Nariaki Sugiura

Further analysts of the data by akaike' s information criterion and the finite corrections

Using Akaikes information criterion, three examples of statistical data are reanalyzed and show reasonably definite conclusions. One is concerned with the multiple comparison problem for the means in normal populations. The second is concerned with the grouping of the categories in a contingency table. The third is concerned with the multiple comparison problem for the analysis of variance by the iogit model in contingency tables, Finite correction of Akaikes information criterionis also proposed.

Testing change-points with linear trend

Tests based on rank statistics are introduced to test for systematic changes in a sequence of independent observations. Proposed tests include a rank test analogous to the parametric likelihood ratio test and others analogous to parametric Bayes tests. The tests are usable with either one- or two-sided alternative hypotheses, and their asymptotic distributions are studied. The results of the general model are applied to two special cases, and their asymptotic distributions are also investigated. A Monte Carlo study verifies the applicability of asymptotic critical points in samples of moderate size, and other simulation studies compare power of the competing tests and their special-case versions. Finally, these tests are applied to a data set of traffic fatalities.

Reference prior bayes estimator for bivariate normal covariance matrix with risk comparison

The explicit form of the reference prior bayes estimator due to Yang and Ber-ger (1994) for bivariate normal covariance matrix under entropy loss is given in terms of Legendre polynomials when degrees of freedom is even and in terms of hypergeometric functions in general case. The finite series expression of the density function of the ratio of latent roots of bivariate Wishart matrix is obtained and the exact risk is compared with those of James-Stein minimax estimator and other orthogonally equivariant estimators. It is found numerically that the reference prior bayes estimator has the smallest risk among the class of equivariant estimators compared, when the ratio of the largest to the smallest population latent roots of covariance matrix lies in the middle of the interval [1, ∞]. It has larger risk than that of James-Stein minimax estimator when the ratio is large. Moreover it has larger risk than that of MLE when, for instance, degrees of freedom is 20 and the ratio lies between 4 and 8.

Estimating common parameters of growth curve models

Suppose that we have two independent random matrices X1 and X2 having multivariate normal distributions with common unknown matrix of parameters ξ (q×m) and different unknown covariance matrices Σ1 and Σ2, given by Np1, N1 (B1ξA1;Σ1, I) and Np2, N2 (B2ξA2;Σ2, I) respectively. Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] (% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\]) be the MLE of ξ based on X1 (X2) only. When q=1, necessary and sufficient conditions that a combined estimator of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] has uniformly smaller covariance matrix than those of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] are given. The k-sample problem as well as one-sample problem is also discussed. These results are extensions of those of Graybill and Deal (1959, Biometrics, 15, 543–550), Bhattacharya (1980, Ann. Statist., 8, 205–211; 1984, Ann. Inst. Statist. Math., 36, 129–134) to multivariate case.

Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(ζ, Σ⊗Ir) and Wp(n, Σ) respectively and let them be independent. The risk of the improved estimator for |Σ| or {ei329-1} based on X and S under entropy loss (=d/|Σ| −log(d/|Σ|)−1 or d|Σ|−log(d|Σ|)−1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.

Risk of Improved Estimators for Generalized Variance and Precision

Let the distributions of observed random matrices X(pxr) and S(pxp) be N(ξ, Σ⊗Іr) and Wp(n, Σ) respectively. Assume that they are independent. The risk of improved estimators for |Σ| or |Σ−1| based on X and S under squared loss is evaluated in terms of incomplete beta function of matrix argument. Numerical comparison for the reduction of risk over the best equivariant estimators is given.

Dominating james-stein positive-part estimator for normal mean with unknown covariance matrix

Assume that we have a random sample of size n from p-variate normal population and we wish to estimate the mean vector under quadratic loss with respect to the inverse of the unknown covariance matrix, A class of superior estimators to James-Stein positive part estimator is given when n>max{9p+10,13p-7}, based on the argument by Shao and Strawderman(1994).

Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.

Delta method approach in a certain irregular condition

Let Xn be a sequence of random ρ-vectors such that , where and Z is a continuously distributed random ρ-vector. Let f(·) be a measurable mapping from a domain of Rp to Rq, where the domain may not include b, i.e., f(b) may not be defined Under this setup, we study the asymptotic distribution of f(XRn). Two theorems are developed to obtain thu asymptotic distribution. Comprehensive examples are provided to show when and where such an irregular situation takes place and to illustrate the usefulness of these theorems. The examples include the problem of choosing the number of components and noniterative estimation in factor analysis.

Exact nonnull distributions of sphericity tests for trivariate normal population with power comparison

SYNOPTIC ABSTRACTExact formulas for the nonnull density function of the locally best invariant test and that of likelihood ratio test for testing sphericity in trivariate normal distribution are derived, extending the exact null density functions obtained by John (1972). Khatri and Srivastava(1971)s formula for the nonnull density of the likelihood ratio test is shown to be useful if adjusted. The graphs of the density functions are shown. Power comparisons are made and Grieve(1984)s conjecture on power of two tests is confirmed. Harris and Peers(1980)s condition on asymptotic powers is also confirmed. Asymptotic formulas by normal and noncentral chisquare distributions are compared.