Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

A comparison of estimators for the proportion in the tail of a normal distribution

Three estimators of the proportion in a tail of the normal distribution are compared using the criteria of mean squared error and mean absolute error. The estimators that we compare are the maximum likelihood estimator, the minimum variance unbiased estimator, and an intuitive estimator that is frequently used in practice. The intuitive estimator is similar to the MLE but uses the usual unbiased estimator of σ² rather than the MLE of σ². We show that the intuitive estimator has low efficiency, and for this reason it is not recommended. For very smallp and for largep the MVUE has the highest efficiency. The MLE is best for moderate values ofp.     

On constructing T 2 control charts for retrospective examination

In this paper we consider the issue of constructing retrospective T ² control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt's control limit and to the standard x ² control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt's control limit performs better that the Bonferroni-adjusted x ² control limit. Furthermore, it appears that the Bonferroni-adjusted Alt's control limit is more than adequate for controlling the overall false alarm probability at a specified value.     

Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology

Taylor's law, which originated in ecology, states that, in sets of measurements of population density, the sample variance is approximately proportional to a power of the sample mean. Taylor's law has been verified for many species ranging from bacterial to human. Here, we show that the variance V(x) and the mean M(x) of the primes not exceeding a real number x obey Taylor's law asymptotically for large x. Specifically, V(x) ～ (1/3)(M(x))² as x → ∞. This apparently new fact about primes shows that Taylor's law may arise in the absence of biological processes, and that patterns discovered in biological data can suggest novel questions in number theory. If the Hardy-Littlewood twin primes conjecture is true, then the identical Taylor's law holds also for twin primes. Taylor's law holds in both instances because the primes (and the twin primes, given the conjecture) not exceeding x are asymptotically uniformly distributed on the integers in [2, x]. Hence, asymptotically M(x) ～ x/2, V(x) ～ x²/12. Higher-order moments of the primes (twin primes) not exceeding x satisfy a generalized Taylor's law. The 11,078,937 primes and 813,371 twin primes not exceeding 2 × 10⁸ illustrate these results.     

Exact Statistical Tests for Heterogeneity of Frequencies Based on Extreme Values

Sophisticated statistical analyses of incidence frequencies are often required for various epidemiologic and biomedical applications. Among the most commonly applied methods is the Pearson's χ² test, which is structured to detect non specific anomalous patterns of frequencies and is useful for testing the significance for incidence heterogeneity. However, the Pearson's χ² test is not efficient for assessing the significance of frequency in a particular cell (or class) to be attributed to chance alone. We recently developed statistical tests for detecting temporal anomalies of disease cases based on maximum and minimum frequencies; these tests are actually designed to test of significance for a particular high or low frequency. The purpose of this article is to demonstrate merits of these tests in epidemiologic and biomedical studies. We show that our proposed methods are more sensitive and powerful for testing extreme cell counts than is the Pearson's χ² test. This feature could provide important and valuable information in epidemiologic or biomeidcal studies. We elucidated and illustrated the differences in sensitivity among our tests and the Pearson's χ² test by analyzing a data set of Langerhans cell histiocytosis cases and its hypothetical sets. We also computed and compared the statistical power of these methods using various sets of cell numbers and alternative frequencies. The investigation of statistical sensitivity and power presented in this work will provide investigators with useful guidelines for selecting the appropriate tests for their studies.     

Inadmissibility of the uniformly minimum variance unbiased estimator of the inverse gaussian variance

The uniformly minimum variance unbiased estimator (UMVUE) of the variance of the inverse Gaussian distribution is shown to be inadmissible in terms of the mean squared error, and a dominating estimator is given. A dominating estimator to the maximum likelihood estimator (MLE) of the variance and estimators dominating the MLE's and the UMVUE's of other parameters are also given.     

Estimating the error variance after a pre-test for an inequality restriction on the coefficients

Estimation of the regression error variance after a preliminary test of an inequality constraint on the coefficient vector is considered. We derive the exact finite sample risk of several inequality restricted and pre-test estimators of σ². These estimators are associated with the maximum likelihood, least squares and minimum mean squared error component estimators. Optimal critical values for the pre-test according to a mini-max regret criterion are numerically calculated. Furthermore, we examine the robustness of the optimal choice of critical values and the risk properties of the estimators of σ² to model mis-specification through the exclusion of relevant regressors.     

Improvement over variance estimation using auxiliary information in sample surveys

This paper addresses the problem of estimating the population variance S²_y of the study variable y using auxiliary information in sample surveys. We have suggested a class of estimators of the population variance S²_y of the study variable y when the population variance S²_x of the auxiliary variable x is known. Asymptotic expressions of bias and mean squared error (MSE) of the proposed class of estimators have been obtained. Asymptotic optimum estimators in the proposed class of estimators have also been identified along with its MSE formula. A comparison has been provided. We have further provided the double sampling version of the proposed class of estimators. The properties of the double sampling version have been provided under large sample approximation. In addition, we support the present study with aid of a numerical illustration.     

A Note on the Predictors of Differenced Sequences

In this note we consider the problems of optimal linear prediction (o.l.p.) and the minimum mean squared error prediction (m.m.s.e.p.) of a sequence X_t, which fits to a stationary and invertible ARMA model through the filter (1 - B^s)^d X_t= Y_t. It is shown that these two predictors are not identical in general from the theoretical point of view. Permitting the degree of differencing d to take any real value, a set of conditions for these commonly applied prediction formulas to be identical is given.     

A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

Statistical Discrimination Analysis Using the Maximum Function

The maximum of k functions defined on R ⁿ , n ≥ 1, by f _max (x) = max{f ₁ (x),…, f _k (x)}, ? x ? R ⁿ , can have important roles in Statistics, particularly in Classification. Through its relation with the Bayes error, which is the reference error in classification, it can serve to compute numerical bounds for errors in other classification schemes. It can also serve to define the joint L¹-distance between more than two densities, which, in turn, will serve as a useful tool in Classification and Cluster Analyses. It has a vast potential application in digital image processing too. Finally, its versatile role can be seen in several numerical examples, related to the analysis of Fisher's classical iris data in multidimensional spaces.     

Higher than second order approximations for sequential point estimation by a two-stage procedure

We consider the sequential point estimation problem of the mean of a normal distribution N(μ, σ²) when the loss function is squared error plus linear cost. It is shown that a two-stage procedure has the asymptotic efficiency of which the order is higher than second order, provided the standard deviation has a known lower bound. We also give a higher than second-order approximation to the risk.     

Efficient computation of the noncentral x2 distribution

In this article we compare and contrast finite algorithms for computing the noncentral x² distribution function for odd or even degrees of freedom with the algorithms proposed recently by Ashour & Abdel-Samad (1990). We also obtain an alternative error bound for Ruben's (1974a) algorithm for even degrees of freedom and analyze the rate of convergence of two common infinite series representations for computing the cdf     

A new stochastic mixed Liu estimator in linear regression model

Abstract

To overcome multicollinearity, a new stochastic mixed Liu estimator is presented and its efficiency is considered. We also compare the proposed estimators in the sense of matrix mean squared error criteria. Finally a numerical example and a simulation study are given to show the performance of the estimators.     

Analytic connections between a double design and its original design in terms of different optimality criteria

In recent years, there has been increasing interest in the study of double designs. Various popular optimality criteria have been proposed from different principles for design construction and comparison, such as E(s²), generalized minimum aberration (GMA), minimum moment aberration (MMA), and minimum projection uniformity (MPU). In this article, these criteria are reviewed, and analytic connections between a double design and its original design in terms of these criteria are investigated. These connections are suitable for general original two-level factorial design, whether regular or non regular. In addition, these results provide strong insight into the relationship between double design and original design from different viewpoints.     

Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.     

Adaptive Hotelling's T 2 Control Charts with Run Rules

This study investigates the statistical properties of the adaptive Hotelling's T ² charts with run rules in which the sample size and sampling interval are allowed to vary according on the current and past sampling points. The adaptive charts include variable sample size (VSS), variable sampling interval (VSI), and variable sample size and sampling interval (VSSI) charts. The adaptive Hotelling's T ² charts with run rules are compared with the fixed sampling rate Hotelling's T ² chart with run rules. The numerical results show that the VSS, VSI, and VSSI features improve the performance of the Hotelling's T ² chart with run rules.     

A comparison of estimators for the mean in the inverse gaussian distribution with a known coefficient of variation

In this paper, we discuss an estimation problem of the mean in the inverse Gaussian distribution with a known coefficient of variation. Two types of linear estimators for the mean, the linear minimum variance unbiased estimator and the linear minimum mean squared error estimator, are constructed by using the squared error loss function and their properties are examined. It is observed that, for small samples the performance of the proposed estimators is better than that of the maximum likelihood estimator, when the coefficient of variation is large.     

Bias Reduction for the Maximum Likelihood Estimator of the Parameters of the Generalized Rayleigh Family of Distributions

We derive analytic expressions for the biases, to O(n^{? 1}), of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap.     

The Tale of Cochran's Rule: My Contingency Table has so Many Expected Values Smaller than 5, What Am I to Do?

In an informal way, some dilemmas in connection with hypothesis testing in contingency tables are discussed. The body of the article concerns the numerical evaluation of Cochran's Rule about the minimum expected value in r × c contingency tables with fixed margins when testing independence with Pearson's X² statistic using the χ² distribution.