A Generalization Of Inverse Marshall-Olkin Family Of Distributions

We introduce a new family of distributions namely inverse truncated discrete Linnik G family of distributions. This family is a generalization of inverse Marshall-Olkin family of distributions, inverse family of distributions generated through truncated negative binomial distribution and inverse family of distributions generated through truncated discrete Mittag-Leffler distribution. A particular member of the family, inverse truncated negative binomial Weibull distribution is studied in detail. The shape properties of the probability density function and hazard rate, model identifiability, moments, median, mean deviation, entropy, distribution of order statistics, stochastic ordering property, mean residual life function and stress-strength properties of the new generalized inverse Weibull distribution are studied. The unknown parameters of the distribution are estimated using maximum likelihood method, product spacing method and least square method. The existence and uniqueness of the maximum likelihood estimates are proved. Simulation is carried out to illustrate the performance of maximum likelihood estimates of model parameters. An AR(1) minification model with this distribution as marginal is developed. The inverse truncated negative binomial Weibull distribution is fitted to a real data set and it is shown that the distribution is more appropriate for modeling in comparison with some other competitive models. inverse truncated negative binomial Weibull distribution distribution. The new proposed distribution is a generalization of Marshall-Olkin extended inverse Weibull, Marshall-Olkin extended inverse Rayleigh, Marshall-Olkin extended inverse exponential, inverse Weibull, inverse Rayleigh and inverse exponential distribution.


Introduction
In the last two decades researchers have greater intention toward the inversion of univariate probability models and their applicability under inverse transformation. The inverse distribution is the distribution of the reciprocal of a random variable. Dubey (1970) proposed inverted beta distribution, Voda (1972) studied inverse Rayleigh distribution, Folks and Chhikara (1978) proposed inverse Gaussian distribution, Prakash (2012) studied the inverted exponential model, Sharma  The inverse Weibull (IW) distribution is commonly used in statistical analysis of lifetime or response time data from reliability experiments. For the situations in which empirical studies indicate that the hazard function might be unimodal, the IW distribution would be an appropriate model. Khan et al. (2008) in their theoretical analysis of IW distribution mention that numerous failure characteristics such as wear out periods and infant mortality can be modeled through IW distribution. They mention the wide range of areas in reliability analysis where IW distribution model can be used successfully. The IW model has been derived as a suitable model for describing the degradation phenomena of mechanical components, such as the dynamic components of diesel engines, see, for example, Murthy et al. (2004). Erto and Rapone (1984) showed that IW model provides good fit to survival data such as the time to breakdown of an insulating fluid subject to the action of constant tension. Interpretation of IW in the context of load strength relationship for a component was provided by Calabria and Pulcini (1994). Furthermore, this distribution is one of the most popular distributions in complementary risk problems. Shafiei et al. (2016) mention that IW distribution is an appropriate model for situation where hazard function is unimodal. The Marshall-Olkin IW distribution and its application in the context of reliability analysis is discussed in Okasha et al. (2017). Hassan and Nassar (2018) studied properties and applications of IW generator of distributions. Compounding IW distribution with zero truncated Poisson and geometric distributions are studied by Chakrabarty and Chowdhury (2018).
Adding parameters to a well-established distribution is a time-honored technique for obtaining more flexible new families of distributions. Marshall and Olkin (1997) discussed a method of adding a new parameter to an existing distribution. It includes the baseline distribution as a special case, and gives more flexibility to model various types of data. One of the important properties of this family is that Marshall-Olkin family of distributions possess stability property in the sense that if the method is applied twice, it returns to the same distribution. Also this family satisfies geometric extreme stability property. Marshall and Olkin (1997) started with a parent survival function F ̅ (x) and considered a family of survival functions given by They described the motivation for the family of distributions (1) as follows: Let X 1 , X 2 , … be a sequence of independent and identically distributed (i.i.d.) random variables with survival function F ̅ (x). Let = min( 1 , 2 , … , ), (2) where N is the geometric random variable with probability mass function (pmf) P(N = n) = α(1 − α) n−1 for n = 1,2, … and 0 < α < 1 and independent of X i 's. Then the random variable U N has the survival function given by (1). If α > 1 and N is a geometric random variable with pmf of the form P(N = n) = 1 α (1 − 1 α ) n−1 , n = 1,2, …, then the random variable V N = max (X 1 , X 2 , … , X N ) also has the survival function as (1).
In the past, many authors have studied various univariate distributions belonging to the Marshall-Olkin family of distributions. We refer to the paper of Tahir and Nadarajah (2015) for a list of univariate Marshall-Olkin distributions. Besides special distributions, three families of distributions, which generalize Marshall-Olkin family of distributions, have been recently introduced.
First, Nadarajah, Jayakumar and Ristić (2013) proposed a new generalization of the Marshal-Olkin family of distributions, by replacing the geometric distribution of N in (2), as truncated negative binomial distribution with pmf given by where > 0 and > 0. The authors showed that the random minimum, = min( 1 , 2 , … , ) has the survival function of the form Note that if → 1, then ̅ ( ; , ) → ̅ ( ). The family of distributions given in (3) is a generalization of Marshall-Olkin family of distributions, in the sense that when = 1, (3) reduces to (1). Pillai and Jayakumar (1995) introduced a class of discrete distributions containing geometric and named it as discrete Mittag-Leffler (DML) distribution, since it arises a discrete analogue of the well known continuous Mittag-Leffler distribution introduced by Pillai (1990). In DML distribution, the probability of success depends on the trail number where as in geometric, it is constant through out the trails. Hence, the model obtained by replacing geometric minimum by truncated DML minimum of i.i.d random variables, may be more realistic.
Second generalized family of distributions have been introduced by Sankaran and Jayakumar (2016). The authors introduced a family of distributions by replacing the distribution of in (2) as the discrete Mittag-Leffler distribution, a generalization of geometric distribution whose probability generating function (pgf) is given by , > 0, 0 < ≤ 1.
Using truncated discrete Mittag-Leffler distribution, they derived a family of distributions with parameters and having survival function Note that, the Marshall-Olkin method applied to ( ), the exponentiated form of a parent distribution function , will also gives rise (4). The family of distributions generated using truncated discrete Mittag Leffler distribution can also be considered as a generalization of Marshall-Olkin family of distributions, since it reduces to Marshall-Olkin family, when = 1 and = 1− .
A non negative integer valued random variable is said to be discrete Linnik distributed, if it has the pgf Third generalized family of distributions introduced by Jayakuamar and Sankaran (2019) using truncated discrete Linnik family of distributions with parameters β, θ and c have the survival function In (5), when = 1 and ≠ 1, we obtain the survival function of the family of distributions generated using truncated discrete Mittag-Leffler distribution. When =1 and ≠ 1 in (5), we obtain the survival function of the family of distributions generated using truncated negative binomial distribution in (3). Also when =1 and = 1 in (5), we obtain the survival function of Marshall-Olkin scheme, in (1).
In this paper, we introduce a new family of distributions which we named as inverse truncated discrete Linnik G family of distributions. In particular, we study inverse truncated negative binomial Weibull distribution distribution. The new proposed distribution is a generalization of Marshall-Olkin extended inverse Weibull, Marshall-Olkin extended inverse Rayleigh, Marshall-Olkin extended inverse exponential, inverse Weibull, inverse Rayleigh and inverse exponential distribution.
The rest of the paper is organized as follows. In Section 2, we introduce a new family of distributions, namely inverse truncated discrete Linnik G distribution and their different sub models such as inverse family of distributions generated through discrete Mittag-Leffler G family of distributions, inverse family of distributions generated through truncated negative binomial G family of distributions and inverse family of distributions generated through Marshall-Olkin G family of distributions. In particular, we study inverse truncated negative binomial Weibull (GIW) distribution in Section 3. The shape properties of density and hazard function are studied. The model identifiability of the distribution is proved. The GIW distribution is represented as a mixture of inverse Weibull distribution. In Section 4, some structural properties of GIW distribution such as moments and generating function, quantiles, mean deviation, entropy, order statistics, stochastic ordering and mean residual life function are studied. Method of generation of random variate from GIW distribution is also discussed. Estimation of stress-strength parameters are discussed in Section 5. Estimation of the model parameters by three methods-maximum likelihood estimation, method of product spacing, method of least squares are performed in Section 6. The existence and uniqueness of maximum likelihood estimates are also established.Simulation study is also carried out in order to establish the consistency property of the maximum likelihood estimates of our proposed model. An autoregressive minification process with GIW marginals is developed in Section 7. Finally, in Section 8, we present an application of a real data set, which exhibits the performance of GIW compared to twelve well known models. The GIW distribution has least -logL, AIC, CAIC, BIC, HQIC, W * , A * , K − S statistic and highest p-value for this data set compared to all other models. , , , > 0; > 0

Inverse family of distributions generated through truncated discrete Linnik distribution
Hence, we obtain a new family of distributions, which we named as inverse family of distributions generated through discrete Linnik G distribution. The probability density function (pdf) and the hazard rate function (hrf) of a random variable from the introduced family are respectively, and ℎ( , , , ) = −1 (1⁄ ) (1⁄ )

Inverse truncated negative binomial G family of distributions
When in equation (6), β = 1 and c = 1−α α , the cdf reduces to inverse truncated negative binomial G family of distributions. So the cdf, pdf and hrf of inverse truncated negative binomial G family of distributions are respectively: and

Inverse Marshall-Olkin G family of distributions
In equation (6), when = 1, = 1 and = 1− , the cdf reduces to inverse Marshall-Olkin G family of distributions. So the cdf, pdf and hrf of inverse Marshall-Olkin G family of distributions are respectively: and Also when = 1, the inverse Marshall-Olkin G family of distribution reduces to inverse family of distributions.

A new generalization of inverse Weibull distribution
Now we consider, generalized inverse Weibull distribution generated through inverse truncated negative binomial and Weibull distribution. Negative binomial is a generalization of the geometric, and Poisson distributions is a limiting particular case. The negative binomial distribution with support over the set of all non-negative integers is also a generalization of the Poisson distribution in the sense that it can deduced as a hierarchial model if X~Poisson (Λ) with Λ being a gamma random variable.

Probability density function
The pdf of the new distribution is given by We refer to this new distribution having cdf (18) as inverted truncated negative binomial Weibull distribution with parameters , , and . We write it as ( ; , , , ).
The graph of ( ) for different values of the parameters is given in Figure 1. Some sub-models of the GIW distribution are listed below: i. When = 1, we have the inverse Marshall-Olkin Weibull distribution. ii.
When → 1, we have the inverse Weibull distribution. v.
When → 1 and = 2, we have the inverse Rayleigh distribution. vi.
When → 1 and = 1, we have the inverse exponential distribution.

Unimodality
The pdf of the GIW model is either decreasing or unimodal. In order to investigate the critical points of density function, its first derivative with respect to y is . ′( ) = 0 implies, Since equation (20) is a nonlinear equation in , there may be more than one positive root to (20). If = 0 is a root of (20), then it corresponds to a local maximum if ′ ( ) > 0 for all < 0 . It corresponds to a local minimum if ′ ( ) < 0 for all < 0 and ′ ( ) > 0 for all > 0 . It corresponds to a point of inflexion if either ′ ( ) > 0 for all ≠ 0 or ′ ( ) < 0 for all ≠ 0 .
The graph of h(y) for different values of the parameters is given in Figure 2. Note that inverse truncated negative binomial Weibull (GIW) distribution and truncated negative binomial inverse Weibull distribution are seems to be similar structure, but little different. For example, let G 1 (y; α, θ, λ, β) and H 1 (y; α, θ, λ, β) are the cdf and hazard function of GIW distribution respectively and G 2 (y; α, θ, λ, β) and H 2 (y; α, θ, λ, β) are the cdf and hazard function of truncated negative binomial inverse Weibull distribution respectively. In Table 1, we consider the values of cdf and hazard function for α = 1.5, θ = 0.5, λ = 1.0 and β = 0.5. Table 1: The values of cdf and hazard function when = . , = . , = . and = . .

Model identifiability
We have to prove model identifiability only with respect to the parameters α and θ, since the other two parameters (λ and β) are from the parent distribution. Let us suppose that G(y; α 1 , θ 1 ) = G(y; α 2 , θ 2 ) for all y > 0. We will show that this condition implies that α 1 = α 2 and θ 1 = θ 2 . For proving model identifiability, we use Theorem 2.4 of Chandra (1977). Proposition : The class of all mixing distribution relative to the GIW distribution is identifiable. Proof : If is truncated negative binomial random variable, truncated at 0, then the probability generating function is From the cdf of , we have 1 < 2 when 1 = 2 and 1 < 2 and and thus the identifiability is proved. Hence the cdf is identifiable with respect to and .

Expansion for distribution function and density function
For |z|<1 and k>0, we have where Γ(. ) is the gamma function. By using (24), the cdf of GIW distribution can be expressed as In similar manner the pdf of GIW distribution can be expressed as

General properties of GIW distribution 4.1 The moments
We know that moments are important in any statistical analysis. In this subsection, we present r th moments of GIW distribution. From the definition of moments, we have By putting = 1 and = 2 in (27), we can easily obtain the mean and variance of GIW distribution.

The moment generating function
The moment generating function is given by By using the Tayler's series of expansion of the function , we obtain, As before, putting = ( + 1) − and simplifying, we have

Simulation and Quantiles
Random variable having GIW distribution can be easily simulated by inverting the cdf. Let has unform (0,1) distribution, then In addition, the ℎ quantile of GIW distribution is given by 0 < < 1.
In particular, the median of GIW distribution is given by The Bowley's skewness is based on quantiles: and the Moors' kurtosis is based on octiles: where (. ) represents the quantile function of . These measures are less sensitive to outliers and they exist even for distributions without moments. Skewness measures the degree of the long tail and kurtosis is a measure of the degree of peakedness. When the distribution is symmetric, S = 0 and when the distribution is left(or right) skewed, < 0 (or > 0). As increases, the tail of the distribution becomes heavier. We compute mean, median, variance, skewness and kurtosis numerically using R software and presented in Table 2 when = 1.0 and = 5.0. From Table 2, we can see that GIW distribution is positively skewed and leptokurtic. When < 1 and is increasing, mean, median and variance are increasing while when > 1 and is increasing, mean, median and variance are decreasing.

Mean deviation
The mean deviation about mean is defined by where μ is the mean, which can be rewritten as Using integration by parts, it simplifies to where G(. ) denote the cdf of GIW distribution. Hence, The mean deviation about median is defined as

Entropy
An entropy is a measure of variation or uncertainty. The Rényi entropy of a random variable with pdf g(. ) is defined as We have Therefore,

Order statistics
Let Y 1 , Y 2 , … , Y n be a random sample of size n from the GIW distribution and Y (1) , Y (2) , … , Y (n) denote the corresponding order statistics. When the population cdf and pdf are G(y) and g(y) respectively, then the r th order (r = 1,2, … , n) cdf and pdf are respectively, given by,

Stochastic ordering
Stochastic orders have been used during the last forty years, at an accelerated rate, in many diverse areas of probability and statistics. Such areas include reliability theory, survival analysis, queueing theory, biology, economics, insurance and actuarial science (see, Shaked and Shanthikumar (2007)). Let X and Y be two random variables having cdf's F and G respectively, and denote by F ̅ = 1 − F and G ̅ = 1 − G their respective survival functions, with corresponding pdf's f, g. The random variable X is said to be smaller than Y in the: i. stochastic order (denoted as X ≤ st Y) if F ̅ ( ) ≤ G ̅ ( ) for all x; ii.
likelihood ratio order (denoted as X ≤ lr Y) if f(x)/g(x) is decreasing in x ≥ 0; iii. hazard rate order (denoted as X ≤ hr Y)if F ̅ ( )/G ̅ ( ) is decreasing in x ≥ 0; iv. reversed hazard rate order (denoted as The four stochastic orders defined above are related to each other, have the following implications (see, Shaked and Shanthikumar (2007)):

Mean residual life function
Given that a component survives up to time t ≥ 0, the residual life is the period beyond t until the time of failure and defined by the conditional random variable Y − t|Y > t. The mean residual life (MRL) function is an important function in survival analysis, actuarial science, economics and other social sciences and reliability for characterizing life time. Although the shape of the failure rate function plays an important role in repair and replacement strategies, the MRL function is more relevant as the latter summarize the entire residual life function, where the former considers only the risk of instantaneous failure. In reliability, it is well known that the MRL function and ratio of consecutive moments of residual life determine the distribution uniquely (Gupta and Gupta (1983)).

A Generalization Of Inverse Marshall-Olkin Family Of Distributions
The r th order moment of the residual life time of the distribution is given by the general formula: Hence the MRL function of the GIW distribution is given by Hence the MRRL function of the GIW distribution is given by where Γ(c, x) = ∫ y c−1 e −y dy ∞ x , c > 0.

Stress-strength parameter
Suppose that the random variable Y is the strength of a component, which is subjected to a random stress Z. The component fails whenever Y < Z and there is no hazard when Y > Z . In the context of reliability, the stress-strength parameter R = P(Y > Z) is a measure of component reliability and its estimation when Y and Z are independent and follow a specified distribution has been discussed widely in the literature. Here, we estimate R = P(Y > Z) for the GIW distribution.

Case I : ≠
Assume that Y~GIW(α, θ 1 , λ, β) and Z~GIW(α, θ 2 , λ, β) are independent. The pdf of Y and the cdf of Z can be expressed respectively as: We have Let us assume that y 1 , y 2 , … , y n and z 1 , z 2 , … , z m are independent observations from Y and Z respectively. The total log-likelihood function L R (Θ * ) where Θ * = (α, θ 1 , θ 2 , λ, β) T , becomes By taking the first partial derivatives of the total log-likelihood function with respect to five parameters in Θ * , we obtain the five normal equations. The maximum likelihood estimates Θ * of Θ * is obtained by solving the system of non linear normal equations numerically. From the solution of these equations, we can estimate R by inserting the estimate of Θ * in equation (33).

Estimation of the parameters 6.1 Maximum likelihood estimation
Several approaches for parameter estimation have been proposed in the literature, but maximum likelihood method is the most commonly employed. We consider estimation of the unknown parameters of GIW distribution by the method of maximum likelihood. Let y 1 , y 2 , … , y n be observed values from the GIW distribution with parameters α, θ, λ and β. The log-likelihood function for (α, θ, λ, β) is given by The derivatives of the log-likelihood function with respect to the parameters α, θ, λ and β are given by respectively, The maximum likelihood estimates of ( α, θ, λ, β ), say (  Maximization of the likelihood function can be performed by using nlm or optim in R statistical package. Now, we study the existence and uniqueness of the maximum likelihood estimates, when the other parameters are known (given).
The normal approximation of the maximum likelihood estimates of the parameters can be adopted for constructing approximate confidence intervals and for testing hypotheses on the parameters ( , , , ). Under conditions that are fulfilled for the parameters in the interior of the parameter space and applying the usual large sample approximation, it can be shown that √n(Θ − Θ) can be approximated by a multivariate normal distribution with zero means and variance-covariance matrix K −1 (Θ) , where K(Θ) is the unit expected information matrix.
As n tends to infinity, we have the asymptotic result where (Θ) is the observed Fisher information matrix. Since K(Θ) involves the unknown parameters of Θ, we may replace it with the MLE Θ. Thus, the average matrix estimated at Θ, say 1 (Θ), can be used to estimate K(Θ). The estimated multivariate normal distribution can thus be used to construct approximate confidence intervals for the unknown parameters and for the hazard rate and survival function.

Method of product spacing
This method was introduced by Cheng and Amin (1983) as an alternative to method of maximum likelihood estimation. The cdf of the GIW distribution is given by equation (18), and the uniform spacing are defined as follows: and the general term of spacing is given by such that ∑ D i = 1. Method of product spacing method choose the estimates which maximizes the product of spacings or in other words, maximizes the geometric mean of the spacing. That is, Taking the logarithm of G, we get, , where S = logG.
We can rewrite S as Differentiating the above equation partially, with respect to the parameters α, θ, λ and β respectively and then equating them to zero, we get the normal equations.
Since the normal equations are non-linear, we can use iterative method to obtain the solution.

Method of least squares
Let y 1 < y 2 < ⋯ < y n be the n ordered random sample of any distribution with cdf G(y), we get E[G(y i )] = i n + 1 .
The least square estimates are obtained by minimizing (41) Putting the cdf of GIW distribution in equation (41), we get (42) In order to minimize equation (42), we have to differentiate it with respect to α, θ, λ and β, which gives the following equations: where A = 1 − (1 − α) −( ) . The above normal equations cannot be solved analytically. So we can use nlm or optim in R statistical package to obtain the solution.

Simulation
We asses the performance of the maximum likelihood estimates of GIW (α, θ, λ, β) distribution by conducting simulation for different sample sizes and parameter values. We use equation (29) to generate random samples from the GIW distribution with parameters α, θ, λ and β. The different sample sizes considered in the simulation are n = 30,70,100 and 200. We have used nlm package in R software to find the estimate. We have repeated the process 1000 times and report the average estimates and associated mean square errors in Table  3. From Table 3, we can see that as the sample size increase, the estimated values are close to the actual values and the mean square errors decreases, which establishes the consistency property of the MLEs.

Autoregressive GIW minification process
We develop a first order autoregressive (AR(1)) minification process with GIW distribution as marginal distribution.
Hence the process {Y n } is stationary with GIW marginals.

Application to real data
In this section, we analyze one data set to demonstrate how the GIW distribution can be a good life time model in comparison with many known distributions. We consider the data set originally reported by Bjerkedal(1960). This data set consists of 72 observations of survival times guinea pigs injected with different doses of tubercle bacilli. The data set has been considered by several authors in the literature, see, Kundu and Howlader (2010) and Cordeiro et al. (2012). The data set follows: The descriptive statistics of the data is presented in Table 4.  (v) Modify 2 into * = 2 (1 + 0.5/ ) and 2 into * = 2 (1 + 0.75/n + 2.25/n 2 ). For further details, see Chen and Balakrishnan (1995).
The values of estimates, −log L, AIC, CAIC, BIC, HQCI are listed in Table 5 and W * , A * , K − S, p-values for all models are listed in Table 6.  From the Table 5 and Table 6, we can see that, GIW distribution has smallest −log L, AIC, CAIC, BIC, HQCI, W * , A * , K − S values. Also the GIW distribution has highest p-value. Hence the new model, that is GIW distribution, yields a better fit than the other models for this data set.
The fitted density and the empirical cdf plot of the GIW distribution are presented in Figure 3. The figure indicates a satisfactory fit for the GIW distribution. To test the null hypothesis H 0 : IMOW versus H 1 : GIW or equivalently H 0 : θ = 1 versus H 1 : θ ≠ 1 we use likelihood ratio test statistic whose value is 4.4706(p-value=0.0345). As a result, the null model IMOW is rejected in favor of alternative model GIW at any level > 0.0345.