Submit manuscript...
eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 10 Issue 3

Adya distribution with properties and application 

Rama Shanker,1 Kamlesh Kumar Shukla,2 Amaresh Ranjan,3 Ravi Shanker4

1 Department of Statistics, Assam University, Silchar, Assam, India
2 Department of Community Medicine, Noida International Institute of Medical Science, India
3 Department of Mathematics, Nalanda Open University, India
4 Department of Mathematics, G.L.A. College, N.P University, India

Correspondence: Rama Shanker, Department of Statistics, Assam University, Silchar, Assam, India

Received: June 18, 2021 | Published: August 11, 2021

Citation: Shanker R, Shukla KK, Ranjan A, et al. Adya distribution with properties and application. Biom Biostat Int J. 2021;10(3):81-88. DOI: 10.15406/bbij.2021.10.00334

Download PDF

Abstract

In the present paper, a new one parameter lifetime distribution named, “Adya distribution’ has been proposed for modeling lifetime data from engineering. Its various statistical properties including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, deviations from the mean and the median, Bonferroni and Lorenz curves, and stress-strength reliability have been studied. Both the method of moment and the maximum likelihood estimation have been discussed for estimating the parameter of the proposed distribution. A numerical example has been presented to test the goodness of fit of the proposed distribution over other one parameter lifetime distributions available in statistical literature.

Keywords: lifetime distributions, statistical and mathematical properties, parameter estimation, goodness of fit

Introduction

The classical one parameter exponential distribution and Lindley distribution proposed by Lindley1 were useful for modeling lifetime data from engineering and biomedical. It has been observed by Shanker et al.2 that exponential and Lindley distributions are not suitable for several lifetime data. In search for better one parameter lifetime distributions, Shanker has introduced several one parameter lifetime distributions including Shanker,3 Aradhana,4 Sujatha,5 Devya.6 The probability density function (pdf) and the cumulative distribution function (cdf) of these distributions are presented in table 1.

Distributions

Probability density functions and Cumulative distribution functions

Shanker

pdf

f(x)=θ2θ2+1(θ+x)eθxf(x)=θ2θ2+1(θ+x)eθx

cdf

F(x)=1[1+θxθ2+1]eθxF(x)=1[1+θxθ2+1]eθx

Aradhana

pdf

f(x)=θ3θ2+2θ+2(1+x)2eθxf(x)=θ3θ2+2θ+2(1+x)2eθx

cdf

F(x)=1[1+θx(θx+2θ+2)θ2+2θ+2]eθxF(x)=1[1+θx(θx+2θ+2)θ2+2θ+2]eθx

Sujatha

pdf

f(x)=θ3θ2+θ+2(1+x+x2)eθxf(x)=θ3θ2+θ+2(1+x+x2)eθx

cdf

F(x)=1[1+θx(θx+θ+2)θ2+θ+2]eθxF(x)=1[1+θx(θx+θ+2)θ2+θ+2]eθx

Devya

pdf

f(x)=θ5θ4+θ3+2θ2+6θ+24(1+x+x2+x3+x4)eθxf(x)=θ5θ4+θ3+2θ2+6θ+24(1+x+x2+x3+x4)eθx

cdf

F(x)=1[1+{θ4(x4+x3+x2+x)+θ3(4x3+3x2+2x)+6θ2(2x2+x)+24θx}θ4+θ3+2θ2+6θ+24]eθxF(x)=1⎢ ⎢1+{θ4(x4+x3+x2+x)+θ3(4x3+3x2+2x)+6θ2(2x2+x)+24θx}θ4+θ3+2θ2+6θ+24⎥ ⎥eθx

Lindley

pdf

f(x)=θ2θ+1(1+x)eθxf(x)=θ2θ+1(1+x)eθx

cdf

F(x)=1[1+θxθ+1]eθxF(x)=1[1+θxθ+1]eθx

Table 1 pdf and cdf of Shanker, Aradhana, Sujatha, Devya, and Lindley distributions for x>0,θ>0x>0,θ>0

The reasons for introducing such lifetime distributions with their advantages and disadvantages, statistical properties, parameter estimation and applications are available in the respective papers.

In this paper, a new lifetime distribution which gives better fit over several one parameter lifetime distributions are introduced. The new one parameter lifetime distribution is defined by its cdf and pdf, respectively

F(x,θ)=1[1+θx(θx+2θ2+2)θ4+2θ2+2]eθx;x>0,θ>0F(x,θ)=1[1+θx(θx+2θ2+2)θ4+2θ2+2]eθx;x>0,θ>0 (1.1)

f(x;θ)=θ3θ4+2θ2+2(θ+x)2eθx;x>0,θ>0f(x;θ)=θ3θ4+2θ2+2(θ+x)2eθx;x>0,θ>0 (1.2)

 We name this distribution, “Adya distribution”. This is a convex combination of exponential (θ)(θ) , gamma (2,θ)(2,θ) and gamma (3,θ)(3,θ) distributions. We have

f(x;θ)=p1g1(x;θ)+p2g2(x;2,θ)+(1p1p2)g3(x;3,θ)f(x;θ)=p1g1(x;θ)+p2g2(x;2,θ)+(1p1p2)g3(x;3,θ)

 Where p1=θ4θ4+2θ2+2p1=θ4θ4+2θ2+2 ,p2=2θ4θ4+2θ2+2p2=2θ4θ4+2θ2+2 ,g1(x;θ)=θeθxg1(x;θ)=θeθx ,g2(x;2,θ)=θ2Γ(2)x21eθxg2(x;2,θ)=θ2Γ(2)x21eθx , and g3(x;3,θ)=θ3Γ(3)x31eθxg3(x;3,θ)=θ3Γ(3)x31eθx ;;x>0,θ>0;x>0,θ>0 .

The pdf and the cdf of Adya distribution for values of the parameter θθ are shown in figures 1 and 2, respectively.

Figure 1 The pdf of Adya distribution .

Figure 2 The cdf of Adya distribution .

Moments and moments based measures

The rr th moment about origin μr of (1.2) can be obtained as

μr=r!{θ4+2(r+1)θ2+(r+1)(r+2)}θr(θ4+2θ2+2);r=1,2,3,... (2.1)

Substitutingr=1,2,3,and4 in (2.1), the first four moments about origin of (1.2) are obtained as

μ1=θ4+4θ2+6θ(θ4+2θ2+2) ,μ2=2(θ4+6θ2+12)θ2(θ4+2θ2+2) ,

μ3=6(θ4+8θ2+20)θ3(θ4+2θ2+2) ,μ4=24(θ4+10θ2+30)θ4(θ4+2θ2+2)

Thus, the central moments of (1.2) are obtained as

μ2=θ8+8θ6+24θ4+24θ2+12θ2(θ4+2θ2+2)2 μ3=2(θ12+12θ10+54θ8+100θ6+108θ4+72θ2+24)θ3(θ4+2θ2+2)3 μ4=3(3θ16+48θ14+304θ12+944θ10+1816θ8+2304θ6+1920θ4+960θ2+240)θ4(θ4+2θ2+2)4

Descriptive measures including coefficient of variation (C.V) , coefficient of skweness (β1) , coefficient of kurtosis (β2) and index of dispersion (γ) of (1.2) are thus obtained as

C.V=σμ1=θ8+8θ6+24θ4+24θ2+12θ4+4θ2+6 β1=μ3μ23/2=2(θ12+12θ10+54θ8+100θ6+108θ4+72θ2+24)(θ8+8θ6+24θ4+24θ2+12)3/2 β2=μ4μ22=3(3θ16+48θ14+304θ12+944θ10+1816θ8+2304θ6+1920θ4+960θ2+240)(θ8+8θ6+24θ4+24θ2+12)2 γ=σ2μ1=θ8+8θ6+24θ4+24θ2+12θ(θ4+2θ2+2)(θ4+4θ2+6)

The natures of these descriptive measures for values of parameter  are shown in figure 3.

Figure 3 Coefficients of variation, skewness, kurtosis and index of dispersion of Adya distribution .

The condition under which Adya distribution is over-dispersed, equi-dispersed, and under-dispersed along with condition under which Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed are presented in table 2.

Distribution

Over-dispersion (μ<σ2)

Equi-dispersion
(μ=σ2)

Under-dispersion
(μ>σ2)

Adya

θ<1.305719841

θ=1.305719841

θ>1.305719841

Shanker

θ<1.171535555

θ=1.171535555

θ>1.171535555

Aradhana

θ<1.283826505

θ=1.283826505

θ>1.283826505

Sujatha

θ<1.364271174

θ=1.364271174

θ>1.364271174

Devya

θ<1.451669994

θ=1.451669994

θ>1.451669994

Lindley

θ<1.170086487

θ=1.170086487

θ>1.170086487

Exponential

θ<1

θ=1

θ>1

Table 2 Over-dispersion, equi-dispersion and under-dispersion of Adya, Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions for parameter θ

 

Hazard rate function and mean residual life function

The hazard rate function and the mean residual life function of a continuous random variableX having pdf and cdf f(x) and F(x) are, respectively, defined as

 

h(x)=limΔx0P(X<x+Δx|X>x)Δx=f(x)1F(x) (3.1)

 

and m(x)=E[Xx|X>x]=11F(x)x[1F(t)]dt (3.2)

 

Thus,h(x) and m(x) of (1.2) are obtained as

h(x)=θ3(θ+x)2θ2x2+2θ(θ2+1)x+(θ4+2θ2+2) (3.3)

 and

m(x)=1[θ2x2+2θ(θ2+1)x+(θ4+2θ2+2)]eθx×x[θ2t2+2θ(θ2+1)t+(θ4+2θ2+2)]eθtdt

=θ2x2+2θ(θ2+2)x+(θ4+4θ2+6)θ[θ2x2+2θ(θ2+1)x+(θ4+2θ2+2)] (3.4)

This givesh(0)=θ5θ4+2θ2+2=f(0) and m(0)=θ4+4θ2+6θ(θ4+2θ2+2)=μ1 . The hazard rate function and mean residual life function of Adya distribution are shown in figure 4.

Figure 4 Graphs of h(x) and m(x) of Adya distribution .

Stochastic orderings

A random variable X is said to be smaller than a random variable in the

  1. stochastic order (XstY) if FX(x)FY(x) for all x
  2. hazard rate order (XhrY) if hX(x)hY(x) for allx
  3. mean residual life order (XmrlY) if mX(x)mY(x) for all x
  4. likelihood ratio order (XlrY) if fX(x)fY(x) decreases inx

Shaked and Shanthikumar7 proposed following results for establishing stochastic ordering of distributions

XlrYXhrYXmrlY (4.1)

XstY

The distribution (1.2) is ordered with respect to the strongest ‘likelihood ratio’ ordering.

Theorem: Suppose X Adya distributon (θ1) and Y Adya distribution (θ2) . If θ1>θ2 , then XlrY and hence XhrY ,XmrlY and XstY .

Proof: We have

fX(x)fY(x)=θ13(θ24+2θ22+2)θ23(θ14+2θ21+2)((θ1+x)2(θ2+x)2)e(θ1θ2)x ; x>0

  Now

lnfX(x)fY(x)=ln[θ13(θ24+2θ22+2)θ23(θ14+2θ21+2)]+2ln(θ1+xθ2+x)(θ1θ2)x

 .

 This givesddx{lnfX(x)fY(x)}=2(θ1θ2)(θ1+x)(θ2+x)(θ1θ2)

 Thus forθ1>θ2 ,ddx{lnfX(x)fY(x)}<0 . This means that XlrY and hence XhrY ,XmrlY and XstY .

Mean deviations

The mean deviation about the mean and the mean deviation about the median are used to measure the amount of scatter in the population from the mean and the median and defined by

δ1(X)=0|xμ|f(x)dx and δ2(X)=0|xM|f(x)dx , respectively, where μ=E(X) and M=Median (X) . The computation of these measures are simplified as

δ1(X)=μ0(μx)f(x)dx+μ(xμ)f(x)dx=2μF(μ)2μ0xf(x)dx (5.1)

and

δ2(X)=M0(Mx)f(x)dx+M(xM)f(x)dx=μ2M0xf(x)dx (5.2)

Using pdf (1.2) and expression for the mean of Adya distribution ,we get

μ0xf(x;θ)dx=μ{θ5μ+θ4(2μ2+1)+θ3(μ3+4μ)+θ2(3μ2+μ)+6θμ+6}eθμθ(θ4+2θ2+2) (5.3)

M0xf(x;θ)dx=μ{θ5M+θ4(2M2+1)+θ3(M3+4M)+θ2(3M2+M)+6θM+6}eθMθ(θ4+2θ2+2)) (5.4)

Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, and the mean deviation about median, of Adya distribution are obtained as

δ1(X)=2{2θ3μ+θ2(μ2+μ)+4θμ+(θ4+6)}eθμθ(θ4+2θ2+2) (5.5)

δ2(X)=2{θ5M+θ4(2M2+1)+θ3(M3+4M)+θ2(3M2+M)+6θM+6}eθMθ(θ4+2θ2+2)μ (5.6)

Bonferroni and lorenz curves

The Bonferroni8 and Lorenz curves are defined as

B(p)=1pμq0xf(x)dx=1pμ[0xf(x)dxqxf(x)dx]=1pμ[μqxf(x)dx] (6.1)

 and L(p)=1μq0xf(x)dx=1μ[0xf(x)dxqxf(x)dx]=1μ[μqxf(x)dx] (6.2)

 respectively or equivalently

B(p)=1pμp0F1(x)dx (6.3)

 and L(p)=1μp0F1(x)dx (6.4)

 respectively, where μ=E(X) and q=F1(p) .

 The Bonferroni and Gini indices are thus defined as

B=110B(p)dp (6.5)

 and G=1210L(p)dp (6.6)

 respectively.

 Using pdf (1.2), we have

qxf(x)dx={θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2(3q2+1)+6θq+6}eθqθ(θ4+2θ2+2) (6.7)

Now using equation (6.7) in (6.1) and (6.2), we have

B(p)=1p[1{θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2(3q2+4)+6θq+6}eθqθ4+4θ2+6] (6.8)

 andL(p)=1{θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2(3q2+4)+6θq+6}eθqθ4+4θ2+6 (6.9)

 Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices are obtained as

B=1{θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2(3q2+4)+6θq+6}eθqθ4+4θ2+6 (6.10)

G=2{θ5q+θ4(2q2+1)+θ3(q3+4q)+θ2(3q2+4)+6θq+6}eθqθ4+4θ2+61 (6.11)

Stress-strength reliability

Suppose X and Y be independent strength and stress random variables having Adya distribution with parameter θ1 and θ2 respectively. Then R=P(Y<X) is known as stress-strength parameter and is a measure of the component reliability.

Thus, R=P(Y<X)=0P(Y<X|X=x)fX(x)dx=0f(x;θ1)F(x;θ2)dx =1θ13[24θ22+12θ2(θ22+θ1θ2+1)(θ1+θ2)+2(θ42+4θ1θ32+θ21θ22+2θ22+4θ1θ2+2)(θ1+θ22)+2θ1(θ24+θ1θ23+2θ2+θ1θ2+2)(θ1+θ2)3+θ21(θ42+2θ22+2)(θ1+θ2)4](θ14+2θ21+2)(θ24+2θ22+2)(θ1+θ2)5 =1θ13[24θ22+12θ2(θ22+θ1θ2+1)(θ1+θ2)+2(θ42+4θ1θ32+θ21θ22+2θ22+4θ1θ2+2)(θ1+θ22)+2θ1(θ24+θ1θ23+2θ2+θ1θ2+2)(θ1+θ2)3+θ21(θ42+2θ22+2)(θ1+θ2)4](θ14+2θ21+2)(θ24+2θ22+2)(θ1+θ2)5

Estimation of parameter

Estimation using method of moment

Since Adya distribution has one parameter, equating the population mean to the corresponding sample mean, the moment estimate ˜θ of θ is the solution of the following fifth degree polynomial equation ˉxθ5θ4+2ˉxθ34θ2+2θˉx6=0

, where ˉx is the sample mean.

Estimation using maximum likelihood estimation

Taking (x1,x2,x3,...,xn) a random sample from (1.2), the natural log likelihood function of Adya distribution is

lnL=nln(θ3θ4+2θ2+2)+2ni=1ln(θ+xi)nθˉx .

This gives,dlnLdθ=3nθ4nθ(θ2+1)θ4+2θ2+2+2ni=11θ+xinˉx ,

 

The MLE ˜θ of θ is the solution of dlnLdθ=0 which is given by

3nθ4nθ(θ2+1)θ4+2θ2+2+2ni=11θ+xinˉx=0 .

This non-linear equation is not in compact form and its solution can be obtained analytically. We have to use non-linear optimization technique such as quasi-Newton algorithm available in R software to maximize the log-likelihood function.

A simulation study

A simulation study has been conducted to examine the performance of the maximum likelihood estimate (MLE) of the parameter of Adya distribution. Random number n=20,40,60 and 80 generated corresponding to the parameter θ=0.1,0.2,0.3,0.4 and 0.5 using Acceptance- Rejection method. Its Bias and Mean Square Error have been calculated and presented in the table 3.

n

θ

Bias  (θ )

MSE(θ )

20

0.1

0.01200

0.00288

0.2

0.021244

0.009028

0.3

0.018753

0.007033

0.4

0.023497

0.0110371

40

0.1

0.004884

0.000954

0.2

0.007587

0.002302

0.3

0.008900

0.003168

0.4

0.010251

0.0042034

60

0.1

0.003484

0.000728

0.2

0.005428

0.001768

0.3

0.005854

0.002056

0.4

0.006242

0.002338

80

0.1

0.0023048

0.000424

0.2

0.0038444

0.001182

0.3

0.004266

0.001456

0.4

0.004735

0.001794

Table 3 Average Bias and Average Mean Square Error of simulated MLE (θ ) for fixed values θ =0.1,0.2,0.3 &0.4

Data analysis

The data set considered for the goodness of fit of Adya distribution is the strength data of glass of the aircraft window reported by Fuller et al.9 and are given as

18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29, 45.381        

The goodness of fit of the distributions are based on the values of 2lnL , AIC (Akaike Information Criterion) and K-S (Kolmogorov-Smirnov) statistic. AIC and K-S are computed using,AIC=2lnL+2k ,K-S=Supx|Fn(x)F0(x)| where k = the number of parameters, n = the sample size and Fn(x) is the empirical distribution function. The distribution having lower 2lnL , AIC , and K-S are said to be best distribution. The MLE (ˆθ) and standard error, S.E(ˆθ) ofθ ,2lnL , AIC, K-S and p-value of the fitted distributions are presented in the table 4.

Distributions

MLE(ˆθ)

S.E(ˆθ)

2lnL

AIC

K-S

p-value

Adya

0.096970

0.01000

240.63

242.63

0.298

0.006

Shanker

0.647164

0.008200

252.35

254.35

0.358

0.0004

Aradhana

0.094319

0.009780

242.22

244.22

0.306

0.0044

Sujatha

0.095613

0.009904

241.50

243.50

0.303

0.0051

Devya

0.160873

0.012916

227.68

229.68

0.422

0.0000

Lindley

0.062992

0.008001

253.98

255.98

0.365

0.0003

Exponential

0.032449

0.005822

274.52

276.53

0.458

0.0000

Table 4 MLE’s, S.E (ˆθ) , AIC and K-S Statistics of the fitted distributions of the given data set

Clearly Adya distribution gives better fit than Shanker, Aradhana, Sujatha, Devya, Lindley and exponential distributions.

Concluding remarks

Adya distribution, a one parameter lifetime distribution, for modeling lifetime data has been presented and studied. The statistical properties including coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves and stress-strength reliability have been discussed. Over-dispersed, equi-dispersed, and under-dispersed of Adya distribution are presented. Method of moment and method of maximum likelihood are explained for estimating parameter. The asymptotic property of the ML estimate of the parameter has been discussed with simulation study. Finally, the goodness of fit test has been presented with a real lifetime data.

NOTE: The paper is named Adya distribution in the name of my loving niece Adya Vedanshi, the daughter of my younger brother Dr. Ravi Shanker.

Acknowledgement

None.

Conflicts of interest

All authors declare that there is no conflict of interest.

References

Creative Commons Attribution License

©2021 Shanker, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.