A Two Parameter Discrete Lindley Distribution

In this article we have proposed and discussed a two parameter discrete Lindley distribution. The derivation of this new model is based on a two step methodology i.e. mixing then discretizing, and can be viewed as a new generalization of geometric distribution. The proposed model has proved itself as the least loss of information model when applied to a number of data sets (in an over and under dispersed structure). The competing models such as Poisson, Negative binomial, Generalized Poisson and discrete gamma distributions are the well known standard discrete distributions. Its Lifetime classification, kurtosis, skewness, ascending and descending factorial moments as well as its recurrence relations, negative moments, parameters estimation via maximum likelihood method, characterization and discretized bi-variate case are presented.


Introduction
For the last few decades discretized distributions have been studied extensively to model the discrete failure time data in statistical literature. Generally, discretized versions are obtained from any continuous distribution defined on the real line R with probability density function (pdf) f (x), and are based on the support; the set of integers Z = {0, ±1, ±2, ±3, . . .} have a probability mass function that takes either of the two forms: or The former in statistical literature is known as the discrete concentration approach in which S(x) is the preserved survival function of continuous distribution at integers, and the latter is called the time discretization approach in which f (x) is the preserved probability density function (pdf) of the continuous distribution at integers.
The discretization phenomenon generally arises when it becomes impossible or inconvenient to measure the life length of a product or device on a continuous scale. Such situations may arise, when the lifetimes need to be recorded on a discrete scale rather than on a continuous analogue. For examples the number of to and for motions of a pendulum or spring device before resting, the number of times devices are switched on/off, the number of days a patient stays in an observation ward, the length of successful stay of a pig (in terms of number of days/weeks, in the laboratory) and the number of weeks/months/years a cancer patient survives after treatment etc. Although there are a number of discrete distributions in the literature to model the above mentioned situations there is still a lot of space left to develop new discretized distribution that is suitable under different conditions.
In this article, a two parameter discrete Lindley distribution is proposed. It does not only have a simple structure, positively skewed and leptokurtic, but it also has more flexibility than the Dȇniz & Ojeda (2011) single parameter discrete Lindley distribution it also has less loss of information compared with standard discrete distributions. Derivation of two parameter discrete Lindley distribution, along with some properties, are given in section two, section three deals with estimation of parameters. In section four characterization issue is addressed, section five addresses the application of the proposed model and in the sixth section we study a discretized bivariate version of the two parameter Lindley distribution.

Definition and Properties of a Two Parameter
Discrete Lindley Distribution

Derivation
The phenomenon of mixing and then discretizing the continuous distributions with the help of proper weights and a set of parameters is so far a new one. In this manuscript, after adopting the mixing and discretization technique, we have proposed a two parameter discrete Lindley distribution, which can be viewed as a new generalization of geometric distribution. Suppose W 1 ∼ Gamma(1, θ) and W 2 ∼ Gamma(2, θ), on mixing these densities with probabilities p 1 = θ θ+β and p 2 = β θ+β so that p 1 + p 2 = 1 and β ≥ 0, the resulting distribution of the random variable X will be a two parameter Lindley distribution, i.e. X ∼ p 1 W 1 + p 2 W 2 . This implies that In order to model the discrete actuarial failure data Dȇniz & Ojeda (2011) proposed asingle parameter discrete Lindley distribution which is not considered a flexible model for analyzing different life time and actuarial data. It is used to model the over dispersed data pattern (see Dȇniz & Ojeda 2011), which is mathematically a complicated one. Therefore, to increase the flexibility for modeling purposes, we developed a two parameter discrete Lindley distribution by inserting equation (    Definition 1. A random variable Y has a two parameter discrete Lindley distribution with parameters 0 < p < 1 and β ≥ 0, denoted by TDL (p, β), is defined as where exp(−θ) = p. For β = 0 the distribution geometrically reduces, and for β = 1 it becomes one a parameter discrete Lindley distribution. The recursive relation between T DL s probabilities is given by for 0 < p < 1,β ≥ 0 and x = 0, 1, 2, 3, . . .

Reliability Characteristics of TDL
Although, most of the statistical research is based on continuous lifetime probability distributions to model the real lifetime phenomena, reliability engineers/anthropologists are looking for solutions, for which time can be interpreted as a discrete variable such as: the number of times a piece of equipment is operated; the life of a switch being measured by the number of strokes; the life of equipment being measured by the number of cycles it completes or the number of times it is operated prior to failure; the life of a weapon is measured by the number of rounds fired until failure; and the number of years of a married couple successfully completed; or the number of miles that a plane is flown before failures etc. In order to do this they started to discretize the continuous lifetime distributions. In reliability theory, the survivor, the hazard, the cumulative hazard,the accumulated hazard and mean residual life functions are important characteristics upon which the classification of discrete lifetime probability distribution is made. These classifications, in turn, point out the nature of the product for the reliability analyst, which could be for example, the increasing (decreasing) failure rate IFR (DFR) class, increasing (decreasing) failure rate average IFRA (DFRA) class, the new better (worse) than used NBU (NWU) class, new better (worse) than used in expectation NBUE (NWUE) class and increasing (decreasing) mean residual lifetime IMRL (DMRL) class etc. (see Kemp 2004). These classes are generally based on reliability/survival functions which give the probability that a component will survive beyond a specified time. The basic definition and formulae of the above mentioned characteristics for TDL are given below. Reliability function of TDL is defined and expressed as S Its failure rate function which gives the probability of failure given that it has not occurred before a specific time, is defined as h

Theorem 1.
If Y ∼ T DL(p, β) then the probability mass function (pmf )of the random variable Y is log-concave for all choices of β and independent of p.
Proof . In order to show that the two parameter discrete Lindley distribution, as defined in equation (4), is log-concave, it is sufficient to show that for β ≥ 0, Generally, it is seen that the log-concave probability mass functions are strongly unimodal (see Kielson & Gerber 1971, Nekoukhou et al. 2012 and have an increasing failure rate (IFR) which suggest an intuitive concept caused by product wearing out.
Therefore, we have the following Corollary.
Corollary 1. If Y ∼ T DL(p, β) then the mode of the random variable Y is located at m and m satisfies pβ−(1−p) Kielson & Gerber (1971), Abouammoh & Mashhour (1981) and Nekoukhou et al. (2012). From this, the following chain of implication for a two parameter discrete Lindley (TDL) distribution (see Kemp 2004) Proof . By definition the pgf can be expressed as On Replacing t by e t we get a moment generating function (mgf) such as β ≥ 0, 0 < p < 1 for 0 < pe t < 1.
The first four derivatives of equation (6), with respect to t at t = 0, yield the first four moments about origin i.e. µ r = d r M Y (t) dt r | t=0 which after simplification are:

Index of Dispersion
Index of dispersion (ID) for discrete distributions is defined as variance to mean ratio, it indicates whether a certain distribution is suitable for under or over dispersed data sets, and is used widely in ecology as a standard measure for measuring clustering (over dispersion) or repulsion (under dispersion) (see Johnson, Kotz & Kemp 1992). If ID≥ 1(≤ 1) the distribution is over dispersed (under dispersed). It is observed that if either p → 1 or β → 0, the distribution will always follow over dispersion and if p → 0, β → ∞ it will follow the under dispersion phenomenon. Moreover, the distribution is positively skewed with a longer tail compared to a one parameter discrete Lindley and leptokurtic in nature, which is evident from the ratio of the square of the third mean moment to the cube of the second mean moment and the ratio of the fourth mean moment to the square of variance. TDL is positively skewed for β → 0 and p → 0. Also, it approaches 2 and zero (0) as p → 1 and β → ∞ respectively, which is evident in Table 1. Moreover, it is leptokurtic in nature, and has high peakedness as p → 0 and β →0 (see Table  2. Its peakedness approaches six as p → 1 and for smaller values of p and higher values of β it becomes equal to 3.  Theorem 3. If Y∼ TDL(p,β) and r th , descending factorial moment of Y is given by where β ≥ 0, 0 < p < 1, r = 0, 1, . . . , (a) n = a(a + 1)(a + 2) · · · (a + n − 1) and µ (0) = 1.

Maximum Likelihood Method
If Y 1 , Y 2 , . . . , Y n be a random sample drawn identically independently from the two parameters discrete Lindley (TDL) distribution with observed values x 1 , x 2 , . . . , x n then the joint probability function for TDL distribution can be expressed as (1 + βx i ), ln(L(p; β)) = 2n ln(1 − p) − n ln((1 + p(β − 1))) + n i=1 x i ln p + n i=1 ln(1 + βx i ). (11) By partially differentiating both sides of equation (11) with respect to p and β, equating them to zero, we get M LE s of p and β respectively, which can be shown as The M LE s are computed using a computational package such as Mathematica [7.0]. In view of the regularity conditions as stated by Rohatgi and Saleh on page 419, the M LE s i.e. (p,β) of TDL has a bivariate normal distribution with mean (p,β) and a variance-covariance matrix (I(p,β)) −1 . Thus (p,β) ∼ BVN((p,β), (I(p,β)) −1 ) where I(p,β) denotes the information matrix and is given below as entropy of a random variable is considered as the measure of the uncertainty of the random variable. It is used to measure the amount of information required to describe the random variable. In this regard, the entropy of the two parameter discrete Lindley (TDL) distribution is defined as:

Characterization
Theorem 7. Let Y be a nonnegative discrete random variable with probability mass function P (Y = x) and x ∈ Z + ; it then will follow the two parameter discrete Lindley distribution with parameters p and β iff where ,0 < p < 1 and β ≥ 0.
Proof . Necessity: According to Kemp (2004) the MRL function is defined as After simplification we get Sufficency: Suppose equation (14) holds then it can be written as on comparing equation (15) with equation (16) we get Theorem 8. The random variable Y ∼ T DL(p, 1) iff it can be written as Y ≡ X 1 + X 2 where X i ∼ G 0 (q) i.e. P (X = x i ) = p xi q, x i = 0, 1, . . . for i = 1, 2 are independent random variables.

Real Data Examples
We used two data sets reported by Chakraborty & Chakravarty (2012), to investigate the competence of the proposed model. Suitsbility of the proposed model is tested via the p-value and the Akaike s information criteria (AIC) proposed by Hirotsugu Akaike in 1971. It was then compared with Poisson, Negative binomial, Generalized Poisson and discrete gamma distributions; as stated in Chakraborty & Chakravarty (2012).
Data set 1: The first data set contains observations on a number of European red mites on apple leaves and is presented in Table 3. Clearly this data set belongs to on over dispersed structure that has ID = 1.9828. We present the MLE, observed and expected frequencies, Log-Likelihood (LL), Chi-square values, p-values and AIC values for two parameter discrete Lindley distribution:β = 0.146;p = 0.479; LL = −222.3; d.f. = 5; χ 2 = 2.514; p-value = 0.774; AIC = 448.76.  It can be observed that in these examples the proposed model not only gives high p-values but also a minimum AIC compared to the distributions mentioned by Chakraborty & Chakravarty (2012). It therefore depicts the situation that the proposed model has the least loss of information in comparison with to the standard distributions.

Discretized Bivariate Case
Bivariate discrete random variables defined on integers or on non-negative values are used to model the paired count data that arise in a number of situations such as: in the analysis of accidents, e.g. the number of accidents in a site before and after infrastructure changes; in epidemiological analysis, e.g. incidents of different diseases in a series of districts; in medical research, e.g. the number of seizures before and after treatment etc. In this regard, literature on bivariate discrete distribution is sparse and worth mentioning particularly in terms of bivariate discretized distributions. Here, we give a bivariate discretized Lindley distribution by discretizing the bivariate continuous distribution, using the following equation:
Proof . Suppose Y = (Y 1 , Y 2 ) follows a two parameter discrete bivariate Lindley distribution as defined in equation (19). Let us consider U = Y 1 + Y 2 and V = Y 1 . Now by using the change of variable technique we can write the joint probability mass function of U and V as: 0 ≤ v ≤ u ∈ Z + , β ≥ 0 and θ > 0. Now, on summing over v we get the probability mass function of U as expressed in equation (25).

Conclusion
A two parameter discrete Lindley distribution has been proposed. Its various distributional properties, reliability characteristics and characterization have been studied. It was found that this distribution has a simple structure, is more mathematically amenable, more flexible and has a longer tail than the one parameter discrete Lindley and other models in modeling actuarial and other count data from various fields such as ecology, health, psychology, sociology and engineering. It also has a less loss of information compared to the standard discrete distributions. Further issues such as characterization and mixtures are currently being researched and may be discussed in the further papers.