Inference on the Common Variance of Correlated Normal Random Variables

ABSTRACT

Scale equivariant estimators of the common variance σ², of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ² which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ²: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.     

Distribution of the quotient of noncentral normal random variables

The Mellin convolution is used to derive in analytical form an exact 3-parameterprobabilitydensity function of the quotient of two noncentral normal random variables. In contrast with the 5-parameter probability density function previously derivedby Fieller (1932) and Hinkley (1969), this 3-parameter probability density function is feasible for computer evaluation of the mean and cumulative distribution function, which are needed, for example, when dealing with estimation and distribution problems in regression analysis and sampling theory. When the normal variables are independent, the probability density function reduces to a 2-parameter function, for which a computer program is operational. An illustrative example is given for one set of parameters when the normal variables are independent, in which themean and functional form of the probability density function are presented, together with a brief tabulation of the probability density function.     

Peculiar bias properties of the ols estimator when applied to a dynamic model with autocorrelated disturbances

This study reveals that contrary to the conventional wisdom among econometricians, the bias of the OLS estimator can be quite small when the estimator is applied to a geometrically distributed lag model, y_t<ce:glyph name="dbnd6"/> α + βx _t+ λy _t-₁. + u_t, with autocorrelated disturbances, be they AR(1), MA(1), MA(2), AR(2), and ARMA(1,1). This happens when λ is large and x_tis smoothly trended (e.g., a real GNP series). In fact, the bias of the OLS estimator becomes zero at one parameter combination, and the OLS estimator performs well over a wide range around this parameter combination. By decomposing the disturbance term into two parts, the paper also explains why OLS shows such an unexpected property. These findings have both pedagogical and practical significance.     

Minimization of model inadequacy when observations subject to shift in means and variances

In this article optimality of experimental design for fitting a lower-order polynomial to a higher order response function for the situation in which observations may be subject to shift in means as well as in variances is considered. It is found that Karson, Manson and Hader‘s (1969) optimum designs provide pro-tection, in some sense, against model inadequacies even when observations are subject to shift in means and variances.     

On fractional moments of quadratic expressions in normal variables1

Fractional moments, product cumulants and product moments of general quadratic expressions in singular and nonsingular normal variables are explicitly evaluated. A general method of deriving such moments is also indicated. Particular cases art; shown to agree with known results.     

Multilevel regression mixture analysis

Summary.  A two-level regression mixture model is discussed and contrasted with the conventional two-level regression model. Simulated and real data shed light on the modelling alternatives. The real data analyses investigate gender differences in mathematics achievement from the US National Education Longitudinal Survey. The two-level regression mixture analyses show that unobserved heterogeneity should not be presupposed to exist only at level 2 at the expense of level 1. Both the simulated and the real data analyses show that level 1 heterogeneity in the form of latent classes can be mistaken for level 2 heterogeneity in the form of the random effects that are used in conventional two-level regression analysis. Because of this, mixture models have an important role to play in multilevel regression analyses. Mixture models allow heterogeneity to be investigated more fully, more correctly attributing different portions of the heterogeneity to the different levels.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

Measuring non-linear dependence for two random variables distributed along a curve

We propose new dependence measures for two real random variables not necessarily linearly related. Covariance and linear correlation are expressed in terms of principal components and are generalized for variables distributed along a curve. Properties of these measures are discussed. The new measures are estimated using principal curves and are computed for simulated and real data sets. Finally, we present several statistical applications for the new dependence measures.     

萨满的“飞翔”与“附体”析论

论及萨满的特性时 ,以著名宗教史学家M·埃利亚德为首的一些学者 ,赞同用“飞翔”的有无作为识别萨满身份的尺度。以社会人类学家M·刘易斯、日本学者佐佐木宏等为首的一些学者 ,则主张“附体”亦为萨满的基本特性。然而 ,“飞翔”与“附体”之间所具有的相互关系并未得到说明。从“飞翔”与“附体”的历史形成入手 ,探讨两者之间存在的内在关系 ,并对萨满教从精灵崇拜趋向神灵信仰 ,这一发展阶段的性质转换作一概略说明。