Marshall-Olkin Discrete Uniform Distribution

We introduce and characterize a new family of distributions, Marshall-Olkin discrete uniform distribution. The natures of hazard rate, entropy, and distribution of minimum of sequence of i.i.d. random variables are derived. First order autoregressive (AR (1)) model with this distribution formarginals is considered.Themaximum likelihood estimates for the parameters are found out. Also, the goodness of the distribution is tested with real data.


Introduction
Marshall and Olkin [1] introduced a new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Jose and Alice [2] discussed Marshall-Olkin family of distributions and their applications in time series modeling and reliability. Jose and Krishna [3] have developed Marshall-Olkin extended uniform distribution. These works and most of the references there in, deal with continuous distribution. Not much work is seen in the discrete case. The reason behind this may be that it is difficult to obtain compact mathematical expressions for moments, reliability, and estimation in the discrete set up.
If ( ) is the survival function of a distribution, then, by Marshall-Olkin method, we get another survival function ( ), by adding a new parameter to it. That is, where ( ) is the p.m.f. corresponding to ( ). The hazard rate of is We consider a new family of distributions by adding an additional parameter by using the Marshall-Olkin scheme (Marshall and Olkin [1] in Section 2). The nature of hazard rate, entropy and expectation are derived. In the third section, distributions of minimum sequence of i.i.d. random variables are found out. An AR (1) model with new Marshall-Olkin discrete uniform distribution is discussed in the next section. In the fifth section, the maximum likelihood estimate (m.l.e.) for the parameters is found out and, in the last section, the goodness of the distribution is tested with a real life data.

Stability Property of the New Family.
If we apply the same method again into the new family, by adding a new parameter " " ( > 0) to the reliability function, we get , which is the reliability function of the new Marshall-Olkin family with parameters ( , ). (2), consider the probability mass function p.m.f.

Probability Mass Function. From
where ( ) is the p.m.f. corresponding to ( ). That is, that is, We know the p.
That is, = 1. But, when = 1, it corresponds to discrete uniform distribution on {1, 2}. The converse is straight forward.  , since its support is on {1, 2, 3, . . . , }, a finite range. It is not log convex, since the class of log convex distribution forms a subclass of the class of i.d. distributions.
This happens only when ≥ 1. Now assume that ( , ) ≤ ( ). That is, Hence, the proof is completed.

Expectation, Standard Deviation, and Entropy of MODU Distribution.
We numerically compute the expectation and standard deviation (Tables 1 and 2) and entropy (Table 3) of the MODU distribution with different and , since compact Table 2: Expectation and standard deviation with different " " and " " ( > 1) for ∼ MODU( , ).
Remark 6. If ∼ MODU( , ), then the sd( ) is decreasing with increasing value of when > 1 and the sd( ) increasing with increasing value of when < 1.
From Remark 1 and Tables 1 and 2, we have the following.   Let ∼ MODU( , ); then the mean, median, and mode of the distribution (in Figure 1 and Tables 1 and 2) for different and are computed in Table 4.
From Figure 1 and from Table 4, the following is clear.

Remark 9.
The MODU distribution is positively skewed, when < 1, since mode < median < mean and the distribution is negatively skewed, when > 1, since mode > median > mean. Also, it is to be noted that the distribution is  Journal of Probability unimodal; that is, when < 1, the mode = 1 and, when > 1, the mode = . Therefore, the MODU distribution can be applied for the data showing positive skewness when > 1 and the data showing negative skewness when < 1.  We illustrate these results graphically ( Figure 3) and numerically (Table 5).

MODU Distribution as the Distribution of Minimum of a Sequence of i.i.d. Random Variables
The following theorem gives a characterization of minimum of a sequence of i.i.d. random variables following MODU distribution. Proof. Proof follows as in the same lines as given in Satheesh et al. [4][5][6] for a similar characterization of minimum of sequence of i.i.d. random variables.
The survival function of which is the survival function of MODU( , ). By retracing the steps we can easily show the converse.

AR (1) Model with MODU Distribution as Innovating Distribution
Arnold and Robertson [7], Chernick [8], and Satheesh et al. [9] studied some properties of the AR (1) models. As Satheesh et al. [9] discussed, construct a first order autoregressive minification process with the following structure: Under stationary equilibrium, this implies If we take ( ) = ( − )/ , then, , which is the survival function of MODU( , ). We can also show that ( ) follows uniform distribution with parameter . By retracing the steps we can easily show the converse.

Maximum Likelihood Estimates (m.l.e.) for the Parameters of MODU( , )
Let 1 , 2 , . . . , be independent random samples from MODU( , ). Then, from the likelihood function of the distribution, we can write We calculated the m.l.e. of and numerically (using Mathematica) as the solution of these two nonlinear equations. But, here, the maximum of the range of observation is . So m.l.e. of the parameter (̂) is the largest value of the observation. Then, from (23), we can find m.l.e. of .
Algorithm 13. Consider the following.
(1) Through simulation, 2500 random samples were generated from inverse function of distribution function ( ) for some given value of the parameters and .
(3) For accuracy, we repeat the same calculation ten times with same values of and .
(4) The mean and the standard error (SE) of these estimates are calculated.     (Table 7). It is clear that the SE of m.l.e. of the parameters " " and is decreasing with increase in sample size.

An Application of MODU
Example 14. The data was collected from the daily attendance register of the science and commerce (nonlanguage) supplementary class (20 working days in a month) of a higher secondary school in Palakkad district, Kerala. We took a sample of 50 students (out of 360) and let be the number of days these students attended in the class for the whole year 2012-2013 (an academic year is 10 months) (see Tables 8, 9, and 10).
We arbitrarily fix 4 months, and the m.l.e.s are computed. Initially we fit discrete uniform to the data (see Tables 11 and  12).
Since mode > median > mean, the data exhibits negative skewness with m.l.e. of > 1 and it is observed that the distribution is unimodal; hence, MODU distribution is supposed to give a better fit than uniform distribution (see Table 13).
Result. From the values, it is seen that MODU distribution is better fit than uniform distribution.
Example 15. The data was collected from the daily attendance register of the supplementary language class (20 working days in a month) from the same higher secondary school in Palakkad district, Kerala. We took a sample of 50 students and let be the number of days these students attended in the same for the whole year 2012-2013 (an academic year is 10 months) (see Tables 14, 15, and 16).
We arbitrarily fix 3 months, and the m.l.e.s are computed. Fit discrete uniform to the above data. We have the results as shown in Tables 17 and 18.    Since mode < median < mean, the data exhibits positive skewness with m.l.e. of < 1 and it is observed that the distribution is unimodal; hence, MODU distribution is supposed to give a better fit than uniform distribution (see Table 19).

Journal of Probability
Result. From the values it is seen that, here, MODU distribution is a better fit than uniform distribution.