Some Multiple Decision Procedures for Comparing Several Population Proportions

In this article, comparison of several population proportions using multiple decision approach is studied. The probability of the order of the sample proportions matching with the order of the population proportions is being controlled. A related multiple comparison procedure with a control is also discussed. For ranking the proportions in multinomial distribution, the simultaneous confidence interval is constructed and used for the ranking. Some examples are used to illustrate the multiple decision procedures discussed in this paper.     

On the Critical Value Behaviour of Multiple Decision Procedures

In this paper we investigate the asymptotic critical value behaviour of certain multiple decision procedures as e.g. simultaneous confidence intervals and simultaneous as well as stepwise multiple test procedures. Supposing that n hypotheses or parameters of interest are under consideration we investigate the critical value behaviour when n increases. More specifically, we answer e.g. the question by which amount the lengths of confidence intervals increase when an additional parameter is added to the statistical analysis. Furthermore, critical values of different multiple decision procedures as for instance step-down and step-up procedures will be compared. Some general theoretic results are derived and applied for various distributions.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

A Multiple Three-Decision Procedures for Comparing Several Treatments with a Control Under Heteroscedasticity

In this article, the design-oriented two-stage multiple three-decision procedure is proposed to classify a set of normal populations with respect to a control under heteroscedasticity. The statistical tables of percentage points and the power-related design constants, to implement our new two-stage procedure, are given. Sometimes when the sample for the second stage is not available, the one-stage data analysis procedure is proposed. Classifying a treatment better than control when it is actually worse (and vice versa) is known as type III error. Both the two-stage and one-stage procedures control the type III error rate at a specified level. The relationship between the two-stage and one-stage procedures is discussed. Finally, the application of the proposed procedures is illustrated with an example.     

A Simpler Approximation for Areas under the Standard Normal Curve

A simple approximation for areas under the standard normal curve is presented that is suitable for use when tables and/or calculators are not available or not permitted.     

On the Normal Inverse Gaussian Stochastic Volatility Model

In this article, the normal inverse Gaussian stochastic volatility model of Barndorff-Nielsen is extended. The resulting model has a more flexible lag structure than the original one. In addition, the second-and fourth-order moments, important properties of a volatility model, are derived. The model can be considered either as a generalized autoregressive conditional heteroscedasticity model with nonnormal errors or as a stochastic volatility model with an inverse Gaussian distributed conditional variance. A simulation study is made to investigate the performance of the maximum likelihood estimator of the model. Finally, the model is applied to stock returns and exchange-rate movements. Its fit to two stylized facts and its forecasting performance is compared with two other volatility models.     

On the Spectral Decomposition in Normal Discriminant Analysis

This article enlarges the covariance configurations, on which the classical linear discriminant analysis is based, by considering the four models arising from the spectral decomposition when eigenvalues and/or eigenvectors matrices are allowed to vary or not between groups. As in the classical approach, the assessment of these configurations is accomplished via a test on the training set. The discrimination rule is then built upon the configuration provided by the test, considering or not the unlabeled data. Numerical experiments, on simulated and real data, have been performed to evaluate the gain of our proposal with respect to the linear discriminant analysis.     

Some Geometric Methods for Constructing Decision Criteria Based On Two-Dimensional Parameters

This paper reviews two types of geometric methods proposed in recent years for defining statistical decision rules based on 2-dimensional parameters that characterize treatment effect in a medical setting. A common example is that of making decisions, such as comparing treatments or selecting a best dose, based on both the probability of efficacy and the probability toxicity. In most applications, the 2-dimensional parameter is defined in terms of a model parameter of higher dimension including effects of treatment and possibly covariates. Each method uses a geometric construct in the 2-dimensional parameter space based on a set of elicited parameter pairs as a basis for defining decision rules. The first construct is a family of contours that partitions the parameter space, with the contours constructed so that all parameter pairs on a given contour are equally desirable. The partition is used to define statistical decision rules that discriminate between parameter pairs in term of their desirabilities. The second construct is a convex 2-dimensional set of desirable parameter pairs, with decisions based on posterior probabilities of this set for given combinations of treatments and covariates under a Bayesian formulation. A general framework for all of these methods is provided, and each method is illustrated by one or more applications.     

The Discrete Normal Distribution

Abstract

The normal distribution has been playing a key role in stochastic modeling for a continuous setup. But its distribution function does not have an analytical form. Moreover, the distribution of a complex multicomponent system made of normal variates occasionally poses derivational difficulties. It may be worth exploring the possibility of developing a discrete version of the normal distribution so that the same can be used for modeling discrete data. Keeping in mind the above requirement we propose a discrete version of the continuous normal distribution. The Increasing Failure Rate property in the discrete setup has been ensured. Characterization results have also been made to establish a direct link between the discrete normal distribution and its continuous counterpart. The corresponding concept of a discrete approximator for the normal deviate has been suggested. An application of the discrete normal distributions for evaluating the reliability of complex systems has been elaborated as an alternative to simulation methods.     

Advantages of the Closed Testing Method in Multiple Comparisons Procedures

This article deals with the use of the Closed Testing approach in Multiple Comparison Procedures (MCPs). MCPs occur when after rejection of a global hypothesis of no effect of a given treatment, a set of pairwise comparisons between levels of that treatment are performed in order to find out significant differences between levels. Given a set of partial hypotheses, such as the pairwise comparisons of an MCP, the Closed Testing approach concentrates on testing the family of all the non empty intersections of these partial hypotheses. The results of our simulation study highlight the advantages of closed testing methods and prove that they are more powerful than other classic MCPs controlling the FWER.     

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.     

Multiple comparisons of exponential location parameters with a control based on doubly censored sample under heteroscedasticity

In this article, a one-sample procedure for multiple comparisons of exponential location parameters with a control under heteroscedasticity is proposed. The observations are obtained by doubly censored samples. A one-sided and two-sided confidence intervals are used to perform such multiple comparisons. Statistical tables of critical values and an example of comparing four drugs in treating leukemia are provided.     

On Simultaneous Estimation of the Number of Signals and Frequencies of Multiple Sinusoids When Some Observations Are Missing

In an earlier article (Bai et al., 1999 Bai , Z. D. , Rao , C. R. , Wu , Y. H. , Zen , M. M. , Zhao , L. C. ( 1999 ). The simultaneous estimation of the number of signals and frequencies of multiple sinusods when some observations are missing: I. Asymptotics . Proc. Natl. Acad. Sci. 96 : 11,106 – 11,110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), the problem of simultaneous estimation of the number of signals and frequencies of multiple sinusoids is considered in the case that some observations are missing. The number of signals is estimated with an information theoretic criterion and the frequencies are estimated with eigenvariation linear prediction. Asymptotic properties of the procedure are investigated but the Monte Carlo simulation is not performed. In this article, a slightly different but scale invariant criterion for detection is proposed and the estimation of frequencies remains the same. Asymptotic properties of this new procedure are provided. Monte Carlo Simulation for both procedures is carried out. Furthermore, comparison on the real signals is also given.     

Some comments on two-stage selection procedures

In this paper we study the procedures of Dudewicz and Dalal ( 1975 ), and the modifications suggested by Rinott ( 1978 ), for selecting the largest mean from k normal populations with unknown variances. We look at the case k = 2 in detail, because there is an optimal allocation scheme here. We do not really allocate the total number of samples into two groups, but we estimate this optimal sample size, as well, so as to guarantee the probability of correct selection (written as P(CS)) at least P^?, 1/2 < P^? < 1 . We prove that the procedure of Rinott is “asymptotically in-efficient” (to be defined below) in the sense of Chow and Robbins ( 1965 ) for any k  2. Next, we propose two-stage procedures having all the properties of Rinott's procedure, together with the property of “asymptotic efficiency” - which is highly desirable.     

Selecting the best normal population better than a standard under unequal variances

In this article we consider a problem of selecting the best normal population that is better than a standard when the variances are unequal. Single-stage selection procedures are proposed when the variances are known. Wilcox (1984) and Taneja and Dudewicz (1992) proposed two-stage selection procedures when the variances are unknown. In addition to these procedures, we propose a two-stage selection procedure based on the method of Lam (1988). Comparisons are made between these selection procedures in terms of the sample sizes.     

Optimal Binary Prediction for Group Decision Making

We address the problem of optimally forecasting a binary variable for a heterogeneous group of decision makers facing various (binary) decision problems that are tied together only by the unknown outcome. A typical example is a weather forecaster who needs to estimate the probability of rain tomorrow and then report it to the public. Given a conditional probability model for the outcome of interest (e.g., logit or probit), we introduce the idea of maximum welfare estimation and derive conditions under which traditional estimators, such as maximum likelihood or (nonlinear) least squares, are asymptotically socially optimal even when the underlying model is misspecified.     

Simultaneous Confidence Intervals for Monotone Contrasts of Normal Means

The one-way ANOVA model is considered. The variances are assumed to be either equal but unknown or unequal and unknown. Two-stage procedures for generating simultaneous confidence intervals (SCI) for the class of monotone contrasts of the means are presented. The intervals are of fixed length and independent of the unknown variances.     

Comparison of Two Distribution-Free Procedures for Multiple Comparisons With a Control

Chakraborti and Desu (1988) presented a distribution-free procedure for testing that k (≥1) distributions are equal to a control distribution. They compared their procedure, a generalization of the test proposed by Mathisen (1943), to the procedure proposed by Slivka (1970). They asserted that their procedure has shorter expected duration than Slivka's procedure in life-testing experiments where observations become available in an ordered manner. Here it is proven that, in fact, Slivka's procedure has shorter duration in such circumstances. Normal approximations are presented which indicate that their procedure requires a smaller sample size to guarantee a specified power for Lehmann alternatives and proportional hazard alternatives when all observations are to be observed.     

Procedures for Selecting Good Exponential Populations

Suppose exponential populations π _i with parameters (μ_i , σ_i ) (i = 1,2,…,k) are given. This article discusses how to select “good” populations in the sense of [Lam (1986 Lam, K. 1986. A new procedure for selecting good populations. Biometrika, 73(1): 201–206. [Crossref], [Web of Science ®] , [Google Scholar]). A new procedure for selecting good populations. Biometrika 73(1):201–206]. Depending on whether the σ _i 's are known or unknown, several one-stage and a two-stage procedure of selection are proposed. The two-stage procedure can be replaced by a one-stage procedure if the second-stage sample is proved intangible. An attracting feature of these procedures is that they need no new statistical tables to implement.