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Biometrics & Biostatistics International Journal

Research Article Volume 7 Issue 3

Area–biased poisson exponential distribution with applications

Ayesha Fazal

Roll of honor Mphil statistics, Kinnaird College for women, Pakistan

Correspondence: Ayesha Fazal, Department of statistics, Roll of honor Mphil statistics, Kinnaird College for women Lahore, Pakistan

Received: May 24, 2018 | Published: June 28, 2018

Citation: Fazal A. Area–biased poisson exponential distribution with applications. Biom Biostat Int J. 2018;7(3):256-261. DOI: 10.15406/bbij.2018.07.00216

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Abstract

In this paper, an area–biased form of the single parameter Poisson exponential distribution (PED) is obtained by area biasing the discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.1 Poisson–exponential distribution is an important discrete distribution which has many applications in countable datasets. The first four moments (about origin) and the central moments (about mean) have been obtained and hence expression for coefficient of variation (CV), skewness, kurtosis and index of dispersion are derived. To estimate the parameters of Area–biased Poisson exponential distribution (ABPED), maximum likelihood method (MLE) and method of moments (MOM) are also developed. The goodness of fit for (ABPED) has been discussed using three real data–sets the fit shows a better fit over size–biased Poisson Lindley distribution (SBPLD).

Keywords: poisson exponential distribution, weighted distributions, moments, estimation of parameters, goodness of fit

Introduction

Fazal & Bashir1 have obtained discrete Poisson exponential distribution (PED) for modelling count data with probability mass function (p.m.f)

P(X=x)=θ(1+θ)x+1,θ>0,x=0,1,2, ...  ...P(X=x)=θ(1+θ)x+1,θ>0,x=0,1,2, ...  ... (1.1)

The Poisson exponential distribution (PED) in (1.1) is a mixture of Poisson and exponential distribution when the parameter of Poisson distribution (λ) follows exponential distribution. The first four moments (about origin) and the variance of (PED) obtained by Fazal & Bashir1 are given as

μ1=1θ

 μ'2=2+θθ2  (1.2)

μ3=θ2+6θ+6θ3

μ4=θ3+14θ2+36θ+24θ4

μ2=1+θθ2

The mathematical properties and estimation of parameter have been discussed by Fazal & Bashir1 and its application proves that it is a good replacement of poisson distribution and Lindley distribution. The size–biased form of (PED) has been discussed by Fazal & Bashir1 and its goodness of fit gives quite satisfactory fit over size–biased poisson distribution, size–biased lindley distribution and size–biased geometric distribution. The mixture of Poisson and Size–biased exponential distribution has been discussed by Fazal & Bashir1 with properties and applications.

The size–biased and Area–biased distributions were discussed earlier by Fisher2 when sample observations have unequal probability of selection, therefore we apply weights to the distribution to model bias.

If the rv ‘x’ had pdf f(x,θ); x=0,1,2,…..,; θ>0 , then the weighted distribution is of the form

P(x;θ)=xmfo(x;θ)μm       (1.3)

For m=1 and m=2 we get the size–biased and area–biased distributions respectively. Area–biased distributions are applicable for sampling in forestry, medical sciences, psychology etc, different discrete mixed distributions have been size–biased and discussed with their applications in real data–sets. Shankar & Kumar3 obtained size–biased Poisson Garima distribution with mathematical properties to analyze genetics datasets. Shankar4 introduced size–biased Poisson Shankar distribution with applications. Shankar & Fasshaye5 considered the size–biased form of Poisson Sujata distribution which was first introduced by Shankar5 for modelling count data in various fields of knowledge. Shakila & Mujahid Rasul6 derived the Poisson Area–Biased Lindley distribution with its applications in biological data to prove that it gives a better fit than Poisson distribution. Shankar & Fassahe7 proposed the size–biased form of Poisson Amarenda distribution and its applications proved that it is a good replacement of size–biased Poisson distribution (SBPD) size–biased Poisson Lindley distribution (SBPLD) and Size–biased Poisson.

Sujhata Distribution (SBPSD).Shakila and Mujahid8 proposed the size–biased form of Poisson Janardhan distribution and derived its mathematical properties, whereas Janardhan distribution is a two parameter distribution obtained by Rama & Mishra9 as a mixture of exponential and gamma distribution. Rama & Mishra9 obtained the size–biased form of Qaussi Poisson–Lindley distribution of which size–biased poisson Lindley distribution is a particular case (SBPLD). Ahmed & Munir10 have discussed few size–biased discrete distributions and their generalizations with properties and application. The size–biased version of Poisson Lindley distribution has been discussed by Ghitanni & Mutairi11 and the new distribution introduced in this paper i.e Area–biased poisson exponential distribution (ABPED) gives more satisfactory fit as compared to size–biased Poisson Lindley distribution. The mathematical properties and estimation of parameters has been discussed and goodness of fit is also presented.12–16

Area–biased poisson exponential distribution

Using (1.1), (1.2), (1.3) the pmf of the Area–biased Poisson exponential distribution can be obtained as

P(x;θ)=x2fo(x;θ)μ2=x2θ/(1+θ)x+1(2+θ)/θ2

Where μ2=(2+θ)θ2  is the second raw moment of discrete poisson exponential distribution.

With simplifications we get the pmf of Area–biased Poisson exponential distribution with parameter θ as

P(X=x)=x2θ3(1+θ)x+1(2+θ)               θ>0  ,   x=1,2,3,4,.    (1.4)

                 

Graphs of Area– biased Poisson exponential distribution for different values of θ are shown in Figure 1 below

Figure 1 Graphs of Area–biased Poisson exponential distribution for different values of θ.

Moments and moment based measures of area–biased poisson exponential distribution

We start the mathematical derivations with moments and moment measures.
The first four raw moments of Area–Biased Poisson exponential distribution (ABPED) are

μ1=θ2+6θ+θ6θ(θ+2)

μ2=θ2+12θ+12θ2

μ3=θ4+30θ3+150θ2+240θ+120θ3(θ+2)

μ4=θ4+60θ3+420θ2+720θ+360θ4

The mean moments of ABPED are obtained by using the relationship between moments about mean and moments about origin

μ2=4θ3+16θ224θ+12θ2(θ+2)2

μ3=4θ5+28θ4336θ3+168θ2+144θ+48θ3(θ+2)3

μ4=760θ7+938θ6+6200θ5+32992θ4+97632θ3+132672θ2+65088θ+720θ4(θ+2)4

The Harmonic mean of Area–biased Poisson exponential distribution is

H.M=θθ+2

The coefficient of variation (C.V), coefficient of skewness (β1) , coefficient of kurtosis (β1)  and index of dispersion (γ) of Area–biased Poisson exponential distribution ABPED are obtained as:

C.V=σμ1=4θ3+16θ224θ+12θ2+6θ+6

β1=μ3μ232=4θ5+28θ4336θ3+168θ2+144θ+48(4θ3+16θ224θ+12)32

β2=μ4μ22=76θ7+938θ6+6200θ5+32992θ4+97632θ3+132672θ2+65088θ+720(4θ3+16θ224θ+12)2

γ=σ2μ1'=4θ3+16θ224θ+12θ(θ+2)(θ2+6θ+6)

For the Area–biased Poisson exponential distribution, (ABPED), from equations (1.6) and (1.7) it can be seen that (γ1,β2)(1.15, 5) as θ→0, the model is positively skewed and leptokurtic.

To study the characteristics and comparative behavior of ABPED and SBPLD, a table of(γ1,β2)(1.15, 5) for varying values of the parameter θ, has been prepared and presented in the table below (Table 1 & 2):

1

2

3

4

5

6

μ1'

4.3333

2.75

2.2

1.9167

1.7429

1.625

μ2

0.8889

0.5625

0.4267

0.3264

0.2547

0.2031

0.2176

0.3521

0.4199

0.4497

0.4614

0.4637

2.4749

2.3754

1.4434

0.7824

0.3155

0.03754

5254.97

621.85

245.63

155.66

121.86

106.53

0.2051

0.3409

0.3879

0.3877

0.37096

0.3494

Table 1 Values of θ for ABPED

 

1

2

3

4

5

6

μ1'

3.66667

2.25

1.8

1.583333

1.457143

1.375

μ2

5.55556

1.9375

1.093333

0.743056

0.556735

0.442708

0.642824

0.61864

0.580903

0.544425

0.512061

0.483901

1.318047

1.49478

1.649924

1.790721

1.921224

2.043701

5.4744

6.057232

6.599941

7.118613

7.625214

8.125813

1.515152

0.861111

0.607407

0.469298

0.382073

0.32197

Table 2 Values of θ for SBPLD

The comparative graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of ABPED and SBPLD are shown in Figure 2.

Figure 2 Graphs of for ABPED and SBPLD.

Reliability measures

Using pmf of ABPED from (1.4), we have

P(x+1,θ)P(x.θ)=(1+1x)2(1+θ)

Which is a decreasing function of x, therefore ABPED is unimodal and has an increasing failure rate.

Generating functions of ABPED

Probability generating function: the probability generating function of ABPED can be obtained as

Px(t)=E(tx)=x=1x2θ3tx(1+θ)x+1(2+θ)

=θ3t2+t(θ4+θ3)θ4+5θ3+9θ2+7θt3(θ+2)+3t2(θ2+3θ+2)3t(θ3+4θ2+5θ+2)+2

Moment generating function: the moment generating function of ABPED is obtained as

Mx(t)=E(etx)=θ3e2t+(θ4+θ3)etθ4+5θ3+9θ2+7θ(θ+2)e3t+(3θ2+9θ+6)e2t(3θ3+12θ2+15θ+6)et

Fisher information matrix
If  x~f(x/θ)  where f(x/θ)=x2θ3(1+θ)x+1(2+θ)  is the pmf of Area–biased Poisson exponential distribution with θ>0, then

Ix(θ)=20θ48θ3192θ2309θ144θ4(1+θ)2(2+θ)3

is the Fisher Information Matrix of ABPED, with

Estimation of parameters

Method of moment (MOM) estimate: let x1, x2,x3…….xn be a sample of size n from ABPED (1.6) then equating the population parameter to the sample mean we obtain the MOM estimate ˜θ  of <θ of ABPED as

˜θ=(ˉx3)+(ˉx3)2+6(ˉx1)(ˉx1)

Where x is the sample mean.

Maximum likelihood estimate (MLE): let x1, x2, x3…….xn be a sample of size n from ABPED (1.6) , the MLE estimate ˜θ  of θ  is obtained as

˜θ=(ˉx3)+(ˉx3)2+6(ˉx1)(ˉx1)

Goodness of fit for ABPED

The Area–biased Poisson exponential distribution has been fitted to a number of countable datasets, and compared with size–biased Poisson lindley distribution. The following examples are used to illustrate a few situations generating the Area–biased distribution and–their applications. Microsoft Excel has been used to facilitate the use of the size–biased models to real life data. The MOM and MLE estimates are used to fit the distributions and is presented in the tables below. The datasets include number of observations of size distributions i:e small groups in various public situations reported by James,17 Coleman and James,18 and Simnoff,19 thunderstorm datasets reported by Shankar et al.,16 for all these datasets the ABPED distribution gives much closer fit than SBPLD (Tables 3–6).

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

1486

1532.5

1480.381

2

694

630.6

704.9073

3

195

191.9

188.8048

4

37

51.3

39.95664

5

10

12.8

7.43203

6

1

3.9

1.273997

TOTAL

2423

2423

2422.756

Estimation of parameters

 

13.766

1.2166

d.f

3

2

p-value

<0.01

0.74903

AICc

3.2732

2.99996

BIC

2.0649

1.7918

Table 3 Counts of group of people in public Places on a spring afternoon in Portland

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

316

323

312.1584

2

141

132.5

148.1144

3

44

40.2

39.53136

4

5

10.7

8.336454

5

4

3.6

1.545125

Total

510

510

Estimation of parameters

 

3.021

0.9728

d.f

2

2

p-value

0.4

0.6148

AICc

3.51453

3.33331

BIC

1.79064

1.60942

Table 4 Counts of shopping groups-Eugene, spring, department store and public market

Size of groups

Observed frequency

Expected frequency

SBPLD

ABPED

1

305

314.4

302.7967

2

144

134.4

150.5768

3

50

42.5

42.1199

4

5

11.8

9.309185

5

2

3.1

1.808334

6

1

0.8

0.323734

Total

507

507

506.9

Estimation of parameters

 

6.415

2.8126

d.f

2

2

p-value

0.043

0.2451

AICc

3.2809

3.000006

BIC

2.0727

1.79177

Table 5 counts of play Groups-Eugene, spring, public playground D

X

Fo

Expected frequency

SBPD

ABPED

1

87

83.2

83.95544

2

25

30.5

29.27444

3

5

5.6

5.741838

4

3

0.7

0.889832

Total

120

120

119.8615

Estimation of parameters

 

1.624

1.0167

d.f

1

1

p-value

0.2025

0.3133

Table 6 Frequency of thunderstorm events containing X thunderstorms at cape kennedy for May

Conclusions

Area–biased Poisson exponential distribution (ABPED) has been derived by area biasing the Discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.1 The discussion on estimation and applications of the area–biased distribution demonstrates that ABPED has a practical use to real life data. Form AIC and BIC measures the proposed Area–biased model appears to offer substantial improvements in fit over the size–biased poisson Lindley model. Also the fitting in these tables reveal that the Area–biased distribution provides us with better fits in the situations where zero–class is missing gives a better fit than size–biased Poisson Lindley distribution (SBPLD) and size–biased Poisson distribution(SBPD) therefore we conclude that area–biased poisson exponential distribution is a good replacement of (SBPLD).

Acknowledgements

The author expresses thankfulness to the learned referee for her valuable suggestions which improved the quality of the paper.

Conflict of interests

None.

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