaGyoaaaa@3C2B@

[0.112]

e λ λ x x! λ>0;x=0,1,2,........ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGLbWaaWbaaeqajuaibaGaeyOeI0Iaeq4UdWgaaKqbakabeU7a SnaaCaaabeqcfasaaiaadIhaaaaajuaGbaGaamiEaiaacgcaaaGaeq 4UdWMaeyOpa4JaaGimaiaacUdacaWG4bGaeyypa0JaaGimaiaacYca caaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6caaaa@5037@

DRayleigh

q=0.90

[0.009]

q x 2 q ( x+1 ) 2    0<q<1;x=0,1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaCaaabeqcfasaaiaadIhajuaGdaahaaqcfasabeaacaaIYaaaaaaa juaGcqGHsislcaWGXbWaaWbaaKqbGeqabaqcfa4aaeWaaKqbGeaaca WG4bGaey4kaSIaaGymaaGaayjkaiaawMcaaKqbaoaaCaaajuaibeqa aiaaikdaaaaaaKqbacbaaaaaaaaapeGaaiiOaiaacckacaGGGcGaaG imaiabgYda8iaadghacqGH8aapcaaIXaGaai4oaiaadIhacqGH9aqp caaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaaaa@56A5@

Geom

q=0.418

[0.03]

q x q ( x+1 )    0<q<1;x=0,1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbWaaWbaaeqajuaibaGaamiEaaaajuaGcqGHsislcaWG XbWaaWbaaKqbGeqabaqcfa4aaeWaaKqbGeaacaWG4bGaey4kaSIaaG ymaaGaayjkaiaawMcaaaaajuaGcaGGGcGaaiiOaiaacckacaaIWaGa eyipaWJaamyCaiabgYda8iaaigdacaGG7aGaamiEaiabg2da9iaaic dacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOl aiaac6cacaGGUaaaaa@5371@

Three Parameter Burr type XII

β=0.003,c=0.72,γ=12.75 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGimaiaac6cacaaIWaGaaGimaiaaiodacaGGSaGaam4y aiabg2da9iaaicdacaGGUaGaaG4naiaaikdacaGGSaGaeq4SdCMaey ypa0JaaGymaiaaikdacaGGUaGaaG4naiaaiwdaaaa@4955@

[0.002, 0.087, 5.06]

β log( 1+ ( x/γ ) c ) β log( 1+( x+1 )/γ ) c )    x=0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGydaahaaqabKqbGeaaciGGSbGaai4BaiaacEgajuaG daqadaqcfasaaiaaigdacqGHRaWkjuaGdaqadaqcfasaaiaadIhaca GGVaGaeq4SdCgacaGLOaGaayzkaaqcfa4aaWbaaKqbGeqabaGaam4y aaaaaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaeqOSdi2aaWbaaeqaju aibaGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbGeaacaaIXaGaey4k aSscfa4aaeWaaKqbGeaacaWG4bGaey4kaSIaaGymaaGaayjkaiaawM caaiaac+cacqaHZoWzcaGGPaqcfa4aaWbaaKqbGeqabaGaam4yaaaa aiaawIcacaGLPaaaaaqcfaOaaiiOaiaacckacaGGGcGaamiEaiabg2 da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGG UaGaaiOlaaaa@66C7@ where 0<β<1;c>0;γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyipaWJaeqOSdiMaeyipaWJaaGymaiaacUdacaWG JbGaeyOpa4JaaGimaiaacUdacqaHZoWzcqGH+aGpcaaIWaaaaa@4353@   

Zero Inflated Poisson

α=3.27,0.57 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde Maeyypa0JaaG4maiaac6cacaaIYaGaaG4naiaacYcacaaIWaGaaiOl aiaaiwdacaaI3aaaaa@3FB1@

[0.049, 0.283]

{ α+( 1α ) e λ λ x /x!λ>0;x=0 ( 1α ) e λ λ x /x!λ>0;x=1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiqaaq aabeqaaiabeg7aHjabgUcaRmaabmaabaGaaGymaiabgkHiTiabeg7a HbGaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH7o aBaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaamiEaaaajuaGcaGGVaGa amiEaiaacgcacqaH7oaBcqGH+aGpcaaIWaGaai4oaiaadIhacqGH9a qpcaaIWaaabaWaaeWaaeaacaaIXaGaeyOeI0IaeqySdegacaGLOaGa ayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU7aSbaajuaGcq aH7oaBdaahaaqabKqbGeaacaWG4baaaKqbakaac+cacaWG4bGaaiyi aiabeU7aSjabg6da+iaaicdacaGG7aGaamiEaiabg2da9iaaigdaca GGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa caGL7baaaaa@6BDB@

Table 5 Estimated parameters by ML method for fitted distributions

X

Observed

DBD-XII

Discrete Lomax

ZIP

Poisson

Discrete Raleigh

Geometric

0

65

63.32

61.89

64.92

27.4

11

45.98

1

14

18.19

21.01

5.82

38.08

26.83

26.76

2

10

9.29

9.65

9.52

26.47

29.55

15.57

3

6

5.49

5.24

10.38

12.26

22.23

9.06

4

4

3.52

3.17

8.48

4.26

12.49

5.28

5

2

2.39

2.06

5.55

1.18

5.42

3.07

6

2

1.69

1.42

3.02

0.27

1.85

1.79

7

2

1.23

1.02

1.41

0.05

0.5

1.04

8

1

0.92

0.76

0.58

0.01

0.11

0.61

9

1

0.7

0.58

0.21

0

0.02

0.35

10

1

0.55

0.46

0.07

0

0

0.21

11

2

2.71

2.73

0.03

0

0

0.29

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaKqbGeqabaGaaGOmaaaaaaa@3947@ P-Values

0.532

0.352

0.0008

0.000

0.000

0.0006

Table 6 Table for goodness of fit

Criterion

Discrete Lomax

DBD-XII

ZIP

Poisson

Discrete Raleigh

Geometric

Neg-Loglike

170.4806

168.7708

182.2449

246.21

277.778

178.7667

AIC

344.9612

343.5415

368.4897

494.42

557.556

359.5333

BIC

350.3622

351.6429

373.8907

497.1205

560.2565

362.2338

Table 7 AIC, BIC and Negative loglikelihood values for fitted distributions

Figure 20 ECD of Counts of cysts of kidneys using steroids.

Figure 21 Overview of fitted distributions.

Figure 22 AIC, BIC and Negative loglikelihood values for fitted distributions. 

Acknowledgments

We don’t have any funding source. In acknowledge, mention, author is thankful reviewers for their construct and valid review which brought the quality of the manuscript up.

Conflicts of interest

None.

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