The Edgeworth approximation and the saddlepoint approximation are common procedures for approximating the distribution of a statistic based on n samples. The former is determined by the method of inverting the characteristic function which is expanded into powers of n-1/2, the latter is the approximate formula of the Fourier inversion integral. In this paper, we show that the third order Edgeworth approximation is obtained by expanding the saddlepoint approximation. This is performed through the perturbated solution for the saddlepoint which gives the latter approximation. Furthermore, a numerical example for an estimator of the Gaussian AR(1) process is provided.
A second order asymptotic lower bound of mean squared error is obtained from a class of estimators, which are third order locally asymptotically regular. The bound is a particularly interesting form in regards to the cases of location and scale parameter. That is, it is expressed by three terms; the statistical curvature, the naming curvature and the derivative of the Fisher information. Moreover, for these cases, the conditions for the attainment of the lower bound are discussed on the basis of a few important properties of the Pitman estimator. As a result, it is proved that the attainment of the bound depends on whether the statistical curvature is zero.
In this paper, we consider the effects of nonnormality on the distributions of the sample roots in MANOVA models. Asymptotic properties of the distributions of the sample roots are presented under nonnormality. We also discuss the distributions of sample roots under the class of elliptical populations. Asymptotic expansion formulas for the distributions are derived by a perturbation method. Finally, simulation results are presented.
In some regression model, the mean square errors of a ratio estimator, a grouped jackknife estimator, and an estimator based on the least square estimators (LSEs) are obtained and compared up to the order O(n-3), where n is the size of the sample. The bias-adjusted ratio estimator and the jackknife estimator are also compared up to the order O(n-3). Then it is concluded that the estimator based on the LSEs is an asymptotically better estimator of ratio up to the order O(n-3). Some examples are given.
The power of Lawley-Hotelling (LH) type criterion is compared with those of likelihood ratio (LR) type, Bartlett-Nanda-Pillai's (BNP) type and largest root criteria. When the sample size is not so large, the power of LH criterion is locally optimum. However, when sample size is large, the power of LH is almost equivalent to that of LR type or BNP type criterion.
An artificial circular data for correspondence analysis has been presented by Iwatsubo and he has obtained the eigenvalues and eigenvectors of the data analytically. This paper first presents an artificial disk data by adding the center of the circle to Iwatsubo's data, and then generalizing it through the use of several concentric circles. The generalized data is solvable analytically and has two parameters; the number of points on each circumference, C, and the number of the circles, R. The former is regarded as a circular trait and the latter a linear trait. Each trait gives rise to the Guttman series; intensity, closure, involution, etc. The series for C are trigonometric functions and those for R are nearly polynomials of the basic linear scores. The ordering of magnitude of the eigenvalues depends on the values of the two parameters. Sometimes it is required to remove the Guttman series of one of the two traits to obtain the scores for the other trait.
The problem of estimation of four commonly used measures of overlap for two normal densities with heterogeneous variances is considered and relations between them are studied. Overlap coefficients are used frequently to describe the degree of interspecific encounter or crowdedness of two species in their resource utilization. Variants of overlap coefficients have also been used to estimate the proportion of genetic deviates in segregating populations and to measure racial segregation. Relations between four measures of overlap are studied and approximate expressions for the bias and the variance of the estimates are presented. The invariance property and a method of statistical inference of these coefficients also are presented. Results of some simulation studies concerning the performance of expressions for the bias and the variance are given.
Let constants c1 and c2 satisfy Pr(X2, (n0)≤c1)=Pr(X2(n0)≥c2)=α/2 and c1<c2, where X2(n0) is distributed as a chi-square distribution with degrees of freedom n0. In this paper we give a proof of the inequality n0 log (c2/c1)/(c2-c1)>1. This inequality is related with the condition for improving on an equal-tails confidence interval of normal variance.
This paper considers the general concept of right spherical distributions with applications to linear models, skewness and kurtosis. Some exact moments of skewness and kurtosis are established. Further, the average k_??_ order efficiency of the least squares estimate is established, which generalizes the results of Iwasaki (1985).
A sufficient condition is given for a countable mixture to be identifiable provided that the supports of mixing distributions are well-ordered sets for a total ordering of the parameter space. The identifiability of some countable mixtures of loggamma distributions and that of some countable mixtures of reversed log-gamma distributions are studied. Both classes of all finite mixtures of log-gamma distributions and all finite mixtures of reversed log-gamma distributions are shown to be identifiable as special cases.
Various patterned methods for construction of variance-balanced ternary (VBT) designs and efficiency-balanced ternary (EBT) designs are developed, along with a few illustrations. Tables of some VBT and EBT designs obtained here are presented under a certain range of design parameters.
Two estimation methods, the maximum likelihood method under the beta-binomial model and the quasi-likelihood moment method based on the mean and variance relation, were applied to G-banding chromosome aberration data from Hiroshima atomic-bomb survivors. The chromosome aberration rate was empirically thought to be overdispersed by the intraindividual correlation or the radiation dose estimation error. Using the results of Pierce, Stram, Vaeth, and Schafer , the mean and variance relationship was formulated under the two variations to apply the quasilikelihood method and the beta-binomial model. The quasi-likelihood moment method allows only single extra-binomial parameter but is robust, whereas the betabinomial model allows for both dose error and intraindividual variations. Dose response parameter estimates obtained using the two methods were similar. However, the quasi-likelihood moment method is computationally less intensive than the betabinomial maximum likelihood method. When observations are perturbed by the dose-error, the quasi-likelihood method is recommended.
A new expression of Cook's distance in ridge regression is proposed by extending a linear regression form to create a ridge regression form. In order to make sensitivity analysis easier, the expression separates the influence measure into three parts, which are least squares, ridge, and shrinkage parts. Therefore this new expression of Cook's distance is hereafter referred to as a separate expression. In addition, two auxiliary influence indices are derived to investigate sensitivity on ridge parameter. Illustrative examples are given to show the effectiveness of the separate expression and the indices for the single case diagnostics. Furthermore an extension of the single case expression into the multiple case version is developed by taking into account the original expression.