Robust durbin's and portmanteau goodness-of-fit test statistics and their limit distributions

Robust analogue of Durbin's(1970)statistic is derived and its limiting distribution is obtained under both null and alternative hypotheses. Also, robust version of the portmanteau goodness-of-fit test statistic for AR(p)model is given and the asymptotic distribution is derived.     

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|^u, u? 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).     

An Efficient Method to Generate Data and Compute Exact P-Values in Goodness-of-Fit Testing

In this article, we use a characterization of the set of sample counts that do not match with the null hypothesis of the test of goodness of fit. Two direct applications arise: first, to instantaneously generate data sets whose corresponding asymptotic P-values belong to a certain pre-defined range; and second, to compute exact P-values for this test in an efficient way. We present both issues before illustrating them by analyzing a couple of data sets. Method's efficiency is also assessed by means of simulations. We focus on Pearson's X ² statistic but the case of likelihood-ratio statistic is also discussed.     

A Comparison of Two Tests for Equality of Two Proportions

Two procedures for testing equality of two proportions are compared in terms of asymptotic efficiency. The comparison favors use of a statistic equivalent to Goodman's Y ² over the usual X ² statistic in some cases including that of equal sample sizes. Numerical comparisons indicate that the asymptotic results have some relevance for moderate sample sizes.     

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.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

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).     

Testing for monotonic trend in time series based on resampling methods

In this paper, we propose several tests for monotonic trend based on the Brillinger's test statistic (1989, Biometrika, 76, 23–30). When there are highly correlated residuals or short record lengths, Brillinger's test procedure tends to have significance level much higher than the nominal level. It is found that this could be related to the discrepancy between the empirical distribution of the test statistic and the asymptotic normal distribution. Hence, in this paper, we propose three bootstrap-based procedures based on the Brillinger's test statistic to test for monotonic trend. The performance of the proposed test procedures is evaluated through an extensive Monte Carlo simulation study, and is compared to other trend test procedures in the literature. It is shown that the proposed bootstrap-based Brillinger test procedures can well control the significance levels and provide satisfactory power performance in testing the monotonic trend under different scenarios.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

On Comparisons of Exact Powers of Bivariate GMANOVA Tests

Exact powers of four classical tests in a GMANOVA model are compared numerically when the order of the error sum of square matrix is 2. The four tests are likelihood ratio (=LR), Pillai's V, Hotelling's T ², and Roy's largest root tests. It turns out that for small sizes, there are a few cases in which Rothenberg's condition for the relative magnitude of asymptotic powers of three standard tests does not hold.     

Asymptotic Expansion of the Non Null Distribution of the Two-Sample t-Statistic Under Non Normality with Application in Power Comparison

This article studies the non null distribution of the two-sample t-statistic, or Welch statistic, under non normality. The asymptotic expansion of the non null distribution is derived up to n ^?1, where n is the pooled sample size, under general conditions. It is used to compare the power with that obtained by normal theory method. A simple technique is recommended to achieve more power through a monotone transformation in practice.     

Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Statistics R ^a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R ^a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R ⁰ is the log likelihood ratio statistic and R ¹ is Pearson's X ² statistic. In this article, we consider improvement of approximation of the distribution of R ^a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R ^a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R ^a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R ^a when a ≤ 0, which include the log likelihood ratio statistic.     

Alternative Tests for Parameter Stability

ABSTRACT

We propose new tests for parameter stability based on estimates computed from a sequence of subsamples moving forward and backward across the sample. We obtain a sequence of moving estimates tests and we derive their asymptotic null distribution based on the functional central limit theorem. The critical values are approximated using Durbin's method. Our simulation results show that these tests have comparable size and slightly higher power in detecting structural change than other competing tests.     

Jackknife empirical likelihood method for testing the equality of two variances

In this paper, we propose a nonparametric method based on jackknife empirical likelihood ratio to test the equality of two variances. The asymptotic distribution of the test statistic has been shown to follow χ² distribution with the degree of freedom 1. Simulations have been conducted to show the type I error and the power compared to Levene's test and F test under different distribution settings. The proposed method has been applied to a real data set to illustrate the testing procedure.     

THE MULTIPLE CORRELATION COEFFICIENT AND FISHER'S A STATISTIC1

Fisher's A statistic, often called the adjusted R² statistic, is shown to be a close approximation to the maximum likelihood estimate of the multiple correlation coefficient, p², based on the marginal distribution of R². Expansions for the estimate are obtained. The same methods lead to maximum marginal likelihood estimators for the noncentrality parameters for noncentral X² and F.     

Likelihood-Based Inference for Weak Exogeneity in I(2) Cointegrated VAR Models

This article develops limit theory for likelihood analysis of weak exogeneity in I(2) cointegrated vector autoregressive (VAR) models incorporating deterministic terms. Conditions for weak exogeneity in I(2) VAR models are reviewed, and the asymptotic properties of conditional maximum likelihood estimators and a likelihood-based weak exogeneity test are then investigated. It is demonstrated that weak exogeneity in I(2) VAR models allows us to conduct asymptotic conditional inference based on mixed Gaussian distributions. It is then proved that a log-likelihood ratio test statistic for weak exogeneity in I(2) VAR models is asymptotically χ² distributed. The article also presents an empirical illustration of the proposed test for weak exogeneity using Japan's macroeconomic data.     

Approximate distribution and test of fit for the clustering effect in the dirichlet multinomial model

The Dirichlet-multinomial model is considered as a model for cluster sampling. The model assumes that the design's covariance matrix is a constant times the covariance under multinomial sampling. The use of this model requires estimating a parameter C, that measures the clustering effect. In this paper, a regression estimate for C is obtained. An approximate distribution of this estimator is obtained through the use of asymptotic techniques. A goodness of fit statistic for testing the fit of the Dirichlet Multinomial model is also obtained, based on those asymptotic techniques. These statistics provide a means of knowing when the data satisfy the model assumption. These results are used to analyze data concerning the authorship of Greek prose.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

New results on the likelihood ratio and score tests for the von Mises distribution

In this paper, we derive Bartlett and Bartlett-type corrections [G.M. Cordeiro and S.L.P. Ferrari 1991, A modified score test statistic having chi-squared distribution to order n ^?1 , Biometrika 78 (1991), pp. 573–582] to improve the likelihood ratio and Rao's score statistics for testing the mean parameter and the concentration parameter in the von Mises distribution. Simple formulae are suggested for the corrections valid for small and large values of the concentration parameter that do not depend on the modified Bessel functions and can be useful in practical applications.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only