Note on an extension of rational bounds for the t-tail area to arbitrary degrees of freedom

The bounds of Soms (1980a) for the tail area of the t-distribution with integral degrees of freedom are extended to arbitrary positive degrees of freedom, Comparisons are made with the bounds of Shenton and Carpenter (1965) and some numerical examples are provided.     

Bounds on the expected maximum

Let (X₁,X₂, …,X_n) be jointly distributed random variables. Define X_n:n = max(X₁,X₂, …,X_n).Bounds on E(X_n:n), obtained by putting constraints on the distributions and/or dependence structure of the X_i's, are surveyed.     

Bounds for the Bayes Error in Classification: A Bayesian Approach Using Discriminant Analysis

We study two of the classical bounds for the Bayes error P _e , Lissack and Fu’s separability bounds and Bhattacharyya’s bounds, in the classification of an observation into one of the two determined distributions, under the hypothesis that the prior probability χ itself has a probability distribution. The effectiveness of this distribution can be measured in terms of the ratio of two mean values. On the other hand, a discriminant analysis-based optimal classification rule allows us to derive the posterior distribution of χ, together with the related posterior bounds of P _e . Research partially supported by NSERC grant A 9249 (Canada). The authors wish to thank two referees, for their very pertinent comments and suggestions, that have helped to improve the quality and the presentation of the paper, and we have, whenever possible, addressed their concerns.     

边限检验理论及几点讨论

检验经济变量之间长期关系的协整技术要求变量是同阶单整的,这不可避免地涉及一定程度的预检验问题,而预检验问题会增加变量间长期关系分析的不确定性。当不能确定变量的单整类型时,边限检验理论提出了一个可以直接检验一个变量和一组解释变量之间长期关系的新方法。在介绍了边限检验方法中基本的VAR模型和假设及边限检验方法中用到的重要统计量——Wald统计量和T统计量及它们各自的渐近分布形式后,说明了边限检验理论在理论和实际运用当中需要注意的几个问题,最后通过实例分析说明了边限检验理论的运用。     

Tables of confidence limits on the tail area of the normal distribution

Tables are given of confidence limits on tail areas, γ, of the normal distribution, where γ = P{Y ≥ L}, and where L is a given number, and Y is normally distributed with unknown mean, μ, and unknown variance, σ².     

Bounds for the variance of the graybill-deal estimator of the common mean of two normal distributions

In this note we derive sharp lower and upper bounds for the variance of the Graybill-Deal estimator of the common mean of two normal distributions with unknown variances when the sample sizes are not necessarily equal. We also derive similar bounds for the variance of the Brown-Cohen (1974) T _a(1) class of unbiased es-timators to which the Graybill-Deal estimator belongs. Further, we illustrate the sharpness of the bounds by numerical computations in the case of the Graybill-Deal estimator.     

Berry-Esseen Bounds for Random Index Non Linear Statistics via Stein's Method

Under the second moment condition, we obtain Berry-Esseen bounds for random index non linear statistics by using a technique discussed in Chen and Shao (2007 Chen, L. H.Y., Shao, Q.-M. (2007). Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13(2):581–599.[Crossref], [Web of Science ®] , [Google Scholar]). A concept in this article is to approximate any random index non-linear statistic by a random index linear statistic. The bounds for random sums of independent random variables are also provided. Applications are the bounds for random U-statistics and random sums of the present values in investment analysis.     

Stochastic Bounds with a Low Rank Decomposition

We investigate how we can bound a discrete time Markov chain (DTMC) by a stochastic matrix with a low rank decomposition. In the first part of the article, we show the links with previous results for matrices with a decomposition of size 1 or 2. Then we show how the complexity of the analysis for steady-state and transient distributions can be simplified when we take into account the decomposition. Finally, we show how we can obtain a monotone stochastic upper bound with a low rank decomposition.     

Bounds on the mean of the maximum of a number of normal random variables

A simple method producing lower and upper bounds on E max(X₁,...,X_n) is presented under assumption that the X_i's are independent normal random variables. Furthermore the upper bounds are determined when the X_i's are normal and positively correlated     

On the distributions of the estimated process capability indices

The exact distributions of the estimated process capability indices are presented and their means, variances, and mean-squared errors are given. The basic assumption is that the process measurements are taken from a normal distribution. Theresults in this article are useful in evaluating process capability.     

Controlled Direct and Mediated Effects: Definition,Identification and Bounds

Abstract. Results are given which provide bounds for controlled direct effects when nounmeasured confounding assumptions required for the identification of these effects do not hold. Previous results concerning bounds for controlled direct effects rely on monotonicity relationships between the treatment, mediator and the outcome themselves; the results presented in this article instead assume that monotonicity relationships hold between the unmeasured confounding variable or variables and the treatment, mediator and outcome. Whereas prior results give bounds that contain the null hypothesis of no direct effect, the results presented here will in many instances yield bounds that do not contain the null hypothesis of no direct effect. For contexts in which a set of variables intercepts all paths between a treatment and an outcome, it is possible to provide a definition for a controlled mediated effect. We discuss the identification of these controlled mediated effects; the bounds for controlled direct effects are applicable also to controlled mediated effects. An example is given to illustrate how the results in the article can be used to draw inferences about direct and mediated effects in the presence of unmeasured confounding variables.     

Bounds for moments through a general orthogonal expansion in a pre-hilbert space-I

Bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube.     

A partitioning of the power in the one sample Student t-test

The power of Student's t-test is partitioned to determine* the proportion of time the sample standard deviation is too small or the mean is misestimated given that rejection of the hypothesis under test has occurred     

Bounds for moments through a general orthogonal expansion in a pre-hilbert space-II

In an earlier article [Canad. J. Statist., Vol, 3, No. 1, 1975, 13–34] bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube. These results are extended to symmetric cases in this article.     

A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

Normal quantile estimation based on minimizing the error in predicted distribution functions

Estimators of the form [Xbar] + kS for estimating the p quantile of a normal distribution are studied when k is chosen to either minimize the mean square error in the predicted distribution function or to make the predicted distribution function unbiased for p. Here, [Xbar] and S are the usual sample mean and standard deviation, respectively, and the predicted distribution function is the true (normal) distribution function evaluated at the estimated quantile.

These k values are presented for various sample sizes and values of p, and application to warranty determination is discussed.     

On the procedure of hodges and lehmann for testing composite hypotheses of the mean of a normal distribution with unknown variance

In 1954 Hodges and Lehmann gave a test procedure for testing the hypothesis that the mean of an identically independently normally distributed random sample with unknown variance is contained within a certain interval [μ1, μ2]. The test is similar on the boundary of the zero-hypothesis and superior in power to the composite t-test usually applied to this problem. However Hodges and Lehmann could prove the unbiasedness of their test only for the special case that the sample consists of two elements. From numerical computations they guessed that unbiasedness would be valid for arbitrary sample sizes. This question is discussed here and partially answered.     

Bounds for permutation arrays

A permutation array (P.A.) defined on an r-set of symbols V is a v×r array of rows each of which is a permutation of the symbols of V and such that any two distinct rows have at most (at least) λ common column entries. We list all known bounds for such arrays and make improvements in certain cases. We consider, at length, the case when every pair of distinct rows of the P.A. have precisely λ common column entries.     

On level and powee of anderson and hauck's peocedure for testing equivalence in compative bioavailability

The paper refers to a new procedure of Anderson and Hauck for the well-known problem of deciding whether two means of normal distributions with common but unknown variance differ within an admissible range. Analytical expressions for level and power of this method are derived and some numerical examples are given. A bound for the difference between real and no¬minal level is obtained which does not involve the variance.