Yoshihide Kakizawa

Comparison of Bartlett-type adjusted tests in the multiparameter case

Considering some Bartlett-type adjusted tests for a simple hypothesis about a multidimensional parameter, this paper clarifies similarities and dissimilarities with the one-parameter case developed in the 1990s, where a major emphasis is put on the issue posed by Rao and Mukerjee [C.R. Rao, R. Mukerjee, Comparison of Bartlett-type adjustments for the efficient score statistic, J. Statist. Plann. Inference 46 (1995) 137-146] on the power under a sequence of local alternatives. Not surprisingly, there is an infinite number of adjustments which extend Chandra-Mukerjee and Taniguchi approaches to the multiparameter case. Revisiting their ideas, this paper presents four specific cases (type K, K=0,1,2,3) and gives a sufficient condition under which our generalized adjustment for each case is uniquely determined, where type 0 is a counterpart of Chandra and Mukerjees original proposal for Raos test statistic, whereas the latter three types are introduced as double adjustments related to the Cordeiro and Ferrari approach. If the adjustment of type 1 is made instead of type K, K=0,2,3, it is shown that Chandra and Mukerjees approach is equivalent to Taniguchis approach in terms of the third-order local power. The same is partially true for type 0, depending on the model under consideration. However, the adjustments of type K, K=2,3, reveal, in general, the non-equivalence of these two approaches in terms of the third-order local power.

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N^-^1 after either Bartletts type adjustment or Cornish-Fishers type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartletts type adjustment and monotone Cornish-Fishers type size adjustment, is examined using simulation studies.

Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat (J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi (J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m-vector processes. It is shown that, for Gaussian vector AR(p) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤k≤p. Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA(q) processes.

Improved chi-squared tests for a composite hypothesis

The Bartlett-type adjustment is a higher-order asymptotic method for improving the chi-squared approximation to the null distributions of various test statistics. Though three influential papers were published in 1991-Chandra and Mukerjee (1991) [8], Cordeiro and Ferrari (1991) [12] and Taniguchi (1991) [36] in alphabetical order, the only CF-approach has been frequently applied in the literature during the last two decades, provided that asymptotic expansion for the null distribution of a given test statistic is available. Revisiting the CM/T-approaches developed in the absence of a nuisance parameter, this paper suggests general adjustments for a class of test statistics that includes, in particular, the likelihood ratio, Raos and Walds test statistics in the presence of a nuisance parameter.

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.

Second-Order Power Comparison of Tests

The second-order local power of a class of tests for a simple hypothesis about a multi-dimensional unknown parameter is considered. It turns out that the test procedure adjusted differently from Mukerjee (1990a) has the identical second-order local power without making use of the average power criterion. The basic principle behind the power identity is that approximate third-order cumulants of the modified square-root version of the test statistic vanish. This represents a substantial extension of the second-order asymptotic results of tests in the 1980s and early 1990s.

Asymptotic Expansions for the Distributions of Maximum and Sum of Quasi-Independent Hotelling's T 2 Statistics Under Non Normality

This article presents asymptotic expansions for the joint characteristic function and the joint distribution of correlated but asymptotically independent (i.e., quasi-independent) Hotellings T2 statistics under nonnormality, which is an extension of Fujikoshi and Seo (1999) under normality. The derivation is based on the differential operator developed by Kakizawa and Iwashita (2005a). Also, asymptotic expansions for the distributions of maximum and sum of quasi-independent Hotellings T2 statistics are derived in order to construct simultaneous confidence intervals of mean vectors in the one-way layout model.

On exponential rates of estimators of the parameter in the first-order autoregressive process

A closed-form expression for the exponential rate of an estimator in the Gaussian AR(1) process is obtained. This shows that the exponential rates of several famous estimators are all identical. Further it is shown that mean-correction does not affect the large deviation asymptotics.

Third-order local power properties of tests for a composite hypothesis

This paper addresses, for a composite hypothesis about a subvector of the parameters in the parametric model, the issues posed by Rao and Mukerjee (1995) [22] and Li (2001) [14] on the power under a sequence of local alternatives. It is shown that a partially adjusted test statistic in a class of test statistics is equally sensitive (up to the third-order) to the change of the nuisance parameters. However, there exist infinitely many ways for improving the chi-squared approximation to the null distribution, which reveal, in general, the non-equivalence of the resulting third-order point-by-point local powers. To make a definitive conclusion, the average local power is then considered, from which the third-order asymptotic optimality of the Bartlett-type adjusted Rao test can be also established.

Bahadur exact slopes of some tests for spectral densities

The large deviation result is proved for two functionals of the empirical spectral process in zero-mean Gaussian stationary processes. As a statistical application, we deal with the Bahadur asymptotic efficiencies of two statistics for testing H: f 1 = f (specified), which are spectral analogue to the Kolmogorov-Smirnov (KS) and Kuiper statistics for testing hypothesis about distribution function in the iid setting. It is shown that the Kuiper type statistic is superior to the KS type statistic in terms of the Bahadur exact slope. We also discuss the a( ≥ 2)-sample problem. Especially, for the two-sample problem, we investigate the Bahadur asymptotic efficiencies of several statistics for testing not only the goodness-of-fit hypothesis H 1: f 1 = f 2 = f (specified) but also the homogeneity hypothesis H 2: f 1 = f 2 (unspecified).