The compound class of extended Weibull power series distributions

The compound class of extended Weibull power series distributions

Rodrigo B. Silva111Email: rodrigobs29@gmail.com Universidade Federal de Pernambuco
Departamento de Estatística, Cidade Universitária, 50740-540 Recife, PE, Brazil
Marcelo B. Pereira222Email: m.p.bourguignon@gmail.com Cícero R. B. Dias333Email: cicerorafael@gmail.com Gauss M. Cordeiro444Corresponding author. Email: gauss@de.ufpe.br
Abstract

In this paper, we introduce a new class of distributions which is obtained by compounding the extended Weibull and power series distributions. The compounding procedure follows the same set-up carried out by Adamidis and Loukas (1998) and defines at least new 68 sub-models. This class includes some well-known mixing distributions, such as the Weibull power series (Morais and Barreto-Souza, 2010) and exponential power series (Chahkandi and Ganjali, 2009) distributions. Some mathematical properties of the new class are studied including moments and generating function. We provide the density function of the order statistics and obtain their moments. The method of maximum likelihood is used for estimating the model parameters and an EM algorithm is proposed for computing the estimates. Special distributions are investigated in some detail. An application to a real data set is given to show the flexibility and potentiality of the new class of distributions.

keywords:
EM algorithm, Extended Weibull distribution, Extended Weibull power series distribution, Order statistic, Power series distribution.
journal: Computational Statistics & Data Analysis

1 Introduction

The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. Several distributions have been proposed in the literature to model lifetime data by compounding some useful lifetime distributions. Adamidis and Loukas (1998) introduced a two-parameter exponential-geometric (EG) distribution by compounding an exponential distribution with a geometric distribution. In the same way, the exponential Poisson (EP) and exponential logarithmic (EL) distributions were introduced and studied by Kus (2007) and Tahmasbi and Rezaei (2008), respectively. Recently, Chahkandi and Ganjali (2009) proposed the exponential power series (EPS) family of distributions, which contains as special cases these distributions. Barreto-Souza et al. (2010) and Lu and Shi (2011) introduced the Weibull-geometric (WG) and Weibull-Poisson (WP) distributions which naturally extend the EG and EP distributions, respectively. In a very recent paper, Morais and Barreto-Souza (2011) defined the Weibull power series (WPS) class of distributions which contains the EPS distributions as sub-models. The WPS distributions can have an increasing, decreasing and upside down bathtub failure rate function.

Now, consider the class of extended Weibull (EW) distributions, as proposed by Gurvich et al. (1997), having the cumulative distribution function (cdf)

(1)

where is a non-negative monotonically increasing function which depends on a parameter vector . The corresponding probability density function (pdf) is given by

(2)

where is the derivative of .

Note that many well-known models are special cases of equation (1) such as:

(i) gives the exponential distribution;

(ii) yields the Rayleigh distribution (Burr type-X distribution);

(iii) leads to the Pareto distribution;

(iv) gives the Gompertz distribution.

In this article, we define the extended Weibull power series (EWPS) class of univariate distributions obtained by compounding the extended Weibull and power series distributions. The compounding procedure follows the key idea of Adamidis and Loukas (1998) or, more generally, by Chahkandi and Ganjali (2009) and Morais and Barreto-Souza et al. (2011). The new class of distributions contains as special models the WPS distributions, which in turn extends the EPS distributions and defines at least new 68 (17 4) sub-models as special cases. The hazard function of our class can be decreasing, increasing, bathtub and upside down bathtub.

We are motivated to introduce the EWPS distributions because of the wide usage of the general class of Weibull distributions and the fact that the current generalization provides means of its continuous extension to still more complex situations.

This paper is organized as follows. In Section 2, we define the EWPS class of distributions and demonstrate that there are many existing models which can be deduced as special cases of the proposed unified model. In Section 3, we provide the density, survival and hazard rate functions and derive some useful expansions. In Section 4, we obtain its quantiles, ordinary and incomplete moments. Further, the order statistics are discussed and their moments are determined. Section 5 deals with reliability and average lifetime. Estimation of the parameters by maximum likelihood using an EM algorithm and large sample inference are investigated in Section 6. In Section 7, we present suitable constraints leading to the maximum entropy characterization of the new class. Three special cases of the proposed class are studied in Section 8. In Section 9, we provide an application to a real data set. The paper is concluded in Section 10.

2 The new class

Our class can be derived as follows. Given , let be independent and identically distributed (iid) random variables following (1). Here, is a discrete random variable following a power series distribution (truncated at zero) with probability mass function

(3)

where depends only on , and is such that is finite. Table 1 summarizes some power series distributions (truncated at zero) defined according to (3) such as the Poisson, logarithmic, geometric and binomial distributions. Let . The conditional cumulative distribution of is given by

i.e., follows a general class of distributions (1) with parameters and based on the same function. Hence, we obtain

The EWPS class of distributions can then be defined by the marginal cdf of :

(4)
Distribution
Poisson
Logarithmic
Geometric 1
Binomial
Table 1: Useful quantities for some power series distributions.
Distribution References
Exponential () 1 Johnson et al. (1994)
Pareto () Johnson et al. (1994)
Rayleigh () Rayleigh (1880)
Weibull () Johnson et al. (1994)
Modified Weibull () Lai et al. (2003)
Weibull extension () Xie et al. (2002)
Log-Weibull () 1 White (1969)
Phani () Phani (1987)
Weibull Kies () Kies (1958)
Additive Weibull () 1 Xie and Lai (1995)
Traditional Weibull () Nadarajah and Kotz (2005)
Gen. power Weibull () 1 Nikulin and Haghighi (2006)
Flexible Weibull extension() 1 Bebbington et al. (2007)
Gompertz () Gompertz (1825)
Exponential power () 1 Smith and Bain (1975)
Chen () Chen (2000)
Pham () 1 Pham (2002)
Table 2: Special distributions and related and functions.

The random variable following (4) with parameters and and the vector of parameters is denoted by . Equation (4) extends several distributions which have been studied in the literature. The EG distribution (Adamidis and Loukas, 1998) is obtained by taking and with . Further, for , we obtain the EP (Kus, 2007) and EL (Tahmasbi and Rezaei, 2008) distributions by taking , and , respectively. In the same way, for , we obtain the WG (Barreto-Souza et al., 2009) and WP (Lu and Shi, 2011) distributions. The EPS distributions are obtained from (4) by mixing with any listed in Table 1 (see Chahkandi and Ganjali, 2009). Finally, we obtain the WPS distributions from (4) by compounding with any in Table 1 (see Morais and Barreto-Souza, 2011). Table 2 displays some useful quantities and respective parameter vectors for each particular distribution.

3 Density, survival and hazard functions

The density function associated to (4) is given by

(5)
Proposition 1.

The EW class of distributions with parameters and is a limiting special case of the EWPS class of distributions when , where .

Proof.

This proof uses a similar argument to that found in Morais and Barreto-Souza (2011). Define . We have

for . ∎

We now provide an interesting expansion for (5). We have . By using this result in (5), it follows that

(6)

where is given by (2). Based on equation (6), we obtain

Hence, the EWPS density function is an infinite mixture of EW densities. So, some mathematical quantities (such as ordinary and incomplete moments, generating function and mean deviations) of the EWPS distributions can be obtained by knowing those quantities for the baseline density function .

The EWPS survival function is given by

(7)

and the corresponding hazard rate function becomes

(8)

4 Quantiles, moments and order statistics

The EWPS distributions are easily simulated from (4) as follows: if has a uniform distribution, then the solution of the nonlinear equation

has the EWPS distribution, where and are the inverse functions of and , respectively. To simulate data from this nonlinear equation, we can use the matrix programming language Ox through SolveNLE subroutine (see Doornik, 2007).

We now derive a general expression for the th raw moment of , which may be determined by using (6) and the monotone convergence theorem. So, for , we obtain

(9)

where is a random variable with pdf .

The incomplete moments and moment generating function (mgf) follow by using (6) and the monotone convergence theorem:

and

where is defined as before.

Order statistics are among the most fundamental tools in non-parametric statistics and inference. They enter in the problems of estimation and hypothesis tests in a variety of ways. Therefore, we now discuss some properties of the order statistics for the proposed class of distributions. The pdf of the th order statistic for a random sample from the EWPS distribution is given by

(10)

where is the pdf given by (5). By using the binomial expansion, we can write (10) as

(11)

where is given by (7). The corresponding cumulative function is

An alternative form for (10) can be obtained from (6) as

(12)

where . So, the th raw moment comes immediately from the above equation

(13)

where is defined before.

5 Reliability and average lifetime

In the context of reliability, the stress-strength model describes the life of a component which has a random strength subjected to a random stress . The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever . Hence, is a measure of component reliability. It has many applications, especially in engineering concepts. The algebraic form for R has been worked out for the majority of the well-known distributions. Here, we obtain the form for the reliability when and are independent random variables having the same EWPS distribution.

The quantity can be expressed as

(14)

Substituting (4) and (5) into equation (14), we obtain

where the integral can be calculated from the baseline EW distribution.

The average lifetime is given by

Given that there was no failure prior to , the residual life is the period from time until the time of failure. The mean residual lifetime can be expressed as

The last integral can be computed from the baseline EW distribution. Furthermore, as .

6 Maximum likelihood estimation

6.1 Preliminaries

Here, we determine the maximum likelihood estimates (MLEs) of the parameters of the EWPS class of distributions from complete samples only. Let be a random sample with observed values from an EWPS distribution with parameters and . Let be the parameter vector. The total log-likelihood function is given by

(15)

The log-likelihood can be maximized either directly by using the SAS (PROC NLMIXED) or the Ox program (sub-routine MaxBFGS) (see Doornik, 2007) or by solving the nonlinear likelihood equations obtained by differentiating (15). The components of the score function are

and

For interval estimation on the model parameters, we require the observed information matrix

whose elements are listed in A. Let be the MLE of . Under standard regular conditions stated in Cox and Hinkley (1974) that are fulfilled for our model whenever the parameters are in the interior of the parameter space, we have that the asymptotic distribution of is multivariate normal , where is the unit information matrix and is the number of parameters of the compounded distribution.

6.2 The EM algorithm

Here, we propose an EM algorithm (Dempster et al., 1977) to estimate . The EM algorithm is a recurrent method such that each step consists of an estimate of the expected value of a hypothetical random variable and then maximizes the log-likelihood for the complete data. Let the complete-data be with observed values and the hypothetical random variables . The joint probability function is such that the marginal density of is the likelihood of interest. Then, we define a hypothetical complete-data distribution for each , with a joint probability function in the form

where and are positive, and . Under this formulation, the E-step of an EM cycle requires the expectation of ; as the current estimate (in the rth iteration) of . The probability function of given , say , is given by

and its expected value is

The EM cycle is completed with the M-step by using the maximum likelihood estimation over , where the missing are replaced by their conditional expectations given before. The log-likelihood for the complete-data is

So, the components of the score function are

and

From a nonlinear system of equations , we obtain the iterative procedure of the EM algorithm

and

where and are obtained numerically. Here, for , we have

Note that, in each step, and are estimated independently. The EWPS distributions can be very useful in modeling lifetime data and practitioners may be interested in fitting one of our models.

7 Maximum entropy identification

Shannon (1948) introduced the probabilistic definition of entropy which is closely connected with the definition of entropy in statistical mechanics. Let be a random variable of a continuous distribution with density . Then, the Shannon entropy of is defined by

(16)

Jaynes (1957) introduced one of the most powerful techniques employed in the field of probability and statistics called the maximum entropy method. This method is closely related to the Shannon entropy and considers a class of density functions

(17)

where , are absolutely integrable functions with respect to , and . In the continuous case, the maximum entropy principle suggests deriving the unknown density function of the random variable by the model that maximizes the Shannon entropy in (16), subject to the information constraints defined in the class . Shore and Johnson (1980) treated axiomatically the maximum entropy method. This method has been successfully applied in a wide variety of fields and has also been used for the characterization of several standard probability distributions; see, for example, Kapur (1989), Soofi (2000) and Zografos and Balakrishnan (2009).

The maximum entropy distribution is the density of the class F, denoted by , which is obtained as the solution of the optimization problem

Jaynes (1957, p. 623) states that the maximum entropy distribution , obtained by the constrained maximization problem described above, “is the only unbiased assignment we can make; to use any other would amount to arbitrary assumption of information which by hypothesis we do not have”. It is the distribution which should not incorporate additional exterior information other than which is specified by the constraints.

We now derive suitable constraints in order to provide a maximum entropy characterization for our class of distributions defined by (4). For this purpose, the next result plays an important role.

Proposition 2.

Let X be a random variable with pdf given by (5). Then,

C1.

C2.

C3.

where Y follows the EW distribution with density function (2).

Proof.

The constraints C1, C2 and C3 are easily obtained and therefore their demonstrations are omitted. ∎

The next proposition reveals that the EWPS distribution has maximum entropy in the class of all probability distributions specified by the constraints stated in the previous proposition.

Proposition 3.

The pdf f of a random variable X, given by (5), is the unique solution of the optimization problem

under the constraints , and presented in the Proposition 2.

Proof.

Let be a pdf which satisfies the constraints C1, C2 and C3. The Kullback-Leibler divergence between and is

Following Cover and Thomas (1991), we obtain

From the definition of and based on the constraints C1, C2 and C3, it follows that

where is defined as before. So, we have with equality if and only if for all , except for a set of measure 0, thus proving the uniqueness. ∎

The intermediate steps in the above proof in fact provide the following explicit expression for the Shannon entropy of the EWPS distribution

(18)

For some EWPS distributions, the above results can only be obtained numerically.

8 Special models

In this section, we investigate some special cases of the EWPS class of distributions. We offer some expressions for moments and moments of the order statistics. To illustrate the flexibility of these distributions, we provide plots of the density and hazard rate functions for selected parameter values.

8.1 Modified Weibull geometric distribution

The modified Weibull geometric (MWG) distribution is defined by the cdf (4) with and leading to

(19)

where . The associated pdf and hazard rate function are

and