The limiting distribution of a test for multivariate structure

We define a chi-squared statistic for p-dimensional data as follows. First, we transform the data to remove the correlations between the p variables. Then, we discretize each variable into groups of equal size and compute the cell counts in the resulting p-way contingency table. Our statistic is just the usual chi-squared statistic for testing independence in a contingency table. Because the cells have been chosen in a data-dependent manner, this statistic does not have the usual limiting distribution. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic when the data is sampled from a multivariate normal distribution. The chi-squared statistic is useful in detecting hidden structure in raw data or residuals. It can also be used as a test for multivariate normality.     

The X2-goodness-of-fit tests with moment type estimators

In the x²-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x²-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x²-distributed. We also propose a statistic Q? for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]? tests are discussed.     

Tests for Symmetry Based on the One-Sample Wilcoxon Signed Rank Statistic

ABSTRACT

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X ² statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

Generalized bartlett correction

This paper provides a general method of modifying a statistic of interest in such a way that the distribution of the modified statistic can be approximated by an arbitrary reference distribution to an order of accuracy of O(n ^-1/2) or even O(n ^-1). The reference distribution is usually the asymptotic distribution of the original statistic. We prove that the multiplication of the statistic by a suitable stochastic correction improves the asymptotic approximation to its distribution. This paper extends the results of the closely related paper by Cordeiro and Ferrari (1991) to cope with several other statistical tests. The resulting expression for the adjustment factor requires knowledge of the Edgeworth-type expansion to order O(n^-1) for the distribution of the unmodified statistic. In practice its functional form involves some derivatives of the reference distribution. Certain difference between the cumulants of appropriate order in n of the unmodified statistic and those of its first-order approximation, and the unmodified statistic itself. Some applications are discussed.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

The Pearson Score Statistic for Multinomial-Poisson Models

The score statistic S² is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X² = ∑(O ? E)²/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S² have been shown to work well in practice and theory and because X² has such a simple and intuitively appealing form, it is of interest to know when S² is identical to X² and when X² has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S² = X² for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

GOODNESS OF FIT FOR THE BINOMIAL DISTRIBUTION

Goodness of fit testing for the binomial distribution can be carried out using Pearson's X²_p statistic and its components. Applications of this technique are considered and compared with recently suggested empirical distribution function tests. Diagnostic use of components is discussed.     

A Permutation Test for Planar Regression

Oja (1987) presents some distribution-free tests applicable in the presence of covariates when treatment values are randomly assigned. The formulas and calculations are cumbersome, however, and implementation of the tests relies on using a x² approximation to the exact null distribution. In this paper a re-formulation of his test statistic is given which has the advantages of ease of calculation, explicit formulas for permutation moments, and allowing a Beta distribution to be fitted to the exact null distribution.     

SMOOTH TESTS FOR THE BIVARIATE POISSON DISTRIBUTION

A theorem of Rayner & Best (1989) is generalised to permit the construction of smooth tests of goodness of fit without requiring a set of orthonormal functions on the hypothesised distribution. This result is used to construct smooth tests for the bivariate Poisson distribution. The test due to Crockett (1979) is similar to a smooth test that assesses the variance structure under the bivariate Poisson model; the test due to Loukas & Kemp (1986) is related to a smooth test that seeks to detect a particular linear relationship between the variances and covariance under the bivariate Poisson model. Using focused smooth tests may be more informative than using previously suggested tests. The distribution of the Loukas & Kemp (1986) statistic is not well approximated by the x²distribution for larger correlations, and a revised statistic is suggested.     

Accuracy of Power-Divergence Statistics for Testing Independence and Homogeneity in Two-Way Contingency Tables

The small-sample accuracy of seven members of the family of power-divergence statistics for testing independence or homogeneity in contingency tables was studied via simulation. The likelihood ratio statistic G ² and Pearson's X ² statistic are among these seven members, whose behavior was studied at nominal test sizes of.01 and.05 with marginal distributions that could be uniform or skewed and with a set of sample sizes that included sparseness conditions as measured through table density (i.e., the ratio of sample size to number of cells). The likelihood ratio statistic G ² rejected the null hypothesis too often even with large table density, whereas Pearson's X ² was sufficiently accurate and only presented a minor misbehavior when table density was less than two observations/cell. None of the other five statistics outperformed Pearson's X ². A nonasymptotic variant of X ² solved the minor inaccuracies of Pearson's X ² and turned out to be the most accurate statistic for testing independence or homogeneity, even with table densities of one observation/cell. These results clearly advise against the use of the likelihood ratio statistic G ².     

A new test for the mean vector in large dimension and small samples

In this article, we consider the problem of testing the mean vector in the multivariate normal distribution, where the dimension p is greater than the sample size N. We propose a new test T_Block and obtain its asymptotic distribution. We also compare the proposed test with other two tests. The simulation results suggest that the performance of the new test is comparable to the existing two tests, and under some circumstances it may have higher power. Therefore, the new statistic can be employed in practice as an alternative choice.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

Three Bartlett-type corrections for score statistics in symmetric nonlinear regression models

We present simple matrix formulae for corrected score statistics in symmetric nonlinear regression models. The corrected score statistics follow more closely a χ ² distribution than the classical score statistic. Our simulation results indicate that the corrected score tests display smaller size distortions than the original score test. We also compare the sizes and the powers of the corrected score tests with bootstrap-based score tests.     

A comparative simulation study of the small sample powers of several goodness fit test

Eight goodness of fit tests are compared with respect to their simulated small sample power of detecting an inbreeding alternative to the Hardy-Weinberg null hypothesis. The Pearson's x ² test is found to be most powerful, and the small rample levels of this test are close to the nominal (x ²) significance levels. The use of conditional expectations, rather than expected frequencies based on ML estimates, increases the power and improves thc x ² fit to the true significance level. The small sample powers are also compared to the asymptotic (Pitman) pourer, based on the noncenlral x ² distribution.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

The weighted compounding of probabilities from three independent hypothesis tests

Tail probabilities from three independent hypothesis tests can be combined to form a test statistic of the form P₁,P₂ ^θ2,P₃ ^θ3.The null distribution of the combined test statistic is presented and critical values for α=0.01 and 0.05 are provided.The power of this test is discussed for the special case ofthree independent F-tests.