The Uniform Truncated Negative Binomial Distribution and its Properties

Corresponding Author: Mohamed G. Sief Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt Email: mgr00@fayoum.edu.eg Abstract: A new three parameters continuous distribution which is named the Uniform Truncated Negative Binomial (UTNB) distribution is introduced. It has the uniform and the Marshall-Olkin extended uniform distributions as sub-models. Some reliability and statistical properties of the new distribution are derived, including the shape behavior of the density and hazard rate functions, the mean residual life, moments and moment generating function, quantiles and related measures. The limiting distributions of the sample extremes, stochastic orderings, entropies and stress-strength reliability are derived. Maximum likelihood estimation is performed. Application to censored data illustrate the potentiality of the proposed distribution is provided.


Introduction
The basic distributions (exponential, Weibull, gamma, log normal, Lomax and others) used widely in reliability and survival analysis, are described by Cox and Oakes (1984), have a limited range of behavior and cannot represent all the situations found in applications. For example, the exponential has a restricted hazard function, being constant. The Weibull is often described as flexible, but its hazard function is in fact restricted to being monotonically increasing or monotonically decreasing, or constant. The limitations of the standard distributions arouse the interest in developing extended or generalized distributions by adding further parameters. Marshall and Olkin (1997) introduced a new technique by adding a parameter to an existing distribution F to generate a family of distributions. The extended family taking the form: In the literature, many new parametric extensions of various univariate distributions were proposed using this method such as: Marshall-Olkin extended Lomax, Ghitany et al. (2007), Marshall-Olkin extended uniform (MOEU), Jose and Krishna (2011), Marshall-Olkin Pareto, Ghitany (2006), Marshall-Olkin logistic, Alice and Jose (2005a), Marshall-Olkin semi-Weibull, Alice and Jose (2005b), Marshall-Olkin q-Weibull, Jose et al. (2010) and Marshall-Olkin Weibull, Ghitany et al. (2005).
Moreover, Jayakumar and Mathew (2008) introduced the exponentiated Marshall-Olkin scheme which has Marshall-Olkin family as a special case given by: where, , 0, x γ α > ∈ R . Also, a new family of distributions was proposed by Nadarajah et al. (2013) by adding an extra parameter to the Marshall-Olkin family of distributions whose survival function is given by: The family (3) is a generalization of the Marshall-Olkin family of distributions (1). Namely, if β = 1, then the introduced family reduces to the Marshall-Olkin family of distributions. Moreover, Nadarajah et al. (2013) introduced three-parameter generalization of the exponential distribution as a particular case of the new family. Also, Jose and Sivadas (2015) used the family (3) to introduce the negative binomial Marshall-Olkin Rayleigh distribution.
The uniform distribution is regarded as the simplest probability model and is related to all distributions by the fact that the cumulative distribution function, taken as a random variable, follows uniform distribution over (0,1). This result is basic to the inverse method of the random variable generation. Some researchers utilized uniform distribution in connection with sample quasiranges. The distribution is also applied to determine power functions of tests of randomness. It is also applied in a power comparison of tests of non-random clustering. There are also numerous applications in nonparametric inference, such as Kolmogorov-Smirnov test for goodness of fit. The uses of uniform distribution to represent the distribution of round off errors and in connection with the probability integral transformations, are also well-known.
We proposed a new version of the family (3) by selecting the survival function of the uniform (0,θ) in where, 0<x<θ, 1 ≠ α, β >0. We refer to the distribution with the survival function (4), as the uniform truncated negative binomial distribution with parameters θ,α,β and will be denoted by UTNB (θ,α,β).
In this article, we introduce a new distribution which is named the Uniform Truncated Negative Binomial (UTNB) distribution. Some statistical characteristics of the proposed distribution are obtained and the maximum likelihood estimation of its parameters is discussed. The article is outlined as following: In section 2, the probability density function (pdf) and its behavior are studied. Section 3 discusses the hazard rate function and obtains the mean residual life function (mrlf). The moment generating function is expressed as an infinite sum and an explicit form for the rth moment is obtained in section 4. In section 5, we give an expression for the quantile function and also some related measures. Section 6 provides the limiting distributions of the sample extreme. The stochastic orderings are derived in section 7. In section 8, the stress-strength reliability R = P(Y<X) of a system is discussed when both the strength X of the system and the stress Y, which act on it, are independent, nonidentical UTNB distributed random variables. Rényi and Shannon entropies are determined in section 9. Section 10 discusses the maximum likelihood estimation of the model's parameters. In section 11, the proposed model has been fitted to randomly censored data to motivate its usefulness. Finally, the concluding remarks are addressed in section 12.

The Shapes of the Density Function
The pdf of the UTNB distribution with is : where, α, β>0, 0<x<θ.

Remark 1
The UTNB distribution having the uniform and MOEU distributions as sub-models as: • If α→1, then the UTNB (θ,α,β)→U (0, θ) • If β→1, we get the MOEU distribution The following proposition gives simple sufficient condition for the shape behavior of the pdf (5)

The Hazard rate and Mean Residual Life Functions
The hazard rate function (hrf) of the UTNB distribution is given by: where, 1≠ α,β > 0,0<x<θ.
The following proposition gives simple condition under which the hrf (6) is increasing or bathtub.

Proof
By taking the first derivative of h(x) we get: is increasing for all 0<x<θ.
If φ(0) <0, thus the function φ(x) has one root say, and it is negative for x<x o and positive for x>x o . This implies that the function h'(x) is negative for x<x o and positive for x>x o . Thus, we conclude that h(x) is bathtub shaped. Figure 2 shows the graph of the hrf for the UTNB distribution for selected values of the parameters α,β,θ. An important ageing property for the UTNB distribution is the mean residual life function (mrlf), which was introduced by Watson and Wells (1961) to analyse burn-in problems. It has been studied by reliabilists, statisticians, survival analysts and others. It is defined simply as the expected value of the remaining lifetime beyond an age t. For X∼ UTNB(α,β,θ), the residual life random variable at age t, is denoted by: The mrlf is defined formally as:

Moment Generating Function and Moments
In this section, we present an infinite sum for the moment generating function (mgf) of the UTNB distribution. Also, we derive an explicit form of the rth moment of the UTNB distribution. In the sequel, we use the substitution The mgf is defined as: The rth moment for the UTNB random variable X is given by:

Quantiles, Kurtosis, Skewness
The qth quantile of the UTNB distribution is given by: where, 0≤q≤1 and G −1 (⋅) is the inverse function of the distribution function. As special cases, we have: • The qth quantile of the uniform distribution (α→1) is given as x q = q θ • The qth quantile of the MOEU distribution (β = 1) is given by In particular, the median of UTNB distribution is given by: Also, we can get the median of the uniform and that of the MOEU as special cases respectively as med 2 A new quantile measures for the dispersion of a distribution around the values µ ± σ will be presented.

Limiting Distribution of Sample Extremes
Let X 1 , X 2 , …, X n be a random sample of size n from the UTNB distribution, then the sample minima and the sample maxima which are a long stand area in application of probability and statistics, are respectively: Using the asymptotic result of X 1:n and X n:n , Arnold et al. (1992), as following: • For the minimal order statistic X 1:n , we have: if and only if: • For the maximal order statistic X n:n , we have: (of the Weibull type) where a n = F −1 (1) and: if and only if: • For all t>0, k>0, we have: In the following proposition we derived the limiting distributions of X 1:n and X n:n from the UTNB distribution.

Proposition 3
Let X 1:n and X n:n be the smallest and the largest order statistic from the UTNB distribution, then we have:
If α → 1, i.e., the uniform distribution, the norming constants will be * 1 , Let the limiting distribution of ( ) * * 1: / n n n X a b − and (X n:n -a n )/b n are, respectively, denoted as G * (t) and G(t). Then, it is well-known from Arnold et al. (1992) that for any finite i>1, the limiting distributions of the random variables (X i:n -a n )/b n and (X n-i+1:n -a n )/b n are, respectively, given by:  (3) we conclude that, for any finite i>1, the limiting distribution of the ith and the (n-i +1) th order statistic from the UTNB distribution, respectively, are given by: where, * , n n b a and b n are given in proposition (3) and Z is discrete random variable follows Poisson distribution with mean t.

Stochastic Orderings
A very important tool for judging the comparative behavior is the stochastic ordering. If X and Y are two continuous random variables, then we can say that X is smaller than Y in the: for all t≥0

decreasing, for all t≥0
According to the well-known result of Shaked and Shanthikumar (1994) for establishing stochastic ordering of distribution we have: In the following proposition, we show that the UTNB distribution is ordered with respect to the strongest "likelihood ratio" ordering.

Proof
It is sufficient to show that (X≤ lr decreasing, for all t≥0 in this cases. Then we get the required results.

Stress-Strength Reliability
If a system subject to a stress, the reliability of the system is defined as the probability that it accomplishes its required function adequately without failure. For stress-strength models both the strength of the system, X and the stress, Y, imposed on it by its operating environments are considered to be random variables. The reliability, R, of the system can be stated as, R = P(Y<X). Now, Let X and Y are two independent but not identically distributed UTNB random variables represent the strength of a system and the stress acting on it respectively. Suppose that X∼UTNB(α,β 1 ,θ) and Y∼UTNB(α,β 2 ,θ). That is the survival functions of X and Y are respectively given by: Then the reliability of the system will be:

Entropies
The entropy of a random variable X is a measure of uncertainty or randomness of dynamical system modeled by this random variable. We derive two popular entropy measures that are Rényi entropy, Song (2001) and Shannon's entropy, Shannon (1951) of the UTNB random variable. The Rényi entropy is defined as: where, γ>0 and γ ≠ 1. If X∼ UTNB(α,β,θ) then we have: , then we have: (1 ) 1 (1 ) 1 1 Therefore, Rényi entropy is given by: Now we consider the Shannon entropy which is defined as E [-log(g(X))]. It is the special case of the Rényi entropy for γ↑1. For the UTNB random variable we have: First, we derive the expectation: Therefore, we get:

Maximum Likelihood Estimation
Suppose that X 1 , X 2 , …, X n be the failure time of individual i and let Y 1 , Y 2 , …, Y n be the corresponding censoring time. Assuming that X i 's and Y i 's are independent. If X i 's ∼UTNB and Y i are assumed to have a non-informative distribution. One observes the pairs (T i ,∆ i ) where T i = min(X i ,Y i ) and ∆ I = I(X i ≤Y i ) is the censoring indicator. Given the data (t 1 ,δ 1 ), (t 2 ,δ 2 ), …, (t n ,δ n ) then the likelihood function is given by: and the log-likelihood function is then given by: Taking the first partial derivative of the loglikelihood function with respect to α,β,θ we have:

Application to Censored Data
In this section, we fit the proposed UTNB model to randomly censored data representing the serum reversal times (in years) of a random sample of 148 children contaminated with HIV from vertical transmission at the university hospital of Ribeirão Preto school of medicine from 1986 to 2001, Silva (2004) The data set contains 84 right censored observations which constitute 56.7% of the sample size (heavy censoring). We have fitted the uniform and the MOEU distributions to the data set using MLE and then compared it with the UTNB distribution. In addition, for model selection, we use the Akiake Information Criterion (AIC), the Bayesian Information Criterion (BIC), the Consistent Akiake Information Criterion (CAIC) and the Hannan-Quinn Information Criterion (HQIC) defined as:  where, k is the number of parameters in the model and n is the sample size. The model with lowest AIC or (BIC, CAIC and HQIC) value is the one that better fits the data. ). This result shows that UTNB is a great improvement over the uniform distribution for this censored data set. Let n be the total number of breaking stress of carbon fibers whose survival times, uncensored data, are available. Retable the n survival times in order of increasing magnitude such that t 1 ≤t 2 ≤…≤t n . The Kaplan and Meier (1958), estimator (KME), also known as the product limit estimator, gave a nonparametric estimator of the survival function of the underlying as follows:

Conclusion
For the first time, we propose a three-parameter continuous distribution which generalizes the uniform and the Marshall-Olkin extended uniform distributions. We refer to the new model as the Uniform Truncated Negative Binomial (UTNB) distribution. Provide several statistical properties for the proposed model including reliability measures: The density, the hazard rate, mean residual life, moment generating function, moments, quantiles, kurtosis, skewness, limiting distribution of sample extremes, stochastic orderings, stress-strength reliability and entropies. Estimation via maximum likelihood is straightforward. Censored data application of the UTNB distribution shows that it could provide a better fit than the uniform and the MOEU distributions. We hope that the proposed extended model may attract wider applications in survival analysis.