Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Inferences on the Among-Group Variance Component in Unbalanced Heteroscedastic One-Fold Nested Design

This article studies the hypothesis testing and interval estimation for the among-group variance component in unbalanced heteroscedastic one-fold nested design. Based on the concepts of generalized p-value and generalized confidence interval, tests and confidence intervals for the among-group variance component are developed. Furthermore, some simulation results are presented to compare the performance of the proposed approach with those of existing approaches. It is found that the proposed approach and one of the existing approaches can maintain the nominal confidence level across a wide array of scenarios, and therefore are recommended to use in practical problems. Finally, a real example is illustrated.     

Exact Size and Power Properties of Five Tests for Multinomial Proportions

Estimators of the expectations of order statistics, suggested by Harrell &; Davis, are used in place of the order statistics in a quantile estimator, proposed by Kappenman , to produce a modification of Kappenman's procedure. Simulation studies indicate that the modification generally results in a reduction of mean squared error     

Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

Comparison of Confidence Intervals for a Poisson Mean – Further Considerations

I make recommendations in choosing a confidence interval for the Poisson mean, from twelve different methods, that are based on four general principles: actual coverage should closely match the nominal coverage; narrower expected widths of confidence intervals are better; the right and left non-coverage should be fairly balanced; and some investigators may prefer closed-form intervals. The interval chosen depends on the relative importance the investigator places on each of these principles. The confidence intervals are examined through graphs of their coverage probability, interval widths and shapes.     

Some aspects of statistical inference on the lagrange (generalized) poisson distribution

A generalization of the Poisson distribution was defined by Consul and Jain (Ann. Math. Statist., 41, (1970)) and was obtained as a particular family of Lagrange distributions by Consul and Shenton (SIAM. J. Appl. Math., 23, (1972)). The distribution is subsequently named the generalized Poisson distribution (GPD). This GPD reduces to the Poisson distribution for ? = 0. When the data have a one-way layout structure, the asymptotically locally optimal Neyman's C(d) test is constructed and compared with the conditional test on the hypothesis H_o? = 0. Within the framework of the generalized linear models an appropriate link function is given, and the asymptotic distributions of the estimated parameters are derived.     

Waiting time paradox and size biased sampling

Feller's (1971) 'Waiting time for the bus paradox1 is explained by the fact that the long Interarrival times have a better chance of covering an arbitrary point in time than do short interarrival times, This gives rise to size biased sampling and the length of the Interarrival time that contains the arbitrary point has a moment distribution, This paper deals with the characterization of the exponential distribution based on this 'Waiting time paradox'. Similar characterizations of the binomial, the negative binomial, the Poisson and the geometric distributions are obtained.     

Interval estimation for a binomial proportion: a bootstrap approach

This paper discusses the classic but still current problem of interval estimation of a binomial proportion. Bootstrap methods are presented for constructing such confidence intervals in a routine, automatic way. Three confidence intervals for a binomial proportion are compared and studied by means of a simulation study, namely: the Wald confidence interval, the Agresti–Coull interval and the bootstrap-t interval. A new confidence interval, the Agresti–Coull interval with bootstrap critical values, is also introduced and its good behaviour related to the average coverage probability is established by means of simulations.     

Estimation in the generalized linear empirical bayes model using the extended quasi-likelihood

A generalized linear empirical Bayes model is developed for empirical Bayes analysis of several means in natural exponential families. A unified approach is presented for all natural exponential families with quadratic variance functions (the Normal, Poisson, Binomial, Gamma, and two others.) The hyperparameters are estimated using the extended quasi-likelihood of Nelder and Pregibon (1987), which is easily implemented via the GLIM package. The accuracy of these estimates is developed by asymptotic approximation of the variance. Two data examples are illustrated.     

The monotone boundary property and the full coverage property of confidence intervals for a binomial proportion

The methodology for deriving the exact confidence coefficient of some confidence intervals for a binomial proportion is proposed in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368]. The methodology requires two conditions of confidence intervals: the monotone boundary property and the full coverage property. In this paper, we show that for some confidence intervals of a binomial proportion, the two properties hold for any sample size. Based on results presented in this paper, the procedure in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368] can be directly used to calculate the exact confidence coefficients of these confidence intervals for any fixed sample size.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

A generalized score confidence interval for a binomial proportion

Constructing a confidence interval for a binomial proportion is one of the most basic problems in statistics. The score interval as well as the Wilson interval with some modified forms have been broadly investigated and suggested by many statisticians. In this paper, a generalized score interval CI_G(a) is proposed by replacing the coefficient 1/4 in the score interval with parameter a. Based on analyzing and comparing various confidence intervals, we recommend the generalized score interval CI_G(0.3) for the nominal confidence levels 0.90, 0.95 and 0.99, which improves the spike phenomenon of the score interval and behaves better and computes more easily than most of other approximate intervals such as the Agresti-Coull interval and the Jeffreys interval to estimate a binomial proportion.     

Multivariate stable exponential families and Tweedie scale

In this paper, we characterize the multivariate stable natural exponential families by a property of homogeneity of the cumulant function of some basis, and by a property of homogeneity of the variance function. We also extend the definition of a Tweedie scale to a finite dimensional space and we give a class of natural exponential families belonging to this scale on the space of symmetric matrices.