Bimal K. Sinha

Chapter 2 – Invariance Approach to Testing

This chapter reviews the invariance principle in the Neyman–Pearson–Wald testing theory and a powerful result for the derivation of the distribution of a maximal invariant. The term invariance is associated with the invariance structure of a testing problem that admits a transformation group. In multivariate hypotheses testing problems, a transformation group is usually a subset of a Euclidean space and carries a topological structure in addition to the algebraic group structure. The topological uniform structure compatible with the group structure enables the utilization of invariant measure, known as the Haar measure, on the group. A topological space may be compact, locally compact, or sigma-compact. It is called locally compact if each point has an open neighborhood with the compact closure, and it is called sigma-compact if it is expressed as a countable union of compact sets.

Tests for Covariance Structures

This chapter presents the problem of developing optimal invariant tests for certain meaningful structures of the scale matrix Σ of X under the distributional assumption. Any invariant test for any hypothesis on Σ is based on the statistic X 3 , X 3 . This is because the problem remains invariant at least under the group G = θ ( n 3 ) × R ( n1 + n2 ) × p and a maximal invariant under this group is X 3 , X 3 . Usually, the group is enlarged under a specific hypothesis on Σ and, accordingly, a more specific maximal invariant, a suitable function of X 3 , X 3 , is determined. It, therefore, follows that given a canonical form of data matrix X with distribution; only X 3 is relevant for considering invariant tests on Σ. Another common problem of covariance structure in applications is testing the sphericity of Σ in the model. It establishes the optimality robustness of the LBI test derived by Sugiura under the normality assumption on Z : m × p .

Two-Population Problems

This chapter presents some testing problems involving the equality of location and scale parameters of two independent univariate populations and discusses the optimality robustness of some familiar tests. Optimality robustness of some tests derived under normal distribution is called the non-normal case, and optimality robustness of some tests derived under exponential distribution is called the non-exponential case. In the case of two independent non-normal populations, for testing the equality of two location parameters without any scale parameter, the test based on the difference between the two sample means is conditional UMPI, conditionally given two ancillary statistics, under certain conditions. In the case of two independent non-exponential populations, the same property is shown to hold for the test based on the difference between two minimums. For testing the equality of two scale parameters with or without location parameters, the standard normal-theory or exponential-theory F -test is still optimal for suitable deviations from normality or exponentiality.

Robustness of t -Test and Tests for Serial Correlation

This chapter focuses on the optimality robustness of the students t -test and tests for serial correlation, mainly without invariance. It also presents some results on the optimalities of the t -test under normality. The tests on serial correlation without invariance proceed in a manner similar to that of the case of the t -tests. The chapter presents an assumption where in one-sided testing problems without invariance, a random vector is z = ( z 1 ,…, z n ) and density is h . The chapter also presents the study of the robustness of the optimality properties of ϕ 1 and ϕ 2 in the class of spherically symmetric distributions. In the consideration of optimality robustness, it is extremely important to pay attention to the class in which the optimality holds.

General Approach to the Robustness of Tests

This chapter focuses on the general approach to the robustness of tests. The critical values of the test statistics are computed under the null hypothesis in the assumed model, whereas the powers of the tests are computed under the non-null hypothesis in the assumed model. In the presence of some uncertainty in the validity of the assumed model, it is important to study the stability or robustness of the critical values of the tests and also the optimality properties of the tests in terms of their power functions, if any, against departures from the assumed model. The robustness of a test can be studied from two aspects: (1) robustness under a null hypothesis, and (2) robustness under an alternative hypothesis.

General Multivariate Analysis of Variance (GMANOVA)

This chapter focuses on general multivariate analysis of variance (GMANOVA). As this model includes the classic multivariate analysis of variance (MANOVA) model and various multivariate regression models, it has a broader application in many fields. The chapter presents the GMANOVA model and the problems, along with some examples. The examples bring out the subtle differences between GMANOVA and MANOVA problems. The chapter also presents the derivation of canonical forms of GMANOVA, MANOVA, and ANOVA. It presents the special cases of the optimality robustness of Hotellings T 2 -test, the ANOVA F -test, and the likelihood ratio test (LRT) in a typical case of the MANOVA. The optimality robustness of the LBI tests of Pillai in general when min ( n 1 , p ) 1 is obtained as special case. It also discusses the GMANOVA problem. The LRT and some related tests are briefly discussed in the chapter. The GMANOVA model refers to an observed data matrix Y : n × p obeying the probability model.

Detection of Outliers

This chapter discusses the detection of outliers. Detection of outliers is vital in the data, if any, with powerful techniques because the presence or absence of such aberrant observations makes a major difference in the subsequent analysis of the data. In test for mean slippage, one is concerned with the problem of developing optimum invariant tests for the detection of outliers with mean slippage. The framework is based on a data matrix X : n × p . Under the null hypothesis H : Δ = 0 , the different rows of X are homogeneous without presence of any outlier. However, under the alternative K : Δ ≠ 0 , rows of X corresponding to non-null rows of A represent the outlying observations. A is a specified matrix except for the permutation of its rows, so that rejection of H does not identify the outlying observations. The chapter also discusses the mean slippage case and the case of dispersion slippage, and a special case of Fergusons univariate normal result.

Spherically Symmetric Distributions

This chapter discusses spherically symmetric distributions. The deviations from normality can be of a different variety; thus, the robustness of tests can be discussed in various ways. Even if a given distribution is close to the normal distribution in the metric, an optimal test under normality is not necessarily optimal, or even approximately optimal, under the given distribution. In finite sample testing problems, it is important to adopt the spherical symmetry and to study how much robust commonly used tests are against departures from normal distribution toward spherically symmetric distributions. One of the most important of the normal distribution is that the normality is preserved under linear transformations. The class of elliptically symmetric distributions contains a heavy-tail distribution such as the multivariate Cauchy distribution with no moments. The chapter also presents some practical implications of the assumption of spherical symmetry.