COMBINING TWO UNBIASED ESTIMATORS OF A COMMON MEAN OF TWO NORMAL POPULATIONS

Consider estimating the common mean μ of two normal populations. Let () and () be the means and variances of two independent samples obtained from these populations. We give sufficient conditions for the choices of α₁ and α₂ in the unbiased estimator     

ESTIMATING THE COMMON MEAN OF A BIVARIATE NORMAL POPULATION

For estimating the common mean of a bivariate normal distribution, Krishnamoorthy & Rohatgi (1989) proposed some estimators which dominate the maximum likelihood estimator in a large region of the parameter space. We consider some modifications of these estimators and study their risk performance.     

A MODIFICATION OF THE SEQUENTIAL PROBABILITY RATIO TEST FOR TESTING A NORMAL MEAN

A modification of the sequential probability ratio test is proposed in which Wald's parallel boundaries are broken at some preassigned point of the sample number axis and Anderson's converging boundaries are used prior to that. Read's partial sequential probability ratio test can be considered as a special case of the proposed procedure. As far as 'the maximum average sample number reducing property is concerned, the procedure is as good as Anderson's modified sequential probability ratio test.     

A PROCEDURE FOR TESTING THE MEAN OF A NORMAL DISTRIBUTION1

A sequential procedure for comparing a null hypothesis concerning the unknown mean of a normal distribution whose variance is known, against a two-sided alternative hypothesis is presented. This procedure incorporates additional safeguards as required by the practicing experimenter. Furthermore, approximate formulae for the O.C. function and the A.S.N. function are derived.     

ESTIMATING THE ANGLE BETWEEN THE MEAN DIRECTIONS OF TWO SPHERICAL DISTRIBUTIONS

This paper considers the problem of calculating a confidence interval for the angular difference between the mean directions of two spherical random variables with rotationally symmetric unimodal distributions. For large sample sizes, it is shown that the asymptotic distribution of 1 – cos α, where α is the sample angular difference, is approximately exponential if the true difference is zero, and approximately normal for a ‘large’ true difference; a scaled beta approximation is determined for the general case. For small sample sizes, a bootstrap approach is recommended. The results are applied to two sets of palaeomagnetic data.     

PITMAN NEARNESS COMPARISONS OF ESTIMATES OF TWO ORDERED NORMAL MEANS

Maximum likelihood estimates of ordered means of two normal distributions having common variance have been shown to be better than the usual maximum likelihood estimates (i.e. corresponding sample means) with respect to Pitman Nearness criterion. The maximum likelihood estimate of common variance taking into consideration the order restriction of the means is shown to have smaller mean square error than the unrestricted maximum likelihood estimate of the common variance. These two estimators have also been compared with respect to Pitman Nearness criterion.     

CHANCE OF MISCLASSIFICATION IN THE CASE OF SEVERAL NORMAL POPULATIONS

When a new observation is to be classified into one of several multivariate normal populations with different means and the same covariance matrix, by Rao's method of scoring, the chance of misclassification is expressed as a multiple integral. This paper gives a practical method of obtaining reasonable approximations to this integral by using tables prepared by Gibbons, Olkin & Sobel (1977) for a different task.     

MINIMAX ESTIMATION OF A MULTIVARIATE NORMAL MEAN UNDER A CONVEX LOSS FUNCTION

Let X = (X₁, - X_p)prime; ˜ N_p (μ, Σ) where μ= (μ₁, -, μ_p)' and Σ= diag (Σ²₁, -, Σ²_p) are both unknown and p3. Let (n_i - 2) w_i/Σ²_i! X²_ni, independent. of w_i (I ≠ j = 1, -, p). Assume that (w₁, -, w_p) and X are independent. Define W = diag (w₁, -, w_p) and ¶ X ¶²_w= X'W^-1Q^-1W^-1X where Q = diag (q₁, -,n q_p), q_i > 0, i = 1, -, p. In this paper, the minimax estimator of Berger & Bock (1976), given by δ (X, W) = [I_p - r(X, W) ¶ X ¶^-2_w Q^-1W^-1] X, is shown to be minimax relative to the convex loss (δ - μ)'[αQ + (1 - α) Σ^-1] δ - μ)/C, where C =α tr (Σ) + (1 - α)p and 0 α 1, under certain conditions on r(X, W). This generalizes the above mentioned result of Berger & Bock.     

ON THE COMMON SCALE PARAMETER OF SEVERAL PARETO POPULATIONS IN CENSORED SAMPLES

The problem of estimation of an unknown common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters in censored samples is considered. A new class of estimators which includes both the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) is proposed and examined under a squared error loss.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

ON THE INDEPENDENCE OF THE SAMPLE MEAN AND TRANSLATION-INVARIANT STATISTICS FOR MATRIX NORMAL DISTRIBUTIONS

We present an explicit characterization of the joint dependency structure of an n×p matrix normal random matrix such that the p-dimensional sample mean vector is independent of all translation invariant statistics.     

INFERENCE ABOUT THE MEAN OF A POISSON DISTRIBUTION IN THE PRESENCE OF A NUISANCE PARAMETER

In this note we examine the problem of estimating the mean of a Poisson distribution when a nuisance parameter is present. Using a condition of Cox (1958) about ancillarity in the presence of a nuisance parameter, we justify that inference about the parameter should be carried out using the conditional distribution given the appropriate ancillary statistics. A small simulation study has been done to compare the performance of the conditional likelihood approach and the standard likelihood approach.     

TESTING THE AGREEMENT OF TWO QUANTITATIVE ASSAYS IN INDIVIDUAL MEANS

ABSTRACT

Existing approaches for the statistical evaluation of the agreement of two quantitative assays in terms of individual means are either based on a linear model and some stringent assumptions or comparisons of averages of individual means. Furthermore, the related statistical tests for some of these approaches are not valid in the sense that the sizes of these tests are not exactly the same as the nominal size even asymptotically. In this paper we propose a new method, which produces exact statistical tests that are easy to compute. When independent replicates are available, the proposed method requires very little or no assumption on the individual error variances. Simulation results show that the proposed tests perform better than some existing tests. Some examples are presented for illustration.     

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.     

MULTIVARIATE TESTS OF NORMAL MEAN VECTORS WITH RESTRICTED ALTERNATIVES

ABSTRACT

In this paper, we consider tests for the hypothesis that the mean vector is zero against one-sided alternatives when the observation vectors are independently and identically distributed as normal with unknown covariance matrix. The exact null-distribution of the tests is derived. The tests generalize the centre-direction test proposed by Tang et al.[1] Tang, D.-I., Gnecco, C. and Geller, N. 1989. An Approximate Likelihood Ratio Test for a Normal Mean Vector with Nonnegative Components with Application to Clinical Trials. Biometrika, 76: 577–583. [Crossref], [Web of Science ®] , [Google Scholar] for known covariance. In addition, the modification is order- and scale-invariant. Power comparisons with some other tests are presented. It can be shown that the null distribution of the test statistic holds for data arising from any elliptical distribution, not just the normal distribution.     

ON THE CHANGE POINT OF THE MEAN RESIDUAL LIFE OF SERIES AND PARALLEL SYSTEMS

This paper considers the mean residual life in series and parallel systems with independent and identically distributed components and obtains relationships between the change points of the mean residual life of systems and that of their components. Compared with the change point for single components, should it exists, the change point for a series system occurs later. For a parallel system, however, the change point is located before that for the components, if it exists at all. Moreover, for both types of systems, the distance between the change points of the mean residual life for systems and for components increases with the number of components. These results are helpful in the determination of optimal burn-in time and related decision making in reliability analysis.     

ON ESTIMATING THE MEAN OF A FINITE POPULATION SAMPLED ON TWO OCCASIONS WITH VARYING PROBABILITIES

The problem considered is one of estimating the current value of a finite population mean from two samples selected with varying probabilities on the current and a previous occasion. Following the generalized least squares approach of Gurney and Daly (1965) we extend the works of Sin& (1968) and others to obtain two sampling strategies which are respectively more efficient than the ones suggested earlier by Raj (1965) and Chotai (1974).     

PERMUTATION TEST FOR EQUALITY OF CORRELATION COEFFICIENTS IN TWO POPULATIONS

ABSTRACT

The purpose of this paper is to investigate the permutation tests for equality of correlation coefficients among two independent populations. We discuss how to apply permutation test to this problem and its asymptotic suitability. We also show some simulation studies and an example of the Iris data.