Estimation in noncentral distributions

This paper investigates alternatives to MIU estimators in noncentral X ² and F distributions. Two directions are pursued. In the first, a general approach for uniformly improving on MVU estimators is described and illustrated. In the second, Bayesian, procedures are characterized and illustrated as well. This effort extends earlier work of Perlman and Rasmussen and of Neff and Strawderman.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T² _R, based on Tiku's (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T² _R is one the whole considerably more powerful than the prominenet Hotelling T² statistics. We also develop a robust statistic T² _D for testing that two multivariate distributions (skew or symmetric) are identical; T² _D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σ_k)f((x-μ_k)/σ_k) and the means and variances of these marginal distributions exist.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

On optimal tests for general interval-hypotheses

Optimal statistical tests, using the normality assumptions for general interval hypotheses including equivalence testing and testing for nonzero difference (or for non-unit) are presented. These tests are based on the decision theory for Polya Type distributions and are compared with usual confidence tests and with ’two one-sided tests’- procedures. A formal relationship between some optimal tests and the Anderson and Hauck procedure as well as a procedure recommended by Patel and Gupta is given. A new procedure for a generalisation of Student's test as well as for equivalence testing for thet-statistics is shown.     

Robustness of R-Chart to Non Normality

This study addresses the appropriate d ₃ values for constructing range control charts (R-charts) when the distributions of the processes are the uniform, triangular, exponential, and Erlang. Comparisons of the range charts are based on Type I error probabilities obtained using simulations. The results reveal that inappropriate use of the d ₃ values strongly affected the performance of the R-charts. Practitioners should be more careful in selecting suitable coefficients when using R-charts methods to process data. The distribution of the processes must be examined before the coefficients are chosen.     

A General Class of Distributions: Properties and Applications

ABSTRACT

In this article, we derive a general class of distributions and establish its relationship to χ² distribution. The proposed class includes normal, inverse Gaussian, lognormal, gamma, Rayleigh, and Maxwell distributions. Various statistical properties of the class are discussed. Some applications of the class are given.     

Multivariate control chart based on the highest possibility region

The T ² control chart is widely adopted in multivariate statistical process control. However, when dealing with asymmetrical or multimodal distributions using the traditional T ² control chart, some points with relatively high occurrence possibility might be excluded, while some points with relatively low occurrence possibility might be accepted. Motived by the thought of the highest posterior density credible region, we develop a control chart based on the highest possibility region to solve this problem. It is shown that the proposed multivariate control chart will not only meet the false alarm requirement, but also ensure that all the in-control points are with relatively high occurrence possibility. The advantages and effectiveness of the proposed control chart are demonstrated by some numerical examples in the end.     

The null distribution of multivariate kurtosis

The paper introduces a x²-approximation to multivariate kurtosis b_2,punder normality. It requires calculating the third moment of b_2,pwhich is obtained. We compare the approximation with simulated percentage points and the normal approximation, and find it to be adequate for p=l and 2. For p=3, the simple average of this estimate and the normal approximation is found to be generally superior to either approximation on its own. For p=4, the normal approximation is best for non-extreme values of ∝     

Goodness of fit for thei ordered categories discrete uniform distribution

Goodness of fit for thei ordered categories discrete uniform distribution can be carried out using Pearson's X² _pstatistic and its components. Applications of this technique are considered and comparisons made with recently suggested empirical uniform distribution     

A stepwise algorithm for selecting category boundaries for the chi-squared goodness-of-fit test

A stepwise algorithm for selecting categories for the chisquared goodness-of-fit test with completely specified continuous null and alternative distributions is described in this paper. The procedure's starting point is an initial partitioning of the sample space into a large number of categories. A second partition with one fewer category is constructed by combining two categories of the original partition. The procedure continues until there are only two categories; the partition in the sequence with the highest estimated power is the one chosen. For illustartive purposes, the performance of the algorithm is evaluated for several hypothesis tests of the from H₀: normal distribution vs. H₁: a specific mixed normal distribution. For each test considered, the partition identified by the algorithm was compared to several equiprobable partitions, including the equiprobable partition with the highest estimated power. In all cases but one, the algorithm identified a parttion with higher estimated power than the best equiprobable partition. Applciations of the procedure are discussed.     

Weighted L1-estimates for the First-order Bifurcating Autoregressive Model

We developed robust estimators that minimize a weighted L₁ norm for the first-order bifurcating autoregressive model. When all of the weights are fixed, our estimate is an L₁ estimate that is robust against outlying points in the response space and more efficient than the least squares estimate for heavy-tailed error distributions. When the weights are random and depend on the points in the factor space, the weighted L₁ estimate is robust against outlying points in the factor space. Simulated and artificial examples are presented. The behavior of the proposed estimate is modeled through a Monte Carlo study.     

The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

On the increasing failure rate criteria

We introduce the t₀ -increasing failure rate (IFR-t₀) class, where the failure rate at age x+t₀is greater then or equal to the failure rate at age x for x≥0. The dual class of t₀- decreasing failure rate (DER-t₀) is defined analogoualy.The relation between the IFR -t₀class and other classes of life distributions is studied. Preservation and nonpreservation properties of the IFR-t₀ and the DFR-t₀ classes under various reliablity operation are presented. The concet of stochastic comparison is utilized to cheracterized the IFR-t ₀ class and to suggest other classes of life distributions for aging.     

On the behaviour of some transforms of the sample correlation coefficient in samples from the bivariate t and the bivariate X2distribution

The present paper studies the normality of five transformations suggested in the literature to normalize the sample correlation coefficient. The parent populations are the bivariate t and the bivariate X ²The results in the previous work of Subrahmaniam and Gajjar are exploited to assess their performance. The density estimation procedure of Tarter and Kronmal is used to provide empiric support to the asymptotic results     

Some Related Minima Stability and Minima Infinite Divisibility of the General Multivariate Pareto Distributions

This article studies the minima stable property of the general multivariate Pareto distributions MP^(k)(I), MP^(k)(II), MP^(k)(III), MP^(k)(IV) which can be applied to characterize the MP^(k) distribution via its weighted ordered coordinates minima and marginal distribution. Also, the multivariate semi-Pareto distribution (denoted by MSP) is discerned in the class of geometric minima infinite divisible and geometric minima stable distributions. If the exponent measure is satisfied by some functional equation, then the geometric minima stable property can be used to characterize the MSP distribution. Finally, the finite sample minima infinite divisible property of the MP^(k)(I), (II), and (IV) distributions is also discussed.     

Identifying Variables Contributing to Outliers in Phase I

When a process is monitored with a T ² control chart in a Phase II setting, the MYT decomposition is a valuable diagnostic tool for interpreting signals in terms of the process variables. The decomposition splits a signaling T ² statistic into independent components that can be associated with either individual variables or groups of variables. Since these components are T ² statistics with known distributions, they can be used to determine which of the process variable(s) contribute to the signal. However, this procedure cannot be applied directly to Phase I since the distributions of the individual components are unknown. In this article, we develop the MYT decomposition procedure for a Phase I operation, when monitoring a random sample of individual observations and identifying outliers. We use a relationship between the T ² statistic in Phase I with the corresponding T ² statistic resulting when an observation is omitted from this sample to derive the distributions of these components and demonstrate the Phase I application of the MYT decomposition.     

Improved testing for the binomial distribution using chi-squared components with data-dependent cells

A power study suggests that a good test of fit analysis for the binomial distribution is provided by a data-dependent Chernoff–Lehmann X ² test with class expectations greater than unity, and its components. These data-dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions and provide a comprehensive assessment of how well the data agree with a binomial distribution. We suggest that a well-performed single test of fit statistic is the Anderson–Darling statistic.