Further results on the trace of a noncentral wishart matrix

In an earlier article Mathai (1980) has given compact representations for the moments and cumulants of the trace of a noncentral Wishart matrix. He has also shown that (trA-ntr;∑)/(2ntri∑²)172. is asymptotically standard normal where A is a noncentral wishart matrix with n degrees of freedom and covariance matrix [0, In the present article explicit expressions for the exact density of the trace are given in terms of confluent hypergeometric functions and in terms of zonal polynomials for the general case and as finite sums when the sample size is odd. As a consequence of some of these representations some summation formulae for zonal polynomials are also given     

Marginally restricted D-optimal designs for correlated observations

Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may not be under the control of the practitioner. Secondly, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to marginally restricted D–optimality and it is applied to a complex situation.     

Permuiation tesis for k-sample binomial data with comparisons of exact and approximate P-levels

Exact ksample permutation tests for binary data for three commonly encountered hypotheses tests are presented,, The tests are derived both under the population and randomization models . The generating function for the number of cases in the null distribution is obtained, The asymptotic distributions of the test statistics are derived . Actual significance levels are computed for the asymptotic test versions , Random sampling of the null distribution is suggested as a superior alternative to the asymptotics and an efficient computer technique for implementing the random sampling is described., finally, some numerical examples are presented and sample size guidelines given for computer implementation of the exact tests.     

On the Optimality and Limitations of Buehler Bounds

In 1957, R.J. Buehler gave a method of constructing honest upper confidence limits for a parameter that are as small as possible subject to a pre‐specified ordering restriction. In reliability theory, these ‘Buehler bounds’ play a central role in setting upper confidence limits for failure probabilities. Despite their stated strong optimality property, Buehler bounds remain virtually unknown to the wider statistical audience. This paper has two purposes. First, it points out that Buehler's construction is not well defined in general. However, a slightly modified version of the Buehler construction is minimal in a slightly weaker, but still compelling, sense. A proof is presented of the optimality of this modified Buehler construction under minimal regularity conditions. Second, the paper demonstrates that Buehler bounds can be expressed as the supremum of Buehler bounds conditional on any nuisance parameters, under very weak assumptions. This result is then used to demonstrate that Buehler bounds reduce to a trivial construction for the location‐scale model. This places important practical limits on the application of Buehler bounds and explains why they are not as well known as they deserve to be.     

Biopharmaceutical applications of group sequental designs

Methods are described for calculating the properties of a general class of group sequential designs. The procedures are based on direct enumeration of a11 possible outcomes thereby giving exact results. The test regions of these designs are closed, may be one- or two-sided and may include a variable number of groups or stages. The methods described can be applied in experimental situations where the responses can be dichotomized or ranked. Specific examples which are given to illustrate the procedures include drug screening, laboratory evaluation, and clinical trials.     

Exact percentage points for the Kolmogorov test on truncated versions of known continuous distributions with unknown truncation parameters

In this paper we deal with Kolmogorov-Smirnov testson uniformity with completely or partly unknown limits. Tables of exact percentage points are presented or referred using the wellknown determinant formula given by Steck (1971). It is shown that these tables also give the percentage points for the analogous statistics of the test on truncated versions of known continuous distributions with completely or partly unknown truncation limits. We will give some examples of these applications. Among these are the tests on exponentiality and on Pareto distribution with known shape parameter and unknown lower terminal.     

Exact inferences of the two-parameter exponential distribution and Pareto distribution with censored data

We develop an exact inference for the location and the scale parameters of the two-exponential distribution and the Pareto distribution based on their maximum-likelihood estimators from the doubly Type-II and the progressive Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. We also provide a bootstrap method for constructing a confidence interval. Some simulation studies are carried out to assess their performances. Using the exact distribution of the scale parameter, we establish an acceptance sampling procedure based on the lifetime of the unit. Some numerical results are tabulated for the illustration. One biometrical example is also given to illustrate the proposed methods.     

An exact two-sample test based on the baumgartner-weiss-schindler statistic and a modification of lepage's test

It is shown that the nonparametric two-saniDle test recently proposed by Baumgartner, WeiB, Schindler (1998, Biometrics, 54, 1129-1135) does not control the type I error rate in case of small sample sizes. We investigate the exact permutation test based on their statistic and demonstrate that this test is almost not conservative. Comparing exact tests, the procedure based on the new statistic has a less conservative size and is, according to simulation results, more powerful than the often employed Wilcoxon test. Furthermore, the new test is also powerful with regard to less restrictive settings than the location-shift model. For example, the test can detect location-scale alternatives. Therefore, we use the test to create a powerful modification of the nonparametric location-scale test according to Lepage (1971, Biometrika, 58, 213-217). Selected critical values for the proposed tests are given.     

The validity of the chi-squared test when expected frequencies are small:A list of recent research references

A bibliography is given containing about 60 recent (chiefly post-1968) items on the validity of the chi-squared test for categorical data when the expected frequencies are small.     

An introduction to designh optimality with an overview of the literature

A brief introduction to oprimal design theorv is given for those who are not familiar with the subject. A list of 312 selected articles on the theory of optimal design is provided. The bibliography should be sufficiently thorough to be of use to researchers in the field.     

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.     

Exact tests for equality of two proportions: fisher v. bayes

Yates (1984) using theoretical and philosophical arguments claims to have proved that the Fisher exact test for comparing the proportions of two binomial experiments is the best exact test. The present article uses objective and practical arguments to confront the Fisher exact test with a Bayes exact test. Using simulated samples we claim to have proved here the inferiority of the Fisher exact test in relation to a Bayes exact test. The comparison is based on the quality concept of Dawid (1982).     

Moments and efficiency of the median and trimmed mean for finite populations

Exact formulas for the expected value and variance of the median and trimmed mean are found as functions of the elements of a finite population under simple random sampling. A simulation study is performed to compare the performance of the median and trimmed mean versus the mean when sampling from various simulated finite populations. Finally, the asymptotic performance of these estimators, when sampling from infinite populations, is compared with the finite populations results.     

Quota fulfillment times

An example of the classical occupancy problem is to sample with replacement from an urn containing several colours of balls and count the number of balls sampled until a given number of “quotas” are filled. This and the corresponding random variable for sampling without replacement will be referred to as quota fulfillment times. Asymptotic and exact methods for computing moments and distributions are given in this paper. Moments of quota fulfillment times are related to moments of order statistics of beta and gamma random variables. Most of the results for sampling without replacement and some of the results for sampling with replacement are believed to be new. Some other known sampling-with-replacement results are given for comparative purposes.     

Some notes on the fisher–bingham family on the sphere

The fisher–Bingham family is potentially useful class of spherical distributions, but its practical application has been hindered by various problems, including; the identification of the form of the distribution, given the parameter values; and finding an effective technique for calculating the normalising constant, a requirement for maximum likelihood estimation. It is explained how these difficulties can be resolved. and then parameter estimation and hypothesis testing are discussed. A practical example is given.     

Suppressing marginal totals from a two-dimensional table to protect sensitive information

The technique of data suppression for protecting sensitive information in a two-dimensional table from exact disclosure raises the computational problems of testing a given table of censored data for security, and searching for a secure suppression pattern of minimum size for a given table. We provide a polynomial security test to solve the former problem, and prove that the latter problem is intractable in the general case, but can be solved in linear time in the special case in which only sensitive cells are to be protected.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

On the distribution-free confidence intervals and universal bounds for quantiles based on joint records

In this paper, we consider three distribution-free confidence intervals for quantiles given joint records from two independent sequences of continuous random variables with a common continuous distribution function. The coverage probabilities of these intervals are compared. We then compute the universal bounds of the expected widths of the proposed confidence intervals. These results naturally extend to any number of independent sequences instead of just two. Finally, the proposed confidence intervals are applied for a real data set to illustrate the practical usefulness of the procedures developed here.     

E-Bayesian estimation of the exponentiated distribution family parameter under LINEX loss function

This paper is concerned with using the E-Bayesian method for computing estimates of the exponentiated distribution family parameter. Based on the LINEX loss function, formulas of E-Bayesian estimation for unknown parameter are given, these estimates are derived based on a conjugate prior. Moreover, property of E-Bayesian estimation—the relationship between of E-Bayesian estimations under different prior distributions of the hyper parameters are also provided. A comparison between the new method and the corresponding maximum likelihood techniques is conducted using the Monte Carlo simulation. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

Two classes of divergence statistics for testing uniform association

The problem of testing uniform association in cross-classifications having ordered categories is considered. Two families of test statistics, both based on divergences between certain functions of the observed data, are studied and compared. Our theoretical study is based on asymptotic properties. For each family, two consistent approximations to the null distribution of the test statistic are studied: the asymptotic null distribution and a bootstrap estimator; all the tests considered are consistent against fixed alternatives; finally, we do a local power study. Surprisingly, both families detect the same local alternatives. The finite sample performance of the tests in these two classes is numerically investigated through some simulation experiments. In the light of the obtained results, some practical recommendations are given.