SHRINKAGE PREDICTION IN THE EXPONENTIAL DISTRIBUTION WITH A PRIOR INTERVAL FOR THE SCALE PARAMETER

The author proposes the best shrinkage predictor of a preassigned dominance level for a future order statistic of an exponential distribution, assuming a prior estimate of the scale parameter is distributed over an interval according to an arbitrary distribution with known mean. Based on a Type II censored sample from this distribution, we predict the future order statistic in another independent sample from the same distribution. The predictor is constructed by incorporating a preliminary confidence interval for the scale parameter and a class of shrinkage predictors constructed here. It improves considerably classical predictors for all values of the scale parameter within its dominance interval containing the confidence interval of a preassigned level.     

AN EXACT FORMULA FOR THE SYMMETRICAL INCOMPLETE BETA FUNCTION WHERE THE PARAMETER IS AN INTEGER OR HALF-INTEGER

A finite series for computing the symmetrical incomplete beta function with parameter q= 0.5, 1, 1.5, is derived. It is more efficient than existing algorithms for the general case, and obviates reference to the Pearson-Johnson tables and subsequent interpolation. This is advantageous in applications to statistical quality assurance, especially in computer simulations of acceptance schemes.     

AN EVALUATION OF NON-ITERATIVE METHODS FOR ESTIMATING THE LINEAR-BY-LINEAR PARAMETER OF ORDINAL LOG-LINEAR MODELS

Parameter estimation for association and log-linear models is an important aspect of the analysis of cross-classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear-by-linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non-iterative procedures for estimating the linear-by-linear parameter for ordinal log-linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non-iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered.     

LINEAR MINIMAX ESTIMATORS FOR ESTIMATING A FUNCTION OF THE PARAMETER

In this note we provide sufficient conditions for the minimaxity of linear estimators of the form aX+b in the one-parameter exponential family for estimating a differentiable function g(θ) with normalized quadratic loss. We provide some examples which show that the natural estimator X is minimax in estimating a function of the parameter (different from the mean).     

A K-SAMPLE EXTENSION TO FLIGNER'S CLASS OF TESTS FOR SCALE

The parameteric tests for equality of variance are well known. The classical F-test is typically used to test the hypothesis of equality of two variances, while tests such as those developed by Bartlett (1937) are commonly used for the k-sample hypothesis. These tests assume an underlying normal distribution and are quite sensitive to departures from normality (Box, 1953). Thus, when considering data that are from non-normal distributions, alternative nonparametric tests must be employed.
Fligner (1979) has proposed a class of two-sample distribution-free tests which possess very desirable properties and are attractive alternatives to other nonparametric tests for scale. The present paper extends the Fligner class of tests to the more general k-sample case.     

PARAMETER ESTIMATION AND APPLICATIONS FOR A GENERALISATION OF THE BETA-BINOMIAL DISTRIBUTION

A three-parameter generalisation of the beta-binomial distribution (BBD) derived by Chandon (1976) is examined. We obtain the maximum likelihood estimates of the parameters and give the elements of the information matrix. To exhibit the applicability of the generalised distribution we show how it gives an improved fit over the BBD for magazine exposure and consumer purchasing data. Finally we derive an empirical Bayes estimate of a binomial proportion based on the generalised beta distribution used in this study.     

DATA-BASED CHOICE OF THE SMOOTHING PARAMETER FOR A KERNEL DENSITY ESTIMATOR

The problem of optimally choosing the smoothing parameter of a Fourier integral density estimator (FIE) is addressed. A new data-based method of so doing is shown to be related to, but an improvement over, the method of Davis (1981). It is shown that Davis' method does not lead to consistent estimation of certain, principally multimodal, densities. In a simulation study involving the bimodal mixture of two normal densities, the new method is seen to represent a substantial improvement.     

ESTIMATION OF THE PARAMETERS OF A NORMAL DISTRIBUTION WHEN THE MEAN IS RESTRICTED TO AN INTERVAL1

Suppose it is known that the mean of a normal distribution is non-negative. Naturally one will use the sample mean truncated at zero as an estimator of the distribution mean. In this paper the properties of such an estimator are investigated.     

ON A STOCHASTIC MODEL FOR AN IRREVERSIBLE SYSTEM OF COMPARTMENTS

A continuous time stochastic model for an irreversible system of kcompartments with nonhomogeneous Poisson inputs is considered. The model allows an arbitrary residence time for each of the compartments. Individuals may enter the system through any compartment and may depart from the system via any compartment.     

The two-stage δ-corrected Kolmogorov-Smirnov test

The delta-corrected Kolmogorov-Smirnov test has been shown to be uniformly more powerful than the classical Kolmogorov-Smirnov test for small to moderate sample sizes. However, the delta-corrected test consists of two tests, leading to a slight inflation of the experimentwise type I error rate. The critical values of the delta-corrected test are adjusted to take into account the two-stage nature of the test, ensuring an experimentwise error rate at the nominal level. A power study confirms that the resulting so-called two-stage delta-corrected test is uniformly more powerful than the classical Kolmogorov-Smirnov test, with power improvements of up to 46 percentage points.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

AN ALGORITHM FOR THE DESIGN OF 2? FACTORIAL EXPERIMENTS ON CONTINUOUS PROCESSES

Saunders & Eccleston (1992) presented an approach to the design of 2-level factorial experiments for continuous processes. It determined sets of contrasts between the observations that could be well estimated, and then selected a design so that those contrasts estimated the parameters of interest. This paper shows that a well-estimated contrast must have a large number of changes of sign or level, and also be ‘paired’ in a particular sense. It develops an algorithm which constructs designs that must have a large number of changes of sign, evenly spread among the contrasts and optimal or near optimal. When such designs exist they are often preferable to those produced by the reverse foldover algorithm of Cheng & Steinberg (1991).     

AN ITERATIVE METHOD FOR ESTIMATION OF PARAMETERS IN A CLUSTERED SAMPLING MODEL

This paper considers an iterative method for obtaining maximum likelihood estimates for a contingency table derived from a clustered sampling model. Comparisons are made with other methods proposed in the literature.     

Bayesian optimization analysis with ML-II ε-contaminated prior

In this paper we derive the predictive density function of a future observation when prior distribution for unknown mean of a normal population is a Type-II maximum likelihood ε-contaminated prior. The derived predictive distribution is applied to the problem of optimization of a regression nature in the decisive prediction framework.     

A RESTRICTED MAXIMUM LIKELIHOOD PROCEDURE FOR ESTIMATING THE VARIANCE FUNCTION OF AN IMMUNOASSAY

Restricted maximum likelihood (REML) is a procedure for estimating a variance function in a heteroscedastic linear model. Although REML has been extended to non-linear models, the case in which the data are dominated by replicated observations with unknown values of the independent variable of interest, such as the concentration of a substance in a blood sample, has not been considered. We derive a REML procedure for an immunoassay and show that the resulting estimator is superior to those currently being used. Some interesting properties of the REML estimator are derived, and its relationship to other estimators is discussed.     

THE OPTIMAL CRITICAL VALUE OF A PRE-TEST FOR AN INEQUALITY RESTRICTION IN A MIS-SPECIFIED REGRESSION MODEL

Optimal critical values are derived for a pre-test of an inequality restriction in a model where relevant regressors are unwittingly omitted. The criterion adopted is either that of the minimum average relative risk, or that of the mini-max regret. The latter approach yields an optimal critical value which is sensitive to the degree of model mis-specification, while the former criterion always leads to the choice of the unrestricted estimator.     

EXISTENCE OF A SEN—YATES—GRUNDY FORM FOR A DISPERSION

The condition of fixed sample size is essential for the existence of a Sen-Yates-Grundy form variance and its design unbiased estimator, in the problem of estimating the mean, variance and covariance of a finite population.     

ESTIMATION OF QUANTILES FOR A FRÉCHET-TYPE DISTRIBUTION

Estimation of high quantiles of a distribution in the domain of attraction of the Fréchet distribution is based on the extremal distribution of the k largest order statistics. The problem is treated by a local maximum likelihood method on a three parameter model. The estimators are shown to be asymptotically consistent for the whole range of the tail index parameter.