Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

Testing the equality of variance-covariance matrices the robust way

The robust statistic T² _Dproposed by Tiku and Singh (1982) for testing the equality of mean vectors of two mu1 t ivariate populations is modified to test the equality of variance-covariance matrices.     

Manova with covariates and an application in profile analysis

Replacing one of the two marginal distributions in a bivariate normal by a family of symmetrical distributions, we obtain a new family of symmetric bivariate distributions. We use the Tiku - Suresh (1990) method to estimate the parameters of this new bivariate family. We define a Hotelling - type statistic to test the mean vector and evaluate the asymptotic power of this statistic relative to the Hotelling T² statistic. We show that the former is considerably more powerful.     

Robustness of MML estimators based on censored samples and robust test statistics

We investigate the efficiences of Tiku's (1967) modified maximum likelihood estimators μ_c and σ_c (based on symmetrically censored normal samples) for estimating the location and scale parameters μ and σ of symmetric non-normal distributions. We show that μ_c and σ_c are jointly more efficient than x? and s for long-tailed distributions (kurtosis $\text{β}{}_2{}^1\text{=}\text{μ}{}_4\text{μ}{}_2{}^2\text{>4.2, β}{}_2{}^1\text{= 4.2}$ for the Logistic), and always more efficient than the trimmed mean μT and the matching sample estimate σ_T of σ. We also show that μ_c and σ_c are jointly at least as efficient as some of the more prominent “robust” estimators (Gross, 1976). We show that the statistic $\text{t}{}_{\mathrm{c}}\text{= μ}{}_{\mathrm{c}}\text{m}\text{σ}{}_{\mathrm{c}}\text{, m = n ?2r + 2rβ}$ (r is the number of observations censored on each side of the sample and β is a constant), is robust and powerful for testing an assumed value of μ. We define a statistic T_c (based on μ_c andσ_c) for testing that two symmetric distributions are identical and show that T_c is robust and generally more poweerful than the well-known nonparametric statistics (Wilcoxon, normal-score, Kolmogorov-Smirnov), against the important location-shift alternatives. We generalize the statistic T_c to test that k symmetric distibutions are identical. The asymptotic distributions of t_c and T_c are normal, under some very general regularity conditions. For small samples, the upper (lower) percentage points of t_c and T_c are shown to be closely approximated by Student's t-distributions. Besides, the statistics μ_c and σ_c (and hence t_c and T_c) are explicit and simple functions of sample observations and are easy to compute.     

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.     

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.     

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.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

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.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

Generalization of the robust bivariate t2 statistic to multivariate populations

The robust bivariate Hotelling–type T² statistics proposed by Tiku and Balakrishnan (1988) is extendend to p–variate (p ≧ 3) populations.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

Asymptotics for multisample statistics

Consider a family of square-integrable R^d-valued statistics S_k = S_k(X_1,k1; X_2,k2;…; X_m,km), where the independent samples X_i,kj respectively have k_i i.i.d. components valued in some separable metric space X_i. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S_k}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S_k is symmetric in the coordinates of each sample X_i,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.     

Measures of lack of fit from tests of chi-squared type

If X² is the Pearson chi-squared statistic for testing fit, then $\text{X}{}^2\text{n}$ has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X² as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of $\text{T}\text{n}$, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

Multivariate analysis of variance with additional data from an unknown subset of the populations

In this paper we consider the problem of testing the means of k multivariate normal populations with additional data from an unknown subset of the k populations. The purpose of this research is to offer test procedures utilizing all the available data for the multivariate analysis of variance problem because the additional data may contain valuable information about the parameters of the k populations. The standard procedure uses only the data from identified populations. We provide a test using all available data based upon Hotelling' s generalized T²statistic. The power of this test is computed using Betz's approximation of Hotelling' s generalized T²statistic by an F-distribution. A comparison of the power of the test and the standard test procedure is also given.     

Goodness-of-Fit Tests for Multivariate Distributions

We propose three new statistics, Z _p , C _p , and R _p for testing a p-variate (p ≥ 2) normal distribution and compare them with the prominent test statistics. We show that C _p is overall most powerful and is effective against skew, long-tailed as well as short-tailed symmetric alternatives. We show that Z _p and R _p are most powerful against skew and long-tailed alternatives, respectively. The Z _p and R _p statistics can also be used for testing an assumed p-variate nonnormal distribution.     

GOODNESS OF FIT STATISTICS BASED ON THE SPACINGS OF COMPLETE OR CENSORED SAMPLES

A goodness-of-fit statistic Z is defined in terms of the spacings generated by the order statistics of a complete or a censored sample from a distribution of the type (l/)f((x-μ)/), μ and unknown. The distribution of Z is studied, mostly through Monte Carlo methods. The power properties of Z for testing Exponential, Uniform, Normal, Gamma and Logistic distributions are discussed; Z is shown to be more powerful than the Smith & Bain (1976) correlation statistic, except for testing Uniform, Normal and Logistic (symmetric distributions) against symmetric alternatives. The statistic Z is generalized to test the goodness-of-fit from κ 2 independent complete or censored samples.