Two New Lifetime Distributions of X–Weibull Family: Theories and Applications

Abouzar Bazyari; Monjed H. M. Samuh

doi:10.2991/jsta.2018.17.2.14

<Previous Article In Issue

Download article (PDF)

Volume 17, Issue 2, June 2018, Pages 375 - 392

Two New Lifetime Distributions of X–Weibull Family: Theories and Applications

Authors

Abouzar Bazyariab_bazyari@yahoo.com

Department of Statistics, Persian Gulf University, Bushehr, Iran

Monjed H. M. Samuhmonjedsamuh@ppu.edu

Department of Applied Mathematics & Physics, Palestine Polytechnic University, Hebron-Palestine

Received 25 May 2016, Accepted 1 August 2017, Available Online 30 June 2018.

DOI: 10.2991/jsta.2018.17.2.14 How to use a DOI?
Keywords: Exponential distribution; Simulation method; T – X distribution family; Weibull distribution
Abstract: In this paper, two new distributions, weibull-rayleigh and weibull-exponential of X–weibull family are introduced. The various properties of theses distributions, for example, the density functions, distribution functions, hazard rate functions, moment functions and Shannon entropy are investigated and capabilities of distributions compared to other distributions. Also by using simulation from these two distributions with various parameters some of the sample statistical indexes are estimated. Two real data sets are used to illustrate the applicability of these distributions.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Statistical distributions are important for parametric inference and applications to fit real world phenomena. Many methods have been developed to generate statistical distributions in the literature. Sometimes fit a distribution to a data set faces poor results. To solve this problem statisticians have always tried to fit a better distribution to data with production or extension the distributions. In this regard, many generalized classes of distributions to describe the different phenomena are obtained. Many types of generalized exponential distributions obtained by Khan and Jain (1978). Gupta and Kundu (1999) introduced the three-parameter “generalized exponential distribution” (location, scale and shape), to study the theoretical properties of this family and compared them with respect to the well study properties of the gamma distribution and weibull distribution. Gupta et al. (1998) proposed to model failure time data by F*(f) = [F(t)] θ where F(t) is the baseline distribution function and θ is a positive real number. The monotonicity of the failure rates were studied, in general, and some order relations are examined. Some examples including exponentiated weibull, exponential, gamma and Pareto distributions were investigated. Gupta and Kundu (2001) studied some properties of a new family of distributions, namely Exponentiated Exponential distribution, discussed in Gupta et al. (1998). They They presented two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to weibull or gamma distribution and in the other data set weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes. Various extensions on the Gamma distribution are given by Stacy (1962). Agarwal and Al-Saleh (2001) defined a new generalized gamma distribution. The trend of this density function and statistical properties are studied. A real life data set used to fit the distribution. A new weibull type distribution was also considered and the hazard rate function of this distribution also has both the properties namely bathtub and monotonicity. Mudholkar et al.onotone hazard rates and can be adopted for testing goodness of fit of weibull as a submodel. The usefulness and flexibility of the family is illustrated by reanalyzing five classical data sets on bus-motor failures from Davis that are typical of data in repair-reuse situations and Efron’s datapertainingtoahead-and-neck-cancerclinical trial.

Eugene et al. (2002) introduced a general class of distributions generated from the logit of the beta random variable. The shape properties of the beta-normal distribution discussed. Also the estimation of parameters of the beta-normal distribution by the maximum likelihood method discussed. Nadarajah and Kotz (2004) introduced a generalization referred to as the beta Gumbel distribution generated from the logit of a beta random variable. They provided a comprehensive treatment of the mathematical properties of this new distribution. Also they derived the analytical shapes of the corresponding probability density function and the hazard rate function and provided graphical illustrations. Akinsete et al. (2008) defined and studied a four-parameter beta-Pareto distribution. They discussed various properties of the distribution. The distribution found to be unimodal and has either a unimodal or a decreasing hazard rate. The expressions for the mean, mean deviation, variance, skewness, kurtosis and entropies obtained. The relationship between these moments and the parameters provided. Also they proposed the method of maximum likelihood to estimate the parameters of the distribution. The distribution applied to two flood data sets. Alzaatreh, et al. (2012a) defined and studied a new distribution, the Gamma-Pareto, and various properties of the distribution are obtained. Results for moments, limiting behavior and entropies are provided. Also the method of maximum likelihood is proposed for estimating the parameters and the distribution is applied to fit three real data sets. Alzaatreh, et al. (2012b) proposed a new method for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions obtained. Alzaatreh, et al. (2013a) a new distribution, namely, weibull-Pareto distribution defined and studied. Various properties of the weibull-Pareto distribution obtained. Results for moments, limiting behavior, and Shannon entropy provided. The method of modified maximum likelihood estimation is proposed for estimating the model parameters. Several real data sets used to illustrate the applications of weibull-Pareto distribution. Alzaatreh, et al. (2013b) proposed the T – X families of distributions. These families of distributions were used to generate a new class of distributions which offer more flexibility in modeling a variety of data sets. Alzaatreh, et al. (2014) discussed some properties of gamma-X family and a member of the family, the gamma-normal distribution, is studied in detail. The limiting behaviors, moments, mean deviations, dispersion, and Shannon entropy for the gamma-normal distribution are provided.

Suppose that X is a random variable with density function f_X(x) and distribution function F_X(x). Also let T be a continuous random variable with density function r(t) and distribution function R(t)defined on the interval [a, b].

The new family of distributions cumulative distribution function is as follows

(1)G(x)=∫aW(F(x))r(t)dt=R(W(F(x))),

where W(F(x) applies to the following conditions:

a)
W(F(x)) ∈ [a, b].
b)
W(F(x) is non-decreasing and differentiable.
c)
limx→−∞W(F(x)=a and limx→∞W(F(x)=b .

In this family, the random variable T is a transformation variable and random variable X is a transformer variable.

By differentiating G(x), the density function of g(x) given by

(2)g(x)=dW(F(x))dxr(W(F(x))).

Since various distributions is introduced and investigated with this method. For example, Alzaatreh and Knight (2013) proposed and studied the new distribution, Gamma-half normal distribution. Various structural properties of the gamma-half normal distribution derived. Results for moments, limit behavior, mean deviations and Shannon entropy provided. To estimate the model parameters, the method of maximum likelihood estimation proposed. Three real-life data sets used to illustrate the applicability of the gamma-half normal distribution. Aljarrah et al. (2014) proposed the use of quantile functions to define the W function and some general properties of this T – X system of distributions studied. Also they derived three new distributions of the T – X family, namely, the normal-weibull based on the quantile of Cauchy distribution, normal-weibull based on the quantile of logistic distribution, and weibull-uniform based on the quantile of log-logistic distribution. Two real data sets applied to illustrate the flexibility of the distributions.

This paper is organized as follows. In Section 2, the weibull-rayleigh distribution is introduced. Results for hazard rate function, moment generating function and Shannon entropy is obtained. Using simulation method from this distribution with various parameters some of the sample statistical indexes are estimated. In Section 3, the weibull-exponential distribution is introduced. Results for hazard rate function, moment generating function and Shannon entropy is obtained. Using simulation method from this distribution with various parameters some of the sample statistical indexes are estimated. In Section 4, two real data sets are used to illustrate the applicability of these distributions. Concluding remarks are given in Section 5.

2. Weibull-rayleigh distribution

Let r(x) be a density function of a random variable with weibull distribution (c, γ) and f_X(x) and F_X(x) be the density function and distribution function with rayleigh distribution with parameter b respectively, and W(F(x)) = − log(1 − F(x)). Then (1) is a new distribution as weibull-rayleigh distribution. The density function and distribution function of weibull-rayleigh distribution is as follows:

(3)GWRD(x)=1−exp{−[−log(1−F(x))γ]c}=1−exp{−(x22b2γ)c},

and

(4)gWRD(x)=cxb2γ(x22b2γ)c−1exp{−(x22b2γ)c}, x≥0,b, c, γ>0.

The following lemmas give the relationship between the weibull-rayleigh distribution and weibull distribution.

Lemma 1.

Suppose that X is a random variable with weibull-rayleigh distribution with parameters (c, γ, b), then the random variable Y=X22b2 is distributed as weibull distribution with parameter (c, γ).

Lemma 2.

Suppose that X is a random variable with weibull distribution with parameters (c, γ), then the random variable V2b2X is distributed as weibull-rayleigh distribution with parameters (c, γ, b).

The proof of these lemmas is easy.

The hazard rate function of weibull-rayleigh distribution can computed as follows:

(5)hWRD(x)=g(x)1−G(x)=cxb2γ(x22b2γ)c−1=cx2c−12c−1(b2γ)c.

If c>12 the rate function of weibull-rayleigh distribution is increasing, if c<12 the rate function is decreasing and if c=12 the rate function is constant and is equal to c(2bγ)c .

Theorem 1.

Suppose that X is a random variable is distributed as weibull-rayleigh distribution with density function given in (4). Then it’s moment generating function is as follows:

MX(t)=c∑j=0∞(−1)jΓ(2c(j+1))(b2γ)c(j+1)2c(j+1)−1t2c(j+1).

Proof.

The moment generating function of weibull-rayleigh distribution is

(6)MX(t)=∫0∞etxg(x)dx=∫0∞c(b2γ)c2c−1x2c−1 exp(−x2c(2b2γ)c+tx) dx.

Using the Taylor expansion, we have

MX(t)=∑j=0∞(−1)jc(b2γ)c2c−1∫0∞x2c−1x2cjj!(2b2γ)cjetxdx=c∑j=0∞(−1)jΓ(2c(j+1))(b2γ)c(j+1)2c(j+1)−1t2c(j+1).

Theorem 2.

Suppose that X is a random variable is distributed as weibull-rayleigh distribution with density function given in (4). Then the first central moment and variance of X are respectively as:

(7)EWRD(X)=Γ(1+12c)2b2γ,

and

(8)VWRD(X)=b2γ (c2c−1Γ(1+1c)−2Γ2(1+12c)).

Proof.

By differentiating from (6) with respect to t, we have

∂∂tMX(t)=c(b2γ)c2c−1∫0∞x2c exp(−x2c(2b2γ)c+tx) dx.

Putting t = 0, then the first central moment is

EWRD(X)=c(b2γ)c2c−1∫0∞x2c exp(−x2c(2b2γ)c) dx=1(b2γ)c2c∫0∞u12c exp(−u(2b2γ)c) du=Γ(1+12c) 2b2γ.

Also by differentiating from(6) with respect to t, the second central moment is

EWRD(X2)=cb2γ2c−1Γ(1+12c).

Therefore the variance of weibull-rayleigh distribution is given by

VWRD(X)=E(X2)−E2(X)=b2γ (c2c−1Γ(1+1c)−2Γ2(1+12c)).

Theorem 3.

Suppose that X is a random variable is distributed as weibull-rayleigh distribution with density function given in (4). Then the mode of distribution is given by

x=2c−1(c−1)(b2γ)cc2c, c>12.

Proof.

The proof of this theorem is easy. One can differentiate from (4) with respect to x and equal it to zero.

Theorem 4.

Suppose that X is a random variable is distributed as weibull-rayleigh distribution with density function given in (4). Then the Shannon entropy is given by

ηX=−E(logXb2)+w(1−1c)+ln(γc)+1,

where w=∫1∞(1[x]−1x) dx is the Euler-Mascheroni constant.

Proof.

Let X be a random variable distributed as weibull-rayleigh distribution. By the definition of Shannon entropy we have

ηX=E (−log g(X))=E (−log(Xb2r (X22b2)))=E (−logXb2)+E (−log r (X22b2)),

where r is the density function of rayleigh distribution with parameter (c, γ). Since the random variable X22b2 by lemma 1 has weibull distribution, therefore E (−log(r (X22b2)) is the Shannon entropy of weibull distribution and is equal to w(1−1c)+ln(γc)+1 . This completes the proof.

To simulate from weibull-rayleigh distribution, first we simulate from rayleigh distribution with parameter (c, γ) for production the samples Y, then by lemma 2 and using the conversion X=2b2Y the simulation from weibull-rayleigh distribution is obtained. The mean, median, variance, skewness and kurtosis of the samples simulation are given in Table 1. From this table we can see that all the samples simulation increases as b increases. Also the mean, median and variance increases as γ increases. It is worth mentioning that the formula in(7) and (8) are conformity with the data given in Table 1.

Kurtosis	Skewness	Variance	Median	Mean	b	γ	c
8.685	1.976	0.251	0.347	0.500	0.5	0.5	0.5
8.233	1.913	0.991	0.697	1	1	0.5	0.5
8.517	1.942	3.947	1.386	1.992	2	0.5	0.5
8.167	1.909	8.967	2.089	3	3	0.5	0.5
8.480	1.941	0.493	0.492	0.708	0.5	0.5	0.5
8.643	1.975	2.022	0.981	1.416	1	0.5	0.5
8.713	1.979	4.056	1.151	1.788	2	0.5	0.5
8.928	1.982	7.955	1.960	2.82	3	0.5	0.5
8.452	1.953	0.992	0.691	0.996	0.5	0.5	0.5
8.774	1.981	3.981	1.378	1.990	1	0.5	0.5
8.387	1.932	15.894	2.779	4	2	0.5	0.5
8.465	1.937	35.716	4.147	5.980	3	0.5	0.5
8.201	1.922	1.498	0.855	1.228	0.5	0.5	0.5
8.534	1.962	6.092	1.718	2.471	1	0.5	0.5
8.171	1.911	23.987	3.407	4.905	2	0.5	0.5
8.481	1.945	54.143	5.117	7.347	3	0.5	0.5
3.189	0.604	0.053	0.418	0.443	0.5	0.5	0.5
3.221	0.617	0.215	0.836	0.887	1	0.5	0.5
3.165	0.608	0.859	1.665	1.770	2	0.5	0.5
3.190	0.612	1.925	2.500	2.660	3	0.5	0.5
3.232	0.632	0.107	0.588	0.625	0.5	0.5	0.5
3.242	0.635	0.432	1.176	1.252	1	0.5	0.5
3.243	0.634	1.710	2.350	2.503	2	0.5	0.5
3.242	3.876	3.876	3.540	3.767	3	0.5	0.5
3.191	0.614	0.214	0.836	0.887	0.5	0.5	0.5
3.239	0.626	0.858	1.671	1.777	1	0.5	0.5
3.249	0.629	3.440	3.325	3.544	2	0.5	0.5
3.239	0.631	7.775	4.997	5.331	3	0.5	0.5
3.126	0.598	0.323	1.022	1.087	0.5	0.5	0.5
3.217	0.620	1.297	2.041	2.170	1	0.5	0.5
3.198	0.618	5.159	4.089	1.349	2	0.5	0.5
3.172	0.610	11.637	6.124	6.520	3	0.5	0.5
2.753	-0.079	0.016	0.456	0.453	0.5	0.5	0.5

Table 1.

Sample statistical from weibull-rayleigh distribution with various parameters

Let x₁, x₂, …, x_n be a random sample from weibull-rayleigh distribution with density function (4). Then the likelihood function is given by

(9)log L(c,γ,b)=∑i=1n(log c+log xi−log γ−2 log b+(c−1) log (xi22b2γ)−(xi22b2γ) c).

By differentiating from(9) with respect to each parameter, we have

(10)∂∂c log L(c,γ,b)=nc−∑i=1n log (xi22b2γ)+∑i=1n(xi22b2γ)c log (xi22b2γ)=0,

(11)∂∂γlog L(c,γ,b)=−ncγ+2b2c(2b2γ)c+1∑i=1n(xi)2c=0

and

(12)∂∂blog L(c,γ,b)=−2ncγ+4γbc(2b2γ)(c+1)∑i=1n(xi)2c=0

By solving the equations (10), (11) and (12) the maximum likelihood estimators is given by ĉ, γˆ and bˆ .

3. Weibull-exponential distribution

Let in formula (1) r(t) and R(t) be the density function and distribution function of a random variable respectively, which is distributed as weibull distribution with parameters (λ, k). Therefore

r(t)=(kλ)(tλ)k−1e−(tλ),λ, k>0,t>0

and

R(t)=1−e−(tλ),λ, k>0,t>0.

The X–weibull family is as

G(x)=∫0W(F(x))(kλ) (tλ)k−1e−(tλ)dt=1−e−(W(F(x))λ)k.

Let f_X(x) and F_X(x) be the density function and distribution function of a random variable respectively, which is distributed as exponential distribution with parameter μ. Also, since X > 0 define W(F(x))=F(x)1−F(x) . Then the density function and distribution function of weibull-exponential distribution is

(13)GWRD(x)=1−exp (−eμx−1λ)k, λ, k, μ>0,x>0

and

(14)gWRD(x)=kμλeμx(−eμx−1λ)k−1exp (−eμx−1λ)k, λ, k, μ>0,x>0.

The following lemmas give the relationship between the weibull-exponential distribution with weibull distribution and weibull-rayleigh distribution.

Lemma 3.

Suppose that X is a random variable with weibull-exponential distribution with parameters (λ, k, μ), then the random variable Y = e^μX − 1 is distributed as weibull distribution with parameter (λ, k). Note that if k = 1, the random variable Y has exponential distribution with parameter λ⁻¹.

Lemma 4.

Suppose that X is a random variable with weibull distribution with parameters (λ, k), then the random variable Y=1μlog(X+1) is distributed as weibull-exponential distribution with parameters (λ, k, μ).

The proof of these lemmas is easy.

Lemma 5.

Suppose that X is a random variable with weibull-exponential distribution with parameters (c, γ, μ), then the random variable Y=2(eμX−1) is distributed as weibull-rayleigh distribution with parameter (c, γ, μ).

Asymptotic behavior of density function of weibull-exponential is

limx→+∞ gWRD(x)=0,

and

limx→0 gWRD(x)={∞0<k<1μλk=10k>1.

The hazard rate function of weibull-exponential distribution can computed as follows:

(15)hWRD(x)=g(x)1−G(x)=kμλeμx(eμx−1λ)k−1, λ, k, μ>0,x>0.

The limit of hazard rate function of weibull-exponential distribution for various values of k is

limx→0 hWRD(x)={∞0<k<1μλk=10k≥1,

and

limx→∞ hWRD(x)=∞.

Theorem 5.

Suppose that X is a random variable with weibull-exponential distribution as (14). The quantile function of X² given by

(16)Q(p)=1μlog(λ−log(1−p))k+1),0<p<1,λ, k, μ>0.

Proof.

The proof is easy by solving G(Q(p)) = p.

If p = 0.5, then the median of distribution given by

(17)Q(0.5)=1μlog(λ0.69k+1),λ, k, μ>0.

Theorem 6.

Suppose that X is a random variable is distributed as weibull-exponential distribution with density function given in (14). Then the mode of distribution is given by

x=1μlog(z+1),

where z the positive roots of the following polynomial

kzk+1+kzk−kλk+1z+(λkz−λkk)=0.

Proof.

By differentiating of logarithm from (14), we have that

(18)∂∂x log gWED(x)=μ+μ(k−1)eμxeμx−1−kμλk(eμx−1)k−1eμx,

With some manipulations we have that

kλkeμx−keμx(eμx−1)k−λk=0.

Let z = e^μx − 1, then

kzk+1+kzk−kλk+1z+(λkz−λkk)=0,

and this completes the proof.

Theorem 7.

Suppose that X is a random variable is distributed as weibull-exponential distribution with density function given in (14). Then a bound for mathematical expectation X given by

E(X)≤1μlog (1+λΓ(1+1k)).

Proof.

The moment generating function of weibull- exponential distribution is

(19)MX(t)=kμλ∫0∞e(μ+t)x (eμx−1λ)k−1e−(eμx−1λ)kdx=kλ∫0∞(y+1)tμ (yλ)k−1e−(yλ)kdy=E(Y+1)tμ,

where Y is distributed as weibull distribution with parameters (λ, k). By differentiating from (19), we have that

∂∂tMX(t)=∂∂t(E (Y+1)tμ)=E ((1μ(Y+1)tμ) log(Y+1)).

Putting t = 0 and using Jensen’s inequality, a bound for E(X) given by

E (X)=E (1μ log(Y+1))≤1μ log (1+E(Y))=1μ log (1+λΓ(1+1k)).

Theorem 8.

Suppose that X is a random variable is distributed as weibull-exponential distribution with density function given in (14). Then a bound for it’s Shannon entropy given by

(20)ηW E≥−log μ−log(1+Γ(1+1k))+ηW,

where η_W the Shannon entropy of weibull distribution with parameters (λ, k).

Proof.

Shannon entropy of a random variable is defined by E_X(− log (g(x))). According to (2) we have that

EX(−log (g(x)))=E(−log (ddx(W(F(x)))r(W(F(x)))))=E(−log (μeμX)+log (r(eμX−1)))=−log μ−μE(X)+ηW≥−log μ−log (1+Γ(1+1k))+ηW.

To simulate from weibull-exponential distribution, first we simulate the samples Y using lemma 4 from weibull distribution, then we use the conversion X=1μlog(Y+1) . The mean, median, variance, skewness and kurtosis of the samples simulation are given in Table 2. From this table we can see that increasing the value of μ causes the decreasing of mean, median and variance.

Kurtosis	Skewness	Variance	Median	Mean	b	γ	c
5.618	1.787	1.367	0.391	0.982	0.5	0.5	0.5
6.265	1.887	0.385	0.237	0.453	1	0.5	0.5
8.746	1.493	0.0866	0.108	0.225	2	0.5	0.5
5.431	1.941	0.037	0.072	0.149	3	0.5	0.5
3.867	1.501	2.227	0.706	1.143	0.5	0.5	0.5
4.583	1.476	0.570	0.371	0.700	1	0.5	0.5
4.016	1.421	0.142	0.129	0.347	2	0.5	0.5
4.788	1.347	0.075	0.122	0.237	3	0.5	0.5
3.515	1.021	3.5656	1.356	2.025	0.5	0.5	0.5
3.360	0.951	1.064	0.708	0.952	1	0.5	0.5
3.804	1.092	0.259	0.340	0.478	2	0.5	0.5
4.201	1.045	0.106	0.259	0.343	3	0.5	0.5
3.170	1.011	4.886	1.675	2.479	0.5	0.5	0.5
3.104	0.889	1.219	0.703	1.213	1	0.5	0.5
2.710	0.846	0.311	0.524	0.556	2	0.5	0.5
4.202	1.045	0.106	0.258	0.340	3	0.5	0.5
4.167	1.112	0.359	0.552	0.745	0.5	0.5	0.5
3.233	1.091	0.084	0.300	0.187	1	0.5	0.5
3.590	0.911	0.021	0.153	0.187	2	0.5	0.5
4.025	1.053	0.008	0.084	0.119	3	0.5	0.5
2.662	0.813	0.668	1.014	1.193	0.5	0.5	0.5
2.988	0.748	0.188	0.519	0.608	1	0.5	0.5
3.126	0.709	0.048	0.257	0.297	2	0.5	0.5
3.261	0.779	0.018	0.185	0.197	3	0.5	0.5
2.557	0.256	1.355	1.713	1.838	0.5	0.5	0.5
2.756	0.351	0.319	0.881	0.929	1	0.5	0.5
2.769	0.403	0.081	0.451	0.476	2	0.5	0.5

Table 2.

Sample statistical indexes from weibull-exponential with various parameters

Increasing the value of k causes the increasing of mean, median and variance. Also increasing the value of λ causes the decreasing of mean and variance and decreasing of median. It is worth mentioning that the formula in (16) is conformity with the data given in Table 2.

Let x₁, x₂, … , x_n be a random sample from weibull-exponential distribution with density function (14). Then the likelihood function is given by

(21)log L(λ,k,μ)=n log k+n log μ−n log λ+μ∑i=0nxi +(k−1)∑i=1n log (eμxi−1λ)−∑i=1n(eμxi−1λ)k.

By differentiating from (21) with respect to each parameter, we have

(22)∂∂c log L(λ,k,μ)=−nλ−(k−1)nλ+kλk+1∑i=1n(eμxi−1λ)k=0,

(23)∂∂k log L(λ,k,μ)=nk+∑i=1nlog (eμxi−1λ)−∑i=1n(eμxi−1λ)k (eμxi−1λ)=0,

and

(24)∂∂μ log L(λ,k,μ)=nk+∑i=1nxi+(k−1)∑i=1nxieμxieμxi−1−kλk∑i=1nxi eμxi(eμxi−1)k−1=0,

By solving the equations (22), (23) and (24) the maximum likelihood estimators is given by λˆ , kˆ and μˆ .

4. Application of distributions

In this section, two real data sets are used to illustrate the applicability of these distributions. The first data set we are considering comes from Smith and Naylor (1987). This data shows the power of glass fibers with length 1.5 mm which are measured in the England international physics laboratory. Alzaatreh, et al. (2014) fitted two gamma- normal distributions (four parametric, GND4p and two parametric, GND2p) to this data set and showed that this distribution Works better than Birnbaum-Saunders distribution (BSD) and Beta Birnbaum-Saunders distribution (BBSD). The maximum likelihood estimators along with comparison between the weibull-rayleigh distribution and gamma- normal distribution and other distributions in terms of Akaike information criterion are computed and given in Table 3.

Distribution	GND4p	GND2p	BBSD	BSD	WRD
Estimated parameters	αˆ=0.587	αˆ=18.084	αˆ=0.363	αˆ=0.269	αˆ=0.989
	βˆ=0.187	βˆ=0.146	βˆ=7857.56	bˆ=1.391	γˆ=2.617
	μˆ=0.2089		â = 1.050		bˆ=1.112
	σˆ=0.391		bˆ=30.478
Akaike information criterion	34.429	34.110	37.552	48.378	33.486

Table 3.

Estimation of parameters for the first data set

From Table 3, we can see that weibull-rayleigh distribution has the minimum value of Akaike information criterion, so that it is the best fit.

The second data set we are considering comes from Cordeiro and Bager (2015). This data shows the production level of milk for 107 Cattles at the beginning of birth. Cordeiro and Bager (2015) fitted the beta power distribution (BPD) to this data and show that the beta power distribution is better than power distribution of this data set. We fit weibull-exponential distribution (WED), weibull distribution (WED), weibull-Logistics distribution, exponential-weibull distribution and generalized-exponential distribution to this data set. The Akaike information criterion and Bayes information criterion along with the maximum likelihood estimators are computed and given in Table 4. It shows that weibull-exponential distribution has the minimum value of Akaike information criterion and Bayes information criterion, so that it is the best fit than the other distributions.

Distribution	WED	BPD	WLD	WD	GE
Estimated parameters	kˆ=1.017	â = 0.270	ĉ = 3.771	kˆ=2.601	â = 3.714
	λˆ=15.288	bˆ=42.022	bˆ=12.126	θˆ=0.523	bˆ=0.238
	μˆ=5.057	â = 6.640	mˆ=0.272
		βˆ=0.775
Akaike information criterion	-52.254	-47.502	-47.154	-38.695	-6.077
Bayes information criterion	-44.235	-39.525	-39.136	-33.349	-0.731

Table 4.

Estimation of parameters for the second data set

5. Concluding remarks

In this study, we introduced weibull-rayleigh distribution and weibull-exponential distribution of X–weibull family. The density functions, distribution functions, hazard rate functions, moment functions and Shannon entropy investigated and capabilities of distributions compared to other distributions. Also by using simulation from these two distributions with various parameters some of the sample statistical indexes estimated. Two real data sets used to illustrate the applicability of these distributions.

Acknowledgment

The authors are deeply grateful to Editor in chief and the referees for their useful comments on an earlier version of this paper.

References

1.A Akinsete, F Famoy, and C Lee, The beta-pareto distribution, Statistics, Vol. 42, 2008, pp. 547-563.

2.SK Agarwarl and JA Al-saleh, Generalized gamma type distribution and its hazard rate function, Communications in Statistics-Theory and Methods, Vol. 30, No. 2, 2001, pp. 309-318.

3.A Alzaatreh, c Lee, and F Famoy, Gamma-pareto distribution and its applications, Journal of Modern Applied Statistical Methods, Vol. 11, No. 1, 2012a, pp. 78-94.

4.A Alzaatreh, C Lee, and F Famoy, On the discrete analogues of continuous distributions, Statistical Methodology, Vol. 9, 2012b, pp. 589-603.

5.A Alzaatreh, F Famoy, and C Lee, Weibull-pareto distribution and its applications, Communications in statistics-Teory and methods, Vol. 42, No. 9, 2013a, pp. 1673-1691.

6.A Alzaatreh, C Lee, and F Famoy, A new method for generating families of continuous distributions, Metron, Vol. 71, No. 1, 2013b, pp. 63-79.

7.A Alzaatreh and K Knight, On the Gamma-half normal distribution and its applications, Journal of Modern Applied Statistical Methods, Vol. 12, No. 1, 2014, pp. 103-119.

8.M Aljarrah, C Lee, and F Famoy, On generating T-X family of distributions using quantile function, Journal of Statistical Distributions and Application, Vol. 1, 2014, pp. 2.

9.GM Cordeiro and RS Bager, The beta power distribution,Brazilian, Journal of Probability and Statistics, Vol. 26, No. 1, 2015, pp. 88-112.

10.N Eugene, C Lee, and F Famoye, The-normal distribution and its applications, Communications in Statistics-Theory and Methods, Vol. 31, No. 4, 2002, pp. 497-512.

11.RD Gupta and D Kundu, Generalized exponential distributions, Australian New Zealand Journal of Statistics, Vol. 41, No. 2, 1999, pp. 173-188.

12.RD Gupta and D Kundu, Exponentiated-exponential family: an alternative to gamma and weibull distributions, Biometrical Journal, Vol. 43, 2001, pp. 117-130.

13.R Gupta, PL Gupta, and RD Gupta, Modeling failure time data by lehman alternatives, Communications in Statistics-Theory and Methods, Vol. 27, No. 4, 1998, pp. 887-904.

14.MSH Khan and GC Jain, A class of distributions in the rst emptiness of a semi-in nite reservoir, Biometrical Journal, Vol. 20, 1978, pp. 243-252.

15.GS Mudholkar, DK Srivastava, and M Fremer, The exponentiated weibull family: A reanalysis of the bus-motor-failure data, Thechnometrics, Vol. 37, No. 4, 1995, pp. 436-445.

16.S Nadarajah and S Kotz, The data Gumbel distribution, Reliability Engineering and System, Vol. 91, No. 6, 2004, pp. 698-697.

17.RL Smith and JC Naylor, A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Journal of Royal Statistical Society Series, Vol. 36, 1987, pp. 358-369.

18.EW Stacy, A generalization of the gamma distribution, Annals of Mathematical Statistics, Vol. 33, 1962, pp. 1187-1192.

<Previous Article In Issue

Download article (PDF)

Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 2
Pages: 375 - 392
Publication Date: 2018/06/30
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.2018.17.2.14 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Abouzar Bazyari
AU  - Monjed H. M. Samuh
PY  - 2018
DA  - 2018/06/30
TI  - Two New Lifetime Distributions of X–Weibull Family: Theories and Applications
JO  - Journal of Statistical Theory and Applications
SP  - 375
EP  - 392
VL  - 17
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.2.14
DO  - 10.2991/jsta.2018.17.2.14
ID  - Bazyari2018
ER  -

download .riscopy to clipboard