Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.     

Thinning operations for modeling time series of counts—a survey

The analysis of time series of counts is an emerging field of science. To obtain an ARMA-like autocorrelation structure, many models make use of thinning operations to adapt the ARMA recursion to the integer-valued case. Most popular among these probabilistic operations is the concept of binomial thinning, leading to the class of INARMA models. These models are proved to be useful, especially for processes of Poisson counts, but may lead to difficulties in the case of different count distributions. Therefore, several alternative thinning concepts have been developed. This article reviews such thinning operations and shows how they are successfully applied to define integer-valued ARMA models.     

Some remarks on classical and bayesian reliability estimation of binomial and poisson distributions

The problems of estimating the reliability function and Pr{X₁+...+X_k ≤ Y} are considered. The random variables X’s and Y are assumed to follow binomial and Poisson distributions. Classical estimators available in the literature are discussed and Bayes estimators are derived. In order to obtain the estimators of these parametric functions, the basic role is played by the estimators of factorial moments of the two distributions.     

Score tests for zero inflation in generalized linear models

The authors develop score tests of goodness of fit for discrete generalized linear models against zero inflation. The binomial and Poisson models are treated as examples, and in the latter case the proposed test reduces to that of Broek (1995). Some simulation results and an illustrative example are presented.     

ALGORITHMS FOR RETURN PROBABILITIES FOR STOCHASTIC FLUID FLOWS

Abstract

We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithms within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the number of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations.     

Inequalities for tail probabilities for the multivariate normal distribution

Inequalities for tail probabilities of the multivariate normal distribution are obtained, as a generalization of those given by Feller (1966). Upper and lower bounds are given in the equi-correlated case. For an arbitrary correlation matrix R, an upper bound is obtained, using a result of Slepian (1962) which asserts that certain multivariate normal probabilities are a non-decreasing function of correlations.     

Prediction Intervals for Time Series: A Modified Sieve Bootstrap Approach

Traditional Box–Jenkins prediction intervals perform poorly when the innovations are not Gaussian. Nonparametric bootstrap procedures overcome this handicap, but most existing methods assume that the AR and MA orders of the process are known. The sieve bootstrap approach requires no such assumption but produces liberal coverage due to the use of residuals that underestimate the actual variance of the innovations and the failure of the methods to capture variations due to sampling error of the mean. A modified approach, that corrects these deficiencies, is implemented. Monte Carlo simulations results show that the modified version achieves nominal or near nominal coverage.     

On the generation of subsets of order statistics

A modification of Schucany's (1972) method for generating order statistics is given which allows for efficient generation of certain subsets of order statistics.     

Testing linear inequality constraints in the standard linear model

In this paper we are concerned with the problem of testing whether the â-parameters of the standard linear model satisfy the linear equality constraints R = r when they are known to satisfy the corresponding linear inequality constraints Râ ? r. In particular we will show that the exact finite sample null distributions of the Likelihood Ratio, Wald and Kuhn-Tucker

statistics are known when R is of full row rank but not known when R has less than full row rank. The less than full row rank problem has not been discussed previously but it is of considerable potential importance.

This paper contains several simple numerical examples which illustrate the computational details of the tests     

Win-probabilities for comparing two binary outcomes

This article considers the problem of choosing between two treatments that have binary outcomes with unknown success probabilities p₁ and p₂. The choice is based upon the information provided by two observations X₁ ～ B(n₁, p₁) and X₂ ～ B(n₂, p₂) from independent binomial distributions. Standard approaches to this problem utilize basic statistical inference methodologies such as hypothesis tests and confidence intervals for the difference p₁ ? p₂ of the success probabilities. However, in this article the analysis of win-probabilities is considered. If X*₁ represents a potential future observation from Treatment 1 while X*₂ represents a potential future observation from Treatment 2, win-probabilities are defined in terms of the comparisons of X*₁ and X*₂. These win-probabilities provide a direct assessment of the relative advantages and disadvantages of choosing either treatment for one future application, and their interpretation can be combined with other factors such as costs, side-effects, and the availabilities of the two treatments. In this article, it is shown how confidence intervals for the win-probabilities can be constructed, and examples of their use are provided. Computer code for the implementation of this new methodology is available from the authors.