Family Of Discrete Distributions Research Articles

Asymptotics of maxima of discrete random variables

If X1, X2,..., Xn are independent and identically distributed discrete random variables and Mn=max (X1,..., Xn) we examine the limiting behavior of (Mn−b(n))/a(n) as n → ∞. It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n → ∞, it is possible to obtain non-degenerate limits for (Mn−b(n))/a(n). We consider four families of discrete distributions and show how this can be done.

Wherefore similar tests?

Similarity of a test is often a necessary condition for a test to be unbiased (in particular for a test to be uniformly most powerful unbiased when such a test exists). Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1986) describes the connection between similar tests and uniformly most powerful unbiased tests. The methods to achieve these properties as outlined in Lehmann are used extensively. In any case, an admissible similar test is frequently one that can be recommended for practical use. In some constrained parameter spaces however, we show that admissible similar tests sometimes completely ignore the constraints. In some of these cases we call such tests constraint insensitive. The tests seem not to be intuitive and perhaps should not be used. On the other hand, there are models with constrained parameter spaces where similar tests do take into account the constraints. In these cases the admissible test is called constraint sensitive. We offer a systematic approach that enables one to determine whether an admissible similar test is constraint insensitive or not. The approach is applied to three classes of models involving order restricted parameters. The models include testing for homogeneity of parameters, testing subsets of parameters, and testing goodness of fit of a family of discrete distributions.

Exact Properties of Demerit Control Charts

A demerit rating system is used to simultaneously monitor counts of several different types of defects in a complex product. The demerit statistic is a linear combination of the counts of these different types of defects. The traditional recommendation is to plot the demerit statistic on a control chart with symmetric 3-sigma control limits. This approach to demerit rating systems is reviewed. An alternative method for determining control limits for the demerit control chart, based on the exact distribution of linear combinations of independent Poisson random variables, is proposed. This method is generalized for use with the exact distribution of linear combinations of independent random variables in a broad family of discrete distributions.

Detecting departures from a poisson model

Poisson models can be widely used for analyzing count data and belong to some broad classes of discrete distributions such as a Katz family, a double exponential family and a family of discrete distributions with a finite first moment. Several procedures for testing goodness of fit of a Poisson model against these classes are discussed and simulations are conducted to compare their performance for detecting departures from Poisson models.

On the monotonic properties of discrete failure rates

As is well known, the monotonicity of failure rate of a life distribution plays an important role in modeling failure time data. In this paper, we develop techniques for the determination of increasing failure rate (IFR) and decreasing failure rate (DFR) property for a wide class of discrete distributions. Instead of using the failure rate, we make use of the ratio of two consecutive probabilities. The method developed is applied to various well known families of discrete distributions which include the binomial, negative binomial and Poisson distributions as special cases. Finally, a formula is presented to determine explicitly the failure rate of the families considered. This formula is used to determine the failure rate of various classes of discrete distributions. These formulas are explicit but complicated and cannot normally be used to determine the monotonicity of the failure rates.

Estimating parameters for discrete distributions via the empirical probability generating function

We consider parameter estimation for a family of discrete distributions characterized by probability generating functions (pgf's). Kemp and Kemp (1988) suggest estimators based on the empirical probability generating function (epgf) the methods involve solving estimating equations obtained by equating functionals of the epgf and pgf on a fixed, finite set of values. We derive asymptotic theory for these estimators and consider some examples. Graphical techniques based on the theory are shown to be useful for exploratory analysis

A family of discrete distributions

Ambagaspitiya and Balakrishnan (1994a) used the identity p n(a,b) = a a + b b + a n p n - 1(a + b,b), n = 1,2,…, with p 0( a, b) = exp(− a) to obtain a recursive formula for computing the distribution function of the compound generalized Poisson distribution. In this paper, we consider the discrete distribution family with the property p n(a,b) = h 1(a,b) + h 2(a,b) n p n - 1(a + b,b), n = 1,2,…, which is a generalization of the first identity. We prove that weighted generalized Poisson distributions and weighted generalized negative binomial distributions with weights of the form w( a + bn; b) are two subclasses in the family. We provide a recursive formula for computation of respective compound distributions. Also we discuss the stability of the recursion as well as handling overflow / underflow problems.

The Asymptotic Probability of a Tie for First Place

Suppose that the scores of $n$ players are independent integer-valued random variables with probabilities $p_j$. We study the probability $P(T_n)$ that there is a tie for the highest score. The asymptotic behavior of this probability is surprising. Depending on the limit of $p_{j+1}/p_j$, we find different limits of different subsequences $P(T_{n(m)})$. These limits are evaluated for several families of discrete distributions.

Parameter Orthogonality for a Family of Discrete Distributions

Parameter Orthogonality for a Family of Discrete Distributions

Parameter Orthogonality for a Family of Discrete Distributions

Abstract The standard contagious distributions (see Douglas 1980) have been used in such varied fields as biology and automobile insurance, often to model various physical phenomena as well as provide a good fit to count data when other models are inadequate. Unfortunately, the parameterizations often used when working with these distributions normally lead to extremely high correlations of the maximum likelihood estimators (MLE's). This tends to lead to mathematical complexities, and causes difficulty or even errors in their interpretation. Furthermore, numerical difficulties may arise when using numerical procedures to locate the estimates. Some of these difficulties were discussed by Douglas (1980, pp. 171, 204-205), who suggested that a reparameterization to reduce or even eliminate such correlation is desirable. If the MLE's are asymptotically uncorrelated, the parameterization is orthogonal. Philpot (1964) derived an orthogonal parameterization for the Neyman Type A distribution; Stein, Zucchini, an...

Sundt and Jewell's Family of Discrete Distributions

AbstractA class of claim frequency distributions discussed by Sundt and Jewell (1981) is completely enumerated. Computational techniques for the associated compound total claims distribution in the presence of policy modifications are then derived.

Characterization of a family of discrete distributions via a chernoff type inequality

A characterization theorem is proved for a family of non-negative integer valued random variables. The result is based on an inequality of the type proved by Chernoff (1981). Applications of the result for various standard distributions are also discussed.

A Generalized Binomial Distribution

Abstract We consider a simple sampling scheme where after each drawing there is a “replacement” whose magnitude is a fixed real number. This gives a unified approach to a family of discrete distributions that includes the hypergeometric, binomial, and Polya distributions as well as their multivariate versions.

Some aspects of the kemp families of distributions

Two families of discrete distributions, one with the probability generating function (p.g.f) and the other with the p.g.f. have been examined. Under certain restrictions on the parameters, some members of the former class can be interpreted as compound distributions. These resulting distributions have the property of being over-, under-, or equi-dispersed. These properties are also examined for the latter class of distributions. Relationships among different members of the latter class are considered graphically by means of a criterion based on successive probabilities.

A Family of Discrete Distributions Defined via Their Factorial Moments

A Family of Discrete Distributions Defined via Their Factorial Moments

A family of discrete distributions defined via their factorial moments

This paper examines the family whose probability generating functions have the form of the generalized hypergeometric function, pFq [(a); (b); λ(s-1)] . It includes a number of matching distributions as well as many classic discrete distributions. Properties may be derived from the differential equations satisfied by the various generating functions e.g. useful recurrence formulae for probabilities, cumulants, and moments about an arbitrary point can be obtained.

Uniformly Most Accurate Upper Tolerance Limits for Monotone Likelihood Ratio Families of Discrete Distributions

Abstract The theory of uniformly most accurate upper tolerance limits is presented for families of discrete distributions having monotone likelihood ratios. Explicit formulae are derived for binomial, negative-binomial and Poisson distributions. The relationship to systems of uniformly most accurate upper confidence limits is exposed.

Note on a certain family of discrete distributions

Note on a certain family of discrete distributions

Note on a Certain Family of Discrete Distributions

Note on a Certain Family of Discrete Distributions