New Family of Logarithmic Lifetime Distributions

In this study a new lifetime class with decreasing failure rate is introduced by compounding truncated logarithmic distribution with any proper continuous lifetime distribution. The properties of the proposed class are discussed, including a formal proof of itsprobability density function, distribution function and explicit algebraic formulae for its reliability and failure rate functions. A simple EM-type algorithm for iteratively computing maximum likelihood estimates is presented. A formal equation for Fisher information matrix is derived in order to obtaining the asymptotic covariance matrix. Thisnew class of distributions generalizes several distributions which have been introduced and studied in the literature.


INTRODUCTION
Multi-parameter distributions to model lifetime data have been introduced by compounding a continuous lifetime and powerseries distributions. The Exponential Geometric (EG), Exponential Poisson (EP) and exponential logarithmic distributions were introduced and studied by Adamidis and Loukas (1998), Kus (2007) and Tahmasbi and Rezaei (2008), respectively. Recently, Chahkandi and Ganjali (2009) introduced the Exponential Power Series (EPS) distributions, which contain these distributions.
Situations where the failure rate function decreases with time have been reported by several authors. Indicative examples are business mortality (Lomax, 1954), failure in the air-conditioning equipment of a fleet of Boeing 720 aircrafts or in semiconductors from various lots combined (Proschan,1963) and the life of integrated circuit modules (Saunders and Myhre, 1983). In general, a population is expected to exhibit Decreasing Failure Rate (DFR) when its behavior over time is characterized by 'work hardening' (in engineering terms) or 'immunity' (in biological terms); sometimes the broader term 'infant mortality' is used to denote the DFR phenomenon. The resulting improvement of reliability with time might have occurred by means of actual physical changes that caused self-improvement or simply it might have been due to population heterogeneity. Indeed, Proschan (1963) provided that the DFR property is inherent to mixtures of distributions with constant failure rate (McNolty et al., 1980 for other properties of exponential mixtures) and Gleser (1989) demonstrated the converse for any gamma distribution with shape parameter less than one. In addition, Gurland and Sethuramm (1994) give examples illustrating that such results may hold for mixtures of distributions with rapidly increasing failure rate. A mixture of truncated geometric distribution and exponential with DFR was introduced. The Exponential-Poisson (EP) distribution proposed by Kus (2007) and genearalized by Hemmati et al. (2011) using Wiebull distribution and the exponential-logarithmic distribution discussed by Tahmasbi and Rezaei (2008). Silva et al. (2010) did a new distribution with decreasing, increasing and upside down bathtub failure rate. A two-parameter distribution family with decreasing failure rate arising by mixing power-series distribution has been introduced by Chahkandi The marginalpdf and cdfof X are given by: We denote a random variable X with pdf and cdf (1)  and (2) by X∼LL (p, θ ). Thisnew class of distributions generalizes several distributions which have been introduced and studied in the literature. For instance using the probability density and its distribution function of exponential distribution in (1), we obtain the logarithmic exponential distribution Tahmasbi and Rezaei (2008) and using Wiebull probability density and its distribution function gives Wiebulllogarithmic distribution Morais and Barreto-Souza (2011). The model is obtained under the concept of population heterogeneity (through the process of compounding). An interpretation of the proposed model is as follows: a situation where failure (of a device for example) occurs due to the presence of an unknown number, Z, of initial defects of same kind (a number of semiconductors from a defective lot, for example). The Ts represent their lifetimes and each defect can be detected only after causing failure, in which case it is repaired perfectly (Adamidis and Loukas, 1998). Then the distributional assumptions given earlier lead to any of the L Ldistributions for modeling the time to the first failure X. Table 1 shows the probability function and the distribution function for some lifetime distributions.
Some of the other lifetime distributions are excluded from this table such as Gamma and lognormal distributions. Those distributions do not have nice forms although they still can be applied in this class numerically.
The q th quantile x q of the LL distribution, the inverse of the distribution function F X (x q ) = q is the same as the inverse of the distribution for any continuous lifetime with distribution function F T (.).

Estimation
In what follows, we discuss the estimation of the LL class parameters. Let x 1 ,…,x n be a random sample with observed values x 1 ,. . . , x n from a L Ldistribution with parameters p and θ . Let Θ = (p, θ ) be the parameters vector. The log log-likelihood function based on the observed random sample size of n, y obs = (x 1 ,…,x n ) is obtained by:  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 n ( ) Θ − Θ is multivariate normal