On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

Asymptotic estimators of the sample size in a record model

The connection between Stirling numbers of the first kind and records is well-known. Applying this relationship, we derive bounds for the maximum likelihood estimator of the sample size based on the number of observed records. The proof proceeds by a remarkable expression of the mode of the unsigned Stirling numbers of the the first kind due to Hammersley. Moreover, this representation of the mode leads to an accurate approximation of the maximum likelihood estimator.     

Usefulness of Asymptotic Distributions in the Classical Occupancy Problem

Approximations to the distribution of a discrete random variable originating from the classical occupancy problem are explored. The random variable X of interest is defined to be how many of N elements selected by or assigned to K individuals when each of the N elements is equally likely to be chosen by or assigned to any of the K individuals. Assuming that N represents the number of cells and each of the K individuals is placed in exactly one of the cells, X can also be defined as the number of cells occupied by the Kindividuals. In the literature, various asymptotic results for the distributions of X and (N ? X) are given; however, no guidelines are specified with respect to their utilization. In this article, these approximations are explored for various values of K and N, and rules of thumb are given for their appropriate use.     

Elementary expressions for moments of truncated negative binomial random variables

Moments of truncated negative binomial random variables arise in many areas. But moments of general order do not appear to be available, even a correct expression for the variance of a truncated negative binomial random variable was derived only in 2016. Here, we derive the elementary expressions for the moments of general order for four different types of truncated negative binomial random variables. Computational issues are discussed for the expressions.     

The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.     

Estimation of the Population Size in the Classical Occupancy Problem

In the classical occupancy problem where the random variable X is the number of N elements selected by K individuals when each element is equally likely to be chosen by any of the individuals, it is desired to estimate N. Three estimators given in the literature are compared with three estimators derived in this article, two of which are based on Bayesian methods, utilizing a simulation study. One of the Bayes estimators appears to perform the best along with one proposed in the literature in 1968. The estimators are also compared using data obtained from a cemetery in Ohio.     

Integral Representation of a Distribution Associated with a Pure Birth Process

ABSTRACT

This article deals with a distribution associated with a pure birth process starting with no individuals, with birth rates λ _n  = λ for n = 0, 2,…, m ? 1 and λ _n  = μ for n ≥ m. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions, and others. Using this representation, the kth factorial moments of the distribution is obtained. Some other forms of this distribution are also given.     

Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q   being the inverse of the proportion λ$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.     

On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

A Lagrangian Non Central Negative Binomial Distribution of the First Kind

A Lagrangian probability distribution of the first kind is proposed. Its probability mass function is expressed in terms of generalized Laguerre polynomials or, equivalently, a generalized hypergeometric function. The distribution may also be formulated as a Charlier series distribution generalized by the generalizing Consul distribution and a non central negative binomial distribution generalized by the generalizing Geeta distribution. This article studies formulation and properties of the distribution such as mixture, dispersion, recursive formulas, conditional distribution and the relationship with queuing theory. Two illustrative examples of application to fitting are given.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

Optimal linear combinations using householder transformations

In pattern classification of sampled vector valued random variables it is often essential, due to computational and accuracy considerations, to consider certain measurable transformations of the random variable. These transformations are generally of a dimension-reducing nature. In this paper we consider the class of linear dimension reducing transformations, i.e., the k × n matrices of rank k where k < n and n is the dimension of the range of the sampled vector random variable.

In this connection, we use certain results (Decell and Quirein, 1973), that guarantee, relative to various class separability criteria, the existence of an extremal transformation. These results also guarantee that the extremal transformation can be expressed in the form (I_k∣ Z)U where I_k is the k × k identity matrix and U is an orthogonal n × n matrix. These results actually limit the search for the extremal linear transformation to a search over the obviously smaller class of k × n matrices of the form (I_k ∣Z)U. In this paper these results are refined in the sense that any extremal transformation can be expressed in the form (I_K∣Z)H_p … H₁ where p ≤ min{k, n?k} and H_i is a Householder transformation i=l,…, p, The latter result allows one to construct a sequence of transformations (L_K∣ Z)H₁, (I_K Z)H₂H₁ … such that the values of the class separability criterion evaluated at this sequence is a bounded, monotone sequence of real numbers. The construction of the i-th element of the sequence of transformations requires the solution of an n-dimensional optimization problem. The solution, for various class separability criteria, of the optimization problem will be the subject of later papers. We have conjectured (with supporting theorems and empirical results) that, since the bounded monotone sequence of real class separability values converges to its least upper bound, this least upper bound is an extremal value of the class separability criterion.

Several open questions are stated and the practical implications of the results are discussed.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

A Condition to Obtain the Same Decision in the Homogeneity Testing Problem from the Frequentist and Bayesian Point of View

We develop a Bayesian procedure for the homogeneity testing problem of r populations using r × s contingency tables. The posterior probability of the homogeneity null hypothesis is calculated using a mixed prior distribution. The methodology consists of choosing an appropriate value of π₀ for the mass assigned to the null and spreading the remainder, 1 ? π₀, over the alternative according to a density function. With this method, a theorem which shows when the same conclusion is reached from both frequentist and Bayesian points of view is obtained. A sufficient condition under which the p-value is less than a value α and the posterior probability is also less than 0.5 is provided.     

The rank distribution of the maximin in a random matrix

Let S be a set of tm distinct real numbers and R a random t × m matrix of these tm numbers with rows {r_i} and columns (c_i}. Define b = Max Min x. l≤i≤t x?r_i. Let c be the event Max Min x = Min Max x. l≤i≤t x?r_i l≤i≤m x?c_i. This paper derives the probability distribution of the rank of b in S, as well as the same distribution conditional on c.     

Discrimination analysis when the variates are grouped and observed in sequential order

Suppose that measurements Math', i = l,....,k, are consecutively taken on an individual at the prescribed costs C_i, i = l,....,k. the individual comes from one of the two populations H₁ and H₂, and it is desired to detect which population the individual belongs to. Given the loss incurreed in selecting population H_r when in fact it belongs to H_s, the prior probability P_r of H_r (r = 1,2), and assuming that H_r has the normal distribution N(µ_r, V), r = 1,2, we derive the sequential Bayesian solution of the discrimination problem when µ₁, µ₂ and V are known. When µ_r, V are unknown and must be estimated, we propose a solution which is asymptotic Bayesian with exponential convergence rate.     

Comment on J. Berkson's paper “in dispraise of the exact test”

This paper rejects the preference expressed by Berkson for the heuristic test statistic T_N with standard normal distribution testing equality of two binomial probabilities in favour of Fisher's conditional exact test statistic T_E. Conditioning upon k₁ + k₂ = k shows that T_N admits too large first kind error probabilities. But also unconditionally T_N is systematically too large compared to T_E, giving too small critical levels at given experimental outcomes and power which is misleadingly too large. This is mainly due to the fact that T_N is not suitably corrected for continuity (at small sample sizes).     

Generalized polya eggenberger family of distributions and its relation to lagrangian katz family

Janardan (1973) introduced the generalized Polya Eggenberger family of distributions (GPED) as a limiting distribution of the generalized Markov-Polya distribution (GMPD). Janardan and Rao (1982) gave a number of characterizing properties of the generalized Markov-Polya and generalized Polya Eggenberger distributions. Here, the GPED family characterized by four parameters, is formally defined and studied. The probability generating function, its moments, and certain recurrence relations with the moments are provided. The Lagrangian Katz family of distributions (Consul and Famoye (1996)) is shown to be a sub-class of the family of GPED (or GPED ₁ ) as it is called in this paper). A generalized Polya Eggenberger distribution of the second kind (GPED ₂ ) is also introduced and some of it's properties are given. Recurrence relations for the probabilities of GPED ₁ and GPED ₂ are given. A number of other structural and characteristic properties of the GPED ₁ are provided, from which the properties of Lagrangian Katz family follow. The parameters of GMPD ₁ are estimated by the method of moments and the maximum likelihood method. An application is provided.