A Discrete Analogue of the Continuous Marshall-Olkin Weibull Distribution with Application to Count Data

In this paper, we introduced the discrete analogue of the continuous Marshall-Olkin Weibull distribution using the discrete concentration approach. Some mathematical properties of the proposed discrete distribution such as the probability mass function, cumulative distribution function, survival function, hazard rate function, second rate of failure, probability generating function, quantile function and moments are derived. The method of maximum likelihood estimation is employed to estimate the unknown parameters of the proposed distribution. The applicability of the proposed discrete distribution was examined using an over-dispersed and under-dispersed data sets.


Introduction
The theory of discretization generally arises when it becomes seemingly difficult to measure the life length of a product or device on a continuous scale. Situation of such arises when the observed lifetimes are preferably recorded on a discrete scale than on a continuous analogue. For examples the number of times devices are switched on/off, the number of days a patient stays in an observation ward, and the number of weeks/months/years a cancer patient survives after treatment etc. Although, classical discrete distributions such as the Poisson, Geometric, Binomial and Negative Binomial distributions have been developed to handle such situation, there is still need to introduce more flexible discrete distributions especially those arising from the discretization of continuous distributions to handle more sophisticated real-life phenomena.

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Several methods of discretizing a continuous distribution have been widely considered in literature. [3] gave a comprehensive survey on the different methods of generating discrete probability distributions as analogues of continuous distributions which include; the discretization method based on the survival function, the discretization method based on probability mass function (infinite series), the discretization method based on the cumulative distribution function, the discretization method based on the hazard function, discretization method based on reverse hazard function, the difference equation analogues of Pearsonian differential equation and the two-stage composite method etc. The method of discretization by survival function also referred to as the discrete concentration approach was proposed by [13].
Let X be a random variable associated to a continuous probability distribution with survival function ( ), X S x [9] defined the probability mass function (pmf) of a discrete where [ ] X indicates the smallest integer part or equal to X, as: Using this method of discretization, [17] introduced the discrete Rayleigh distribution, [10] examined the discrete half-normal distribution, [11] studied the discrete Burr distribution, [8] introduced the discrete inverse Weibull distribution, [5] proposed the discrete Lindley distribution, [15] developed the discrete Type II generalized exponential distribution, [7] discretized the inverse Rayleigh distribution, [16] introduced the discrete analogue of the continuous Power Lindley distribution, and [2] recently developed the discrete Marshall-Olkin generalized exponential distribution, among others.
In this paper, using the same method of discretization defined in equation (1), we introduced the discrete analogue of the continuous Marshall-Olkin Weibull (DMOW) distribution. The rest Sections of this paper are as follows: Section 2 proposes the new discrete Marshall-Olkin Weibull (DMOW) distribution and presents some of its mathematical properties, the estimation of the unknown parameters of the proposed distribution is presented in Section 3, while Section 4 presents the application of the proposed distribution to two count data sets. Section 5 concludes the paper.  [12] introduced the Marshall-Olkin family of distributions obtained by adding a scale parameter ( ), α which they called the "tilt parameter" to an existing distribution. They derived the Marshall-Olkin Weibull distribution with the survival function given by

< α <
Using the series representation in [4], (2) can be rewritten as The graphical representation of the probability mass function of the discrete Marshall-Olkin Weibull (DMOW) distribution for varying parameters of t is shown in Figure 1.

Figure 1. Probability
The plots in Figure 1 reveal that the pmf of the DMOW distribution can be decreasing, left-skewed, right in handling any nature of real data sets.

Distribution and survival functions
The cumulative distribution function of the DMOW distribution is defined by the corresponding survival function of the DMOW distribution is obtained as The graphical representation of the probability mass function of the discrete Olkin Weibull (DMOW) distribution for varying parameters of t

Probability mass function (pmf) of the DMOW distribution
The plots in Figure 1 reveal that the pmf of the DMOW distribution can be skewed, right-skewed unimodal and symmetric which proves in handling any nature of real data sets.

survival functions
The cumulative distribution function of the DMOW distribution is defined by the corresponding survival function of the DMOW distribution is obtained as The graphical representation of the probability mass function of the discrete Olkin Weibull (DMOW) distribution for varying parameters of the distribution distribution.
The plots in Figure 1 reveal that the pmf of the DMOW distribution can be skewed unimodal and symmetric which proves flexibility The cumulative distribution function of the DMOW distribution is defined by (6) the corresponding survival function of the DMOW distribution is obtained as , 0, , 0, 0 1.
A Discrete Analogue of the Continuous Marshall Earthline J.

Hazard rate, reversed hazard rate
The hazard rate, reversed hazard rate and the second rate of failure of the Marshall-Olkin Weibull distribution are respectively defined by  The hazard rate, reversed hazard rate and the second rate of failure of the Olkin Weibull distribution are respectively defined by The hazard rate, reversed hazard rate and the second rate of failure of the discrete cumulative distribution function Olkin Weibull distribution for distribution.

Figure 3
Clearly, Figure 3 shows that the hazard rate of the DMOW distribution exhibits a decreasing, increasing, bathtub

Probability generating function, moments and quantile functio
Let X be a discrete random variable defined in the non The probability generating function Using equation (11) Using equation (11), the probability generating function of the DMOW distribution is Using equation (11), the probability generating function of the DMOW distribution is The mean of the discrete Marshall-Olkin Weibull distribution can be obtain by taking the first derivative of equation (12) and setting 1 S = , yielding ( )   The recurrence relation for generating probabilities of the DMOW distribution is given by where, The pth quantile of the DMOW distribution denoted by ( ) X Q p is given by The median of the DMOW distribution is obtained from equation (14) by substituting

Parameter Estimation
is a random sample of size n from the discrete Marshall-Olkin Weibull distribution with probability mass function defined in equation (3), then the log-likelihood function is given by

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The maximum likelihood estimates say ( )ˆ, , , φ = α β γ can be obtained by solving the system of non-linear equation 0 ∂ = ∂φ ℓ . Hence, taking the first derivative of equation (16) with respect to the parameter vector φ , we have ( 1) ( 1) Since there is no close form expression for the MLE of the discrete Marshall-Olkin Weibull distribution, it becomes difficult to obtain analytical solutions for the unknown parameter estimates, thus a standard numeric optimization algorithm such as the Newton-Raphson Iterative Scheme is employed to optimize the log-likelihood function. The fitdistrplus package in R statistical software is used to evaluate the maximum likelihood estimates of the DMOW distribution.

Applications
In this section, we attempt to illustrate the flexibility of the proposed DMOW distribution using two real data sets. The first data set records the number of strikes in UK coal mining industries in four successive week periods during 1948-1959 reported in [7]. The data set is under-dispersed with µ = 0.99 and 2 σ = 0.74. The second data set is a biological data set which records the number of Hemocytometer yeast cell counts per square reported in [1]. The data set is over-dispersed with µ = 0.68 and 2 σ = 0.81.
The goodness of fit of the DMOW distribution is compared with the following existing discrete distributions:

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(1) Discrete Weibull (DW) distribution ( [13]) with pmf given by ( 1) , (2) Exponentiated Discrete Weibull (EDW) distribution ( [14]) with pmf given by ( 1) 1 1 , ( 1)    Tables 2 and 3 present the parameter estimates, log-likelihood, observed and expected frequencies, and 2 χ statistics with the corresponding p-value of each distribution for the two data sets. The better distribution corresponds to the one having the maximized log-likelihood and p-value and the least 2 χ statistics value. Thus, the results obtained from these Tables suggest that the DMOW distribution performs reasonably better than the compared discrete distributions for the two data sets.

Conclusion
A discrete analogue of the continuous Marshall-Olkin Weibull distribution has been introduced using the discrete concentration approach. Some mathematical properties of the proposed DMOW distribution were derived and it observed that the probability mass function of the distribution accommodates a decreasing, left-skewed, right-skewed unimodal and symmetric shapes while the hazard rate function exhibits a decreasing, an increasing, bathtub and inverted bathtub shapes properties. The method of maximum likelihood estimation was employed to estimate the unknown parameters of the proposed distribution and finally, two count data sets (over-dispersed and under-dispersed) was used to examine the applicability of the DMOW distribution.