Characterizations of Some Probability Distributions with Completely Monotonic Density Functions

For a non-negative continuous random variable X, Chaudhry and Zubair (2002) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed using the proposed characterization techniques.

The objective of this paper is to discuss various basic distributional properties of a probability distribution with a completely monotonic probability density function introduced by Chaudhry and Zubair (2002), and based on these properties, to establish its some new characterization results by truncated moments, order statistics and record values. Chaudhry and Zubair (1994) introduced the following generalized gamma function: In view of Gradshteyn and Rhyzhik (1980), (1.2) is expressed in terms of the Macdonald function (or the modified Bessel function of the second kind) by Further, Chaudhry and Zubair (2002) showed that ∫ ( ) which is given in Figure 1. The effects of the parameters can easily be observed from Figure 1 that the distribution of the random variable has a pdf which is decreasing and is positively right skewed with longer and heavier right tails.

Distributional properties of Chaudhry and Zubair's Macdonald probability distribution
In this section, for the sake of completeness, we independently derive various basic distributional properties, viz., expressions for the cumulative distribution function (cdf), the survival and hazard functions, and the expressions for the ℎ moment and the ℎ incomplete moment. We also sketch the corresponding graphs of the cdf and the hazard function.

Cumulative distribution function:
The cumulative distribution function (cdf) corresponding to the pdf (1.4) is given by which easily follows by evaluating the above integral in terms of the generalized gamma function, ( ), that is, Chaudhry and Zubair (2002), and Gradshteyn and Rhyzhik (1980). Please note that since , which is the pdf (1.4) under question.

Survival and Hazard functions:
Using (1.4) and (2.1), we compute the corresponding survival function (sf) and hazard function (hf) which are respectively given by For some selected values of the parameters, we have sketched the graphs of the cdf (2.1) and the hf (2.3), which are respectively given in Figures 2 and 3. The effects of the parameters can easily be observed from Figure 3 that the hazard function of distribution of the random variable is a decreasing function and has a bathtub shape with longer and heavier right tails.  , . , , .

Moments:
In what follows, we give the expressions for the ℎ moment and the ℎ incomplete moment.

First Moment:
When = 1 in Eq. (2.5), the 1stmoment is given by

Percentiles
The percentage points of a given distribution are also important for any statistical applications, for example, we may be interested in knowing the median (50%), or 75% quartiles, or 95%, or 99% confidence levels, to assess the statistical significance of an observation whose distribution is known. For any 0 < < 1, the 100 ℎ percentile or the quantile of order of a distribution with the pdf ( ) is defined as a number such that the area under ( ) to the left of is , that is, is any solution of the equation

Characterization Results
The problems of characterizations of probability distributions have been investigated by many authors and researchers. As pointed out by Nagaraja (2006), "A characterization is a certain distributional or statistical property of a statistic or statistics that uniquely determines the associated stochastic model". Similarly, according to Koudou and Ley (2014), "In probability and statistics, a characterization theorem occurs when a given distribution is the only one which satisfies a certain property". Furthermore, Koudou and Ley (2014) points out that "characterization theorems also deepen our understanding of the distributions under investigation and sometimes open unexpected paths to innovations which might have been uncovered otherwise".
In order to apply a particular probability distribution to some real world data, many authors and researchers recommend characterizing it first subject to certain conditions. See, for example, Galambos and Kotz (1978)

Characterization by Truncated Moment:
In this subsection, we provide two new characterization results of the Chaudhry and Zubair's Macdonald probability distribution by truncated moment. The first characterization result (Theorem 3.1) is based on a relation between left truncated moment and failure rate function. The second characterization result (Theorem 3.2) is based on a relation between right truncated moment and reversed failure rate function. For this, we will need the following assumption and lemmas. Assumption 3.1.
Suppose the random variable is absolutely continuous with the cumulative distribution function ( ) and the probability density function ( ). We assume that = { | ( ) > 0}, and = { | ( ) < 1}. We also assume that ( ) is a differentiable for all , and ( ) exists.    , where 1 ( ) is given by (2.9). Consequently, the proof of "if" part of the Theorem 3.1 follows from Lemma 3.1.
Conversely, suppose that , (0 ≤ < ∞, > 0), which is the required pdf of the random variable . This completes the proof of Theorem 3.2.

Characterizations by Order Statistics:
If 1 , 2 , . . . , be the independent copies of the random variable with absolutely continuous distribution function ( ) and pdf ( ), and if 1, ≤ 2, ≤ . . . ≤ , be the corresponding order statistics, then it is known from Ahsanullah et al.

Characterization by Upper Record Values:
For details on record values, see Ahsanullah (1995). Let 1 , 2 , . .. be a sequence of independent and identically distributed absolutely continuous random variables with distribution function ( ) and pdf ( ).

Remark 2: Pareto Distribution:
A continuous random variable is said to have the Pareto distribution, if its pdf ( ) is given by ( ) = + 1 , where ≥ 1, > 1. For details on Pareto distribution, see Johnson et al. (1994). It is easily seen that, for the Pareto distribution's pdf, we have (−1) ( ) ( ) ≥ 0, and hence it is completely monotonic for all ≥ 1. For the characterizations of the Pareto distribution by truncated moment, the interested readers are referred to Ahsanullah, et al. (2016), and by upper records, please refer to Ahsanullah and Shakil (2012).

Remark 3: Exponential Distribution:
A random variable is said to have the exponential distribution if its pdf is given by ( ) = ( − ), where > 0, > 0 and > 0, It is easy to see that , for the exponential distribution's pdf, we have (−1) ( ) ( ) ≥ 0, and hence it is completely monotonic for all > 0. For the characterizations of the exponential distribution by truncated moment, please refer to , where the characterizations of the Boltzmann distribution by truncated moment have been discussed since the Boltzmann distribution and the exponential distribution coincide by simple transformation of the parameters.

Concluding Remarks
In this paper, we have considered, for a non-negative continuous random variable , a probability distribution with a completely monotonic probability density function introduced by Chaudhry and Zubair (2002), called the Macdonald probability density function.. We have established some new characterization results of Chaudhry and Zubair's Macdonald probability distribution by truncated moments, order statistics and record values. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different sets of values of the parameters. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed. We hope the findings of the paper will be quite useful for the practitioners in various fields of sciences.