Multiple decision rules for comparing several populations with a fixed known standard

Independent observations are available from k univariate distributions indexed by a real parameter θ. It is desired to select that distribution with the largest parameter value unless this value is smaller than some fixed standard θ₀ in which case no distribution is to be selected. Various single-stage procedures for this (k+l)-decision problem are discussed, using indifference zone, decision theoretic, Bayesian, and subset selection approaches.     

Bounded multinomial distribution

A "bounded" multinomial distribution is introduced. The probability distribution function and the size of the sample space are investigated     

Adaptive sequential procedures for comparing new treatments with a standard

In this paper we propose and study two sequential elimination procedures for selecting all new treatments better than a standard or control treatment. These procedures differ from those previously proposed in that we assume variances are unequal and unknown. Expressions for asymptotic expected sample sizes are given. Confidence intervals associated with the procedures are also discussed.     

Minimax subset selection for the multinomial distribution

Let (X₁,…,X_k) be a multinomial vector with unknown cell probabilities (p₁,?,p_k). A subset of the cells is to be selected in a way so that the cell associated with the smallest cell probability is included in the selected subset with a preassigned probability, P¹. Suppose the loss is measured by the size of the selected subset, S. Using linear programming techniques, selection rules can be constructed which are minimax with respect to S in the class of rules which satisfy the P¹-condition. In some situations, the rule constructed by this method is the rule proposed by Nagel (1970). Similar techniques also work for selection in terms of the largest cell probability.     

Selecting the t-best cells in a multinomial distribution

The problem of selecting the t-best cells in a multinomial distribution with t + k cells, k > 1, 2 <= t is considered under the fixed sample-size indifference zone approach. The least favourable configuration is derived for the usual procedure of selection, for large values of N (the sample size). The result settles Conjecture I (for large N) and Conjecture IV of Chen and Hwang (Commun. Statist. - Theory Meth. 13 (10), 1289-1298, 1984) in the affirmative.     

A new confidence region for the multinomial distribution

A simple confidence region is proposed for the multinomial parameter. It is designed for situations having zero cell counts. Simulation studies as well as a real data application show that it performs at least as well as than at least two of the most common confidence regions.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

Bayes a-opttmal and efficient block designs for comparing test treatments with a standard treatment

A sufficient condition for the Bayes A-optimality of block designs when comparing a standard treatment with v test treatments is given by Majumdar. (In:Optimal Design and Analysis of Experiments, Y. Dodge, V. V. Fedorov and H. P. Wynn (Eds.), 15-27, North-Holland, 1988). The priors that he considers depend on a constant α ε [0, ∞), with α - 0 corresponding to no prior information at all. The given sufficient condition, consequently, also depends on a. Large families of optimal and highly efficient designs are only known for the case α - 0. We will show how some of the results for α - 0 can be extended to obtain large families of optimal and highly efficient designs for arbitrary values of α. In addition, these results are useful when considering design robustness against an improper choice of α.     

Simple nonparametric tests for a known standard survival based on censored data

It is often of interest in survival analysis to test whether the distribution of lifetimes from which the sample under study was derived is the same as a reference distribution. The latter can be specified on the basis of previous studies or on subject matter considerations. In this paper several tests are developed for the above hypothesis, suitable for right-censored observations. The tests are based on modifications of Moses' one-sample limits of some classical two-sample rank tests. The asymptotic distributions of the test statistics are derived, consistency is established for alternatives which are stochastically ordered with respect to the null, and Pitman asymptotic efficiencies are calculated relative to competing tests. Simulated power comparisons are reported. An example is given with data on the survival times of lung cancer patients.     

Tests for comparing a sequence of bernoulli random variables to a sequence of known probabilities

The problem of determining whether a sequence of observed Bernoulli variates is consistent with a hypothesized underlying sequence of known probabilities is considered. A family of asymptotically normal test statistics is proposed, members of which are shown to be asymptotically locally optimal against specific types of alternatives. For small samples, a skewness correction is shown to improve greatly the adequacy of the asymptotic approximations to the null distributions of the proposed test statistics. The application of testing for increased cancer risk in families is considered, and modifications to the test statistics which adjust for the method of family ascertainment are indicated     

Inference with a contrast-based posterior distribution and application in spatial statistics

The likelihood function is often used for parameter estimation. Its use, however, may cause difficulties in specific situations. In order to circumvent these difficulties, we propose a parameter estimation method based on the replacement of the likelihood in the formula of the Bayesian posterior distribution by a function which depends on a contrast measuring the discrepancy between observed data and a parametric model. The properties of the contrast-based (CB) posterior distribution are studied to understand what the consequences of incorporating a contrast in the Bayes formula are. We show that the CB-posterior distribution can be used to make frequentist inference and to assess the asymptotic variance matrix of the estimator with limited analytical calculations compared to the classical contrast approach. Even if the primary focus of this paper is on frequentist estimation, it is shown that for specific contrasts the CB-posterior distribution can be used to make inference in the Bayesian way.The method was used to estimate the parameters of a variogram (simulated data), a Markovian model (simulated data) and a cylinder-based autosimilar model describing soil roughness (real data). Even if the method is presented in the spatial statistics perspective, it can be applied to non-spatial data.     

Estimation of a distribution function bounded by two known distribution functions

The problem of estimation of a cumulative distribution function (cdf), bounded by two known cdf's, is considered. An estimator satisfying the desired restriction has been obtained by suitably adjusting the empirical cdf. Consistency of the adjusted estimator has been established and its mean square error (MSE) has been shown to be smallerthan that of the empirical cdf. The new estimator has been comparedwith the empirical cdf for some special cases.     

Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component‐wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max‐stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.     

Some MV-optimal group divisible type block designs for comparing a set of test treatments with a set of standard treatments

In this paper, we consider experimental situations in which it is desired to optimally compare t-test treatments to s standard treatments using a block design in which the experimental units are arranged in b blocks of size k. A method is given for generating an MV-optimal block design for such situations and sufficient conditions are derived which can often be used to establish the MV-optimality of reinforced group divisible designs which are often obtained using the process given.     

Inference of non-centrality parameter of a truncated non-central chi-squared distribution

Non-central chi-squared distribution plays a vital role in statistical testing procedures. Estimation of the non-centrality parameter provides valuable information for the power calculation of the associated test. We are interested in the statistical inference property of the non-centrality parameter estimate based on one observation (usually a summary statistic) from a truncated chi-squared distribution. This work is motivated by the application of the flexible two-stage design in case–control studies, where the sample size needed for the second stage of a two-stage study can be determined adaptively by the results of the first stage. We first study the moment estimate for the truncated distribution and prove its existence, uniqueness, and inadmissibility and convergence properties. We then define a new class of estimates that includes the moment estimate as a special case. Among this class of estimates, we recommend to use one member that outperforms the moment estimate in a wide range of scenarios. We also present two methods for constructing confidence intervals. Simulation studies are conducted to evaluate the performance of the proposed point and interval estimates.     

Inference about parameters in Binomial-Poisson distribution with additional incomplete data

ABSTRACT

This article considers the distribution of Binomial-Poisson random vector which has two components and includes two parameters: one is the rate of a Poisson distribution, the other is the proportion in a Binomial distribution. The inference about the two parameters is usually made based on only paired observations. However, the number of paired observations is, in general, not large enough because of either technical difficulty or budget limitation, and so one can not make efficient inferences with only paired data. Instead, it is often much easier and not too costly to have incomplete observation on only one component independently. In this article we will combine both the paired complete data and unpaired incomplete data for estimating the two parameters. The performances of various estimators are compared both analytically and numerically. It is observed that fully using the unpaired incomplete data can always improve the inference, and the improvement is very significant in the case when there are only a few paired complete observations.     

A multinomial model for cross classified data with a measure of dependency

This paper presents a modified multinomial model for analyzing behaviour among wildlife populations. It assumes that the covariance matrix of the observed proportions is a multiple of the covariance matrix under simple random sampling. The model also allows a measure of dependency among the clusters within subpopulations, a type of dependency that assumes the relationships among units are the same for any two units. In addition, this paper illustrates the fact that the incorrect application of the Pearson chi-square statistic based on simple random sampling can produce misleading results when frequencies are obtained from a non-multinomial sampling scheme. Data obtained from a study of wild turkeys are analyzed using the proposed multinomial model.     

Inference for the extreme value distribution under progressive Type-II censoring

The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. In this paper, we discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error. The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.