An Inferential Aptness of a Weibull Generated Distribution and Application

Brijesh P. Singh and Utpal Dhar Das*

Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005, India


*Corresponding Author

Received 31 July 2021; Accepted 11 November 2021; Publication 06 December 2021


In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kth order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

Keywords: Bonferroni and Gini coefficient, K-S test, MLE, Moments, MRLF, Rényi and Shannon entropy.

1 Introduction

The exponentiated exponential, Weibull, Gamma, Lognormal distribution and their weighted version have an extensive usage in the fields of medical and engineering sciences. Some weighted distributions defined in the statistical literature, for example the weighted inverted exponential distribution [18], weighted Weibull distribution [19] and [25], weighted multivariate normal distribution [15], weighted inverse Weibull distribution [17], weighted three parameter Weibull distribution [2]. A two parameter weighted exponential distribution introduced [21] based on a modified weighted version of Azzalini’s approach [3]. A two parameter general class of distribution based on Lindley and a compounded exponential distribution with Lindley distribution for decreasing hazard has been discussed and apply to the real data sets [6] and [7].

The Rayleigh distribution is a particular form of two parameter Weibull distribution and widely used to model, events that occur in different fields of natural sciences. The generalized Rayleigh distribution is studied [14], [27] and [20]. Recently [16] observed that the two parameter generalized Rayleigh distribution that can be used quite effectively in modeling strength and life time data. Different methods to estimate the unknown parameters of the generalized Rayleigh and discussed several interesting properties [11]. The Weibull Rayleigh distribution developed [12] and derived its statistical properties. Exponentiated inverse Rayleigh distribution (EIRD) was introduced [13] and discussed its various statistical properties and it is a generalized form of inverse Rayleigh distribution [24]. A two parameter model introduced [1] as a competitive extension for Rayleigh distribution using the TIHL-G distributions and defined it as type 1 half-logistic Rayleigh distribution (TIHLR) and discussed its statistical properties and simulation studies.

A random variable X is said to have a mixture of two distributions ϕ1(x) and ϕ2(x) if its probability distribution is given by


where η1 and η2 are two positive number such that η1+η2=1.

In this paper an attempt has been made to develop a single parameter continuous distribution on the same logic what has been used in the process of development of Lindley distribution. Therefore in this study, exponential and Rayleigh distribution have been mixed with a suitable mixing parameter. Its first four moments, mean residual life function hazard function and various entropy has been discussed. Estimation of the parameter has been discussed and the suitability of distribution is tested on some real data set.

2 Proposed Continuous Distribution

The probability density function (pdf) of Weibull distribution is given by

fw(x;k,λ)=λkxk-1e-λxk (1)

In the above equation, if we put k=1, the distribution become exponential distribution and for k=2, the distribution become Rayleigh distribution.

Now we consider mixing parameter as p=λλ+α, we have

f(x) =pf(x;1,λ)+(1-p)f(x;2,λ)
=λλ+αe-λx(λ+2αxe-λx(x-1));α>0,λ>0 (2)

If α=0 in the Equation (2), we have an exponential distribution and if α=1 in the Equation (2), we have a mixture of exponential and Rayleigh distribution with mixing proportion λλ+1. This distribution is named as Rayleigh-exponential distribution (RED) and the pdf is given as

f(x)=λλ+1e-λx(λ+2xe-λx(x-1));λ>0 (3)

The plot of pdf of RED is given as


Figure 1 Probability density function of RED.

The cdf of RED is given by

F(x;λ)=0xf(t)dt=1-λe-λx+e-λx2λ+1 (4)


Figure 2 Cumulative distribution function of RED.

The survival function S(t), which is a probability that a patient or item will survive beyond any specified time t.

S(t)=1-F(t)=λe-λt+e-λt2λ+1 (5)


Figure 3 Survival function of RED.

and the corresponding hazard function of RED distribution is given by

h(x)=f(x)1-F(x)=λ(λ+2xe-λx(x-1))(λ+e-λx(x-1)) (6)

Now from Equation (6)

limx0h(x) =λ2λ+1 (7)
limx1λh(x) =(λ2+2e-(1λ-1))(λ+e-(1λ-1)) (8)
limxh(x) =λ (9)


Figure 4 Hazard function of RED.

From Equations (7), (8), (9) and Figure 4, we can say that the hazard of RED is first increasing then decreasing and finally it become constant.

3 Moments

The rth order moments is given by

E(Xr) =0xrf(x)dx=0xrλλ+1e-λx(λ+2xe-λx(x-1))dx
=λ2λ+1Γ(r+1)λr+1+Γ(r2+1)(λ+1)λr2=λλ+1(r!λr)+(r2)!(λ+1)λr2 (10)

Now the moments of the distribution is obtained as

E(X)= 1λ+1(1+12πλ) (11)
E(X2)= 3λ(λ+1) (12)
E(X3)= 3λ(λ+1)(2λ+14πλ) (13)
E(X4)= 2λ2(λ+1)(1+12λ) (14)
V(X)=E(X2)-(E(X))2= 1λ+1[3λ-1λ+1(1+12πλ)2] (15)

Median of the distribution is given by the equation

0Mλλ+1e-λx(λ+2xe-λx(x-1))=12 (16)λMλ+1(λ+e-λM(M-1))=12 (17)

This is a non linear equation we can solve it by numerically.

4 Quantile Function

The quantile function xq of RED is the real solution of the equation given below

F(xq) =p
(λ+1)(1-p)eλxq =λ+e-λ[(xq-12)2-14] (18)

The equation is not in closed form thus the solution of xq may obtain iteratively. If q=0.5 in the above equation, we can get median of the distribution.

5 Generating Function

Theorem 1 Moment generating function of RED is given by





= λ2(λ+1)(λ-t)+1λ+10e-x(λx-t)d(x(λx-t))

After simplification on last integral we get,

  +tet24λλ(λ+1)0e-(xλ-t2λ)2d[(xλ-t2λ)] (19)

Now let (xλ-t2λ)2=z,we have 2(xλ-t2λ)d(xλ-t2λ)=dz, also x0,zt24λ and x,z.

λ2(λ+1)(λ-t)+1λ+1+tet24λ2λ(λ+1)t24λz12-1e-zdz (20)

Where Γ(s,x)=xts-1e-tdt is upper incomplete gamma function. Finally from (20) we get our required results as

λ2(λ+1)(λ-t)+1λ+1+tet24λ2λ(λ+1)Γ(12,t24λ) (21)

Corollary 1 If we replace it for t in equation number (21) we get the characteristic function as

Φx(t) =0eitxf(x)dx
=λ2(λ+1)(λ-it)+1λ+1+ite-t24λ2λ(λ+1)Γ(12,-t24λ) (22)

6 Bonferroni and Lorenz Curves

The Bonferroni, Lorenz curves and Bonferroni, Gini indices have applications not only in economics to study the income and poverty, but also in other fields like reliability, insurance, medical and demography. The Bonferroni [8] and Lorenz curves are defined by

B(p)=1pμ0qxf(x)dxandL(p)=1μ0qxf(x)dx (23)

Respectively where, μ=E(x) and q=F-1(p). The Bonferroni and Gini indices are defined by

B=1-01B(p)dpandG=1-201L(p)dp (24)


B(p) =1pμλλ+10qxe-λx(λ+2xe-λx(x-1))dx
=1pμ1λ+1[1-(1+λq)e-(λq)]+1pμ1λ+1I1 (25)

Now let λx2=z; 2λxdx=dz2dx=dzλz and x0,z0; xq,zλq2

I1=1λ0λq2ze-zdzI1=-qe-λq2+1λ0λq2e-z2zdz (26)

Let z=u;dz2z=du;z0,u0;zλq2,uqλ, then

I1=-qe-λq2+1λ0qλe-u2du=-qe-λq2+1λπ2erf(qλ) (27)



Now from (6) and (27) we get the expression of Bonferroni curve

B(p) =1pμ1λ+1[[1-(1+λq)e-(λq)]+(12πλerf(qλ)-qe-(λq2))]
=[[1-(1+λq)e-(λq)]+(12πλerf(qλ)-qe-(λq2))]p(1+12πλ) (28)

where μ=1λ+1(1+12πλ), mean of the distribution and the Lorenz curve is obtained as

L(p)=[[1-(1+λq)e-(λq)]+(12πλerf(qλ)-qe-(λq2))](1+12πλ) (29)

7 Mean Residual Life Function

The mean residual life function is defined by

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

Now let λt2=z; 2λtdt=dz2dt=dzλz and tx,zλx2; t,z

e-λxλ+1+12λ(λ+1)λx2z-12e-zdz (30)

Now from (7) the MRLF obtained as

m(x)=e-λxλ+1+Γ(λx2,12)2λ(λ+1) (31)

Now if we put x=0 in equation number (31) then we get m(0)=1λ+1(1+12πλ), which is mean of the distribution and Γ(*,*) is the upper incomplete gamma function.

8 Rényi Entropy

We know that the entropy is a measure of uncertainty. In 1960, Rényi [5] defined a generalization of Shannon entropy which depends on a parameter and it is defined by,

e(η) =11-ηlog[0fη(x)dx]

Now applying binomial expansion (1+x)n=k=0n(nk)xk from above equation we get


Using e-x=k=0(-x)kk!, we get


After simplification we get our required expression.

e(η) =11-ηlog[λ2η(λ+1)ηk=0ηl=0(-1)l(ηk)(2λ)k

8.1 Cumulative Residual Entropy

Cumulative residual entropy is defined as

ΥCR =-0Pr(X>x)logPr(X>x)dx

Applying logarithmic expansion log(1+x)=k=1(-1)k-1xkk on last part of integrand of equation number (33), we get


After simplification we obtained the cumulative residual entropy as

ΥCR =-1λ+1log(λλ+1)(1+12πλ)

8.2 Shannon Entropy

Shannon entropy introduced by Shannon [9] is a limiting case of Rényi entropy it is widely used in Physics. The Rényi entropy tends to Shannon entropy as η0.

E(-logf(x))=-0f(x)logf(x)dx (34)

Now from (2) we get


Applying logarithmic expansion log(1+x)=k=1(-1)k-1xkk we get


Now applying e-x=m=0(-1)mxmm!, we get

  0xkm=0(-1)m(λkx(x-1))mm![λe-λx+2xe-λx2]dx (35)

Again applying binomial theorem (1-x)n=k=0n(nk)(-x)k in equation number (8.2) we get


After simplification we obtained Shannon entropy as

  +λλ+1ζλ;k,m,n[Γ(k+m+n+1)λk+m+n+1+Γ(k+m+n+22)λk+m+n2] (36)



9 Order Statistics

Let x1,x2,xn be a random sample of size n from the RED. Let X(1)<X(2)<<X(n) denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k th order statistic, say Y=X(k) are given by



fY(y)=n!(k-1)!(n-k)!l=0n-k(n-kl)(-1)lFk+l-1(y)f(y) (37)




FY(y)=i=knl=0n-j(nj)(n-il)(-1)lFi+l(y) (38)

Now, using equation number (2) and (4) in Equations (37) and (38) we get the corresponding pdf and the cdf of k-th order statistics of the RED are obtained as

fY(y) =n!(k-1)!(n-k)!l=0n-km=0k+l-1(n-kl)(k+l-1m)
(-1)l+mλe-λ(m+1)x(λ+1)(m+1)[λ+e-λx(x-1)]m(λ+2xe-λx(x-1)) (39)


FY(y) =i=knl=0n-im=0i+lu=0m(ni)(n-il)(i+lm)(mu)
(-1)l+mλm-ue-λux(x-1) (40)

10 Maximum Likelihood Estimation

The proposed distribution RED is a single parameter distribution and may estimate using method of maximum likelihood. The likelihood function for the proposed distribution can be written as




Now, log-likelihood can be given as


Differentiating the above equation with respect to λ partially, we get,

logλ=nλ-nλ+1-i=1nxi+i=1n1-2xi2(xi-1)e-λxi(xi-1)(λ+2xie-λxi(xi-1)) (41)

This is a non-linear equation we solve this by Newton Raphson method.

11 Simulation Study

In this section, an extensive numerical investigation will be carried out to evaluate the performance of MLE for RED. Performance of estimators is evaluated through their biases, and mean square errors (MSEs), variances (MLEs) for different sample sizes. Different sample of sizes are considered as n=10, 30, 50, 100, 200 and 500 in addition with different values of λ=0.25, 0.5, 0.75, 1, 1.5, 2, 2.5 and 3. The experiment will replicate with 10,000 times.

Table 1 Simulation results for different values of the parameter λ

λ=0.25 n Bias MSE Var. Est. λ=0.5 n Bias MSE Var. Est.

10 0.0355 0.0617 0.0170 0.2855 10 0.0481 0.0688 0.0486 0.5481
30 0.0100 0.0046 0.0040 0.2600 30 0.0148 0.0136 0.0137 0.5148
50 0.0060 0.0023 0.0022 0.2560 50 0.0112 0.0080 0.0079 0.5112
100 0.0025 0.0011 0.0011 0.2528 100 0.0073 0.0040 0.0039 0.5073
200 0.0019 0.0005 0.0005 0.2519 200 0.0035 0.0018 0.0019 0.5035
500 0.0009 0.0002 0.0002 0.2509 500 0.0034 0.0007 0.0007 0.5034

λ=0.75 n Bias MSE Var. Est. λ=1.0 n Bias MSE Var. Est.

10 0.0615 0.0981 0.0909 0.8115 10 0.0819 0.1609 0.1454 1.0819
30 0.0215 0.0263 0.0264 0.7715 30 0.0311 0.0444 0.0420 1.0311
50 0.0124 0.0156 0.0153 0.7624 50 0.0257 0.0252 0.0248 1.0257
100 0.0038 0.0074 0.0074 0.7539 100 0.0096 0.0136 0.0120 1.0096
200 0.0032 0.0038 0.0038 0.7532 200 0.0134 0.0067 0.0060 1.0134
500 -0.0035 0.0020 0.0020 0.7465 500 -0.0038 0.0024 0.0023 0.9961

λ=1.5 n Bias MSE Var. Est. λ=2.0 n Bias MSE Var. Est.

10 0.1108 0.3189 0.2836 1.6108 10 0.1518 0.5312 0.4755 2.1518
30 0.0435 0.0829 0.0819 1.5435 30 0.0578 0.1382 0.1358 2.0578
50 0.0353 0.0501 0.0482 1.5353 50 0.0366 0.0804 0.0790 2.0366
100 0.0160 0.0252 0.0234 1.5160 100 0.0263 0.0384 0.0387 2.0263
200 0.0048 0.0120 0.0115 1.5047 200 0.0139 0.0188 0.0191 2.0138
500 -0.0008 0.0062 0.0045 1.4992 500 -0.0027 0.0076 0.0075 1.9973

λ=2.5 n Bias MSE Var. Est. λ=3.0 n Bias MSE Var. Est.

10 0.2094 0.8579 0.7365 2.7094 10 0.2452 1.1826 1.0440 3.2452
30 0.0572 0.2110 0.2028 2.5572 30 0.0715 0.3035 0.2881 3.0715
50 0.0317 0.1176 0.1178 2.5317 50 0.0493 0.1707 0.1679 3.0493
100 0.0213 0.0595 0.0579 2.5213 100 0.0216 0.0826 0.0817 3.0216
200 0.0242 0.0283 0.0289 2.5242 200 0.0097 0.0401 0.0403 3.0097
500 0.0149 0.0129 0.0114 2.5149 500 0.0041 0.0157 0.0160 3.0041

In each experiment the estimate of the parameter λ will be obtained by methods of maximum likelihood estimation. The Biases, MSEs, Variances and estimates are reported in Table 1. We clearly observe from the Table 1, the values of bias and MSE of the parameter decreases as the sample size n increases, it proves the consistency of the estimator.

12 Real Data Application

The application of RED have been discussed with the following real data sets. The first data is about failure and service times for a particular model windshield of aircraft from [10], originally given in [22]. The data consist 153 observations. Among them 88 are classified as failed windshields and the remaining 65 are censored i.e. working at the time of taking observations. The unit for measurement is 1000 hours. The second data set represents 40 patients suffering from blood cancer (leukemia) from one of ministry of health hospitals in Saudi Arabia [4] and the third data set consists of survival times of guinea pigs injected with different amount of tubercle bacilli and was studied [26], the data represents the survival times of Guinea pigs in days. Summary measures of data sets are given in Table 2. The pp-plot and TTT plot are shown in the Figures 5, 8, 11 for respective data sets. The fitted pdf plots for EE, EIRD, TIHLR, Lindley, Exponential and proposed distribution (RED) is display in the Figures 6, 9, 12 also the empirical cdf and fitted cdf of respective data sets are shown in the Figures 7, 10, 13.

It is reveals that all the data sets are under dispersed and positively skewed except second data set.

Table 2 Summary of three data sets

Data Sets n Mean Sd Median Skewness Kurtosis Min Max
Aircraft windsheild 65 2.081 1.230 2.065 0.449 2.784 0.046 5.140
Leukaemia 40 3.141 1.359 3.348 -0.417 2.274 0.315 5.381
Guinea pigs 72 1.754 1.044 1.450 1.328 4.914 0.100 5.550


Figure 5 pp-plot and TTT plot for the aircraft windshield data.

The above data sets used for checking the suitability of proposed distribution RED along with some other distributions viz. exponentiated exponential distribution (EE) proposed by [23] , exponentiated Inverse Rayleigh distribution (EIRD) introduced by [13], type 1 half-logistic Rayleigh distribution (TIHLR) proposed [1], exponential and Lindley distribution. The ML estimates, value of -2LL, Akaike Information criteria (AIC), Corrected Akaike Information criteria (AICc), Hannan-Quinn information criterion (HQIC) are presented in the Tables 3, 5 and 7 and also K-S statistic, A-D statistic and there associated p-value of the considered distributions are presented in Tables 4, 6 and 8. The AIC, BIC, AICc, HQIC, K-S and A-D Statistics are computed using the following formulae:

AIC =-2LL+2k,BIC=-2LL+klogn
AICc =AIC+2k2+2kn-k-1,HQIC=-2LL+2klog(log(n))
D =supx|Fn(x)-F0(x)|A2=-N-S;
S =i=1N2i-1N[logF(Yi)+log(1-F(YN+1-i))]

where k= the number of parameters, n= the sample size, and the Fn(x) is empirical distribution function F0(x) is the theoretical cumulative distribution function and Yi are the ordered data. The best distribution is the distribution corresponding to lower values of -2LL, AIC, BIC, AICc, K-S and A-D statistics and there corresponding higher p-values respectively.

Table 3 -2LL and information criterion for aircraft windshield data


Distribution α θ -2LL AIC BIC AICc HQIC
RED 0.1930 212.59 214.59 216.77 214.66 215.45
EE 1.9458 0.7024 212.63 216.63 220.98 216.82 218.35
EIRD 0.2148 0.1591 322.60 326.60 330.95 326.79 328.31
TIHLR 0.5920 0.3880 215.20 219.20 223.55 219.39 220.91
Lindley 0.7543 215.32 217.32 219.49 217.38 218.17
Exponential 0.4804 225.30 227.30 229.47 227.36 228.15

Table 4 Kolmogorov-Smirnov and Anderson-Darling Statistic for aircraft windshield data

Distribution K-S p-value A-D p-value
RED 0.0719 0.8663 0.9309 0.3953
EE 0.1405 0.1397 1.3609 0.2134
EIRD 0.3639 0.0000 13.122 0.0000
TIHLR 0.1422 0.1308 2.6584 0.0411
Lindley 0.1588 0.0671 2.3349 0.0607
Exponential 0.2132 0.0045 4.1845 0.0071


Figure 6 Fitted pdf for the aircraft windshield data.


Figure 7 Fitted cdf for the aircraft windshield data.


Figure 8 pp-plot and TTT plot for the blood cancer (leukaemia) data.

Table 5 -2LL and information criterion for blood cancer (leukaemia) data


Distribution α θ -2LL AIC BIC AICc HQIC
RED 0.0839 145.35 147.35 149.04 147.46 147.96
EE 3.5189 0.6141 149.92 153.92 157.30 154.25 155.15
EIRD 0.4437 0.9562 196.05 200.05 203.43 200.37 201.27
TIHLR 0.2737 0.4364 137.41 144.79 144.79 141.73 142.63
Lindley 0.5269 160.50 162.50 164.19 162.60 163.11
Exponential 0.3184 171.56 173.56 175.24 173.67 174.17

Table 6 Kolmogorov-Smirnov and Anderson-Darling Statistic for blood cancer (leukaemia) data

Distribution K-S p-value A-D p-value
RED 0.1318 0.4903 1.1906 0.2709
EE 0.1612 0.2495 1.7137 0.1330
EIRD 0.7730 0.0000 6.3646 0.0006
TIHLR 0.1181 0.6315 0.6944 0.5625
Lindley 0.2405 0.0195 3.6452 0.0132
Exponential 0.3002 0.0015 5.4782 0.0017


Figure 9 Fitted pdf for the blood cancer (leukaemia) data.


Figure 10 Fitted cdf for the blood cancer (leukaemia) data.


Figure 11 pp-plot and TTT plot for survival times of guinea pigs data.

Table 7 -2LL and information criterion for survival times of guinea pigs data


Distribution α θ -2LL AIC BIC AICc HQIC
RED 0.3252 195.23 197.23 199.51 197.29 198.14
EE 3.4932 1.1181 188.95 192.95 197.51 193.13 194.77
EIRD 0.4077 0.4584 277.57 281.57 286.12 281.74 283.38
TIHLR 0.6602 0.4906 204.51 208.51 213.07 208.68 210.33
Lindley 0.8744 213.05 215.05 217.33 215.11 215.96
Exponential 0.5702 224.89 226.89 229.17 226.95 227.79

Table 8 Kolmogorov-Smirnov and Anderson-Darling Statistic survival times of guinea pigs data

Distribution K-S p-value A-D p-value
RED 0.1200 0.2508 1.0113 0.3511
EE 0.0883 0.6290 0.4572 0.7901
EIRD 0.4213 0.0000 10.437 0.0000
TIHLR 0.1866 0.0133 3.7647 0.0114
Lindley 0.1866 0.0133 3.7647 0.0114
Exponential 0.2832 0.0000 6.8837 0.0004


Figure 12 Fitted pdf for the survival times of guinea pigs data.


Figure 13 Fitted cdf for the survival times of guinea pigs data.

13 Conclusion

In this paper, we propose and explore the properties of the proposed distribution named as Rayleigh-Exponential Distribution (RED). We investigate some of its statistical properties like rth order moment, quantile function, moment generating function, characteristics function, Bonferroni, Lorenz curves, mean residual life function. Some entropy has been discussed like Rényi, Shannon entropy and cumulative residual entropy. The maximum likelihood method is employed to estimate the parameter. We fit the real data sets to demonstrate the flexibility and aptness of the proposed distribution. The RED performs better than other distributions for the first data set but in other two data set its rank is second. This shows that the RED is a competent model to some other two parameters models also. We hope that the RED distribution will attract wider application in areas such as engineering, survival and lifetime data, hydrology, economics and other areas.


Authors extend sincere thanks to the anonymous referees for their valuable suggestions to improve the quality of initial draft of the paper presented by Utpal Dhar Das in the International conference on “Recent Advances in Statistics and Data Science for Sustainable Development” in conjunction with 34th Convention of India Society for Probability and Statistics (ISPS) on 21–23rd December, 2019 in Department of Statistics, Utkal University, Bhubaneswar, Odisha.


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Brijesh P. Singh, is currently working as Professor in the Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi, India. He has obtained Ph. D. degree in Statistics from Banaras Hindu University, Varanasi and has more than 20 years’ experience of teaching and research in the area of Statistical Demography and modeling. His research interests are in statistical modeling and analysis of demographic data specially fertility, mortality, reproductive health and domestic violence with its reason and consequences.


Utpal Dhar Das, is presently working as research scholar in the Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi, India. He is a bright fellow in Mathematics and Statistics and was awarded gold medal in M. Sc. (Statistics) from Assam University, Silchar. He has published 12 research articles in reputed both international and national journals. His research interests are in the areas of Generalized Probability distributions, transformed probability distributions.


1 Introduction

2 Proposed Continuous Distribution





3 Moments

4 Quantile Function

5 Generating Function

6 Bonferroni and Lorenz Curves

7 Mean Residual Life Function

8 Rényi Entropy

8.1 Cumulative Residual Entropy

8.2 Shannon Entropy

9 Order Statistics

10 Maximum Likelihood Estimation

11 Simulation Study

12 Real Data Application










13 Conclusion