On a Dualization of Graphical Gaussian Models: A Correction Note

Abstract.  We correct two proofs concerning Markov properties for graphs representing marginal independence relations.     

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

A Note of Wavelet Variance

This paper proposes a definition of wavelet variance in integral form and explores some properties of the variance. By this definition, a method of parameter estimation for long memory processes is established. An example of the estimation is discussed.     

A Note on Change-Over Designs

A change-over model with correlated errors is discussed. In particular the results of Patterson (1950) and Lucas (1951) for balanced change-over designs are extended to more than four treatments and more general variance matrices. A connection with recent work on neighbour models for field trials is made.     

A Note On Weighted Distributions

A representation of the "original" random variable (r.v.) in terms of the "weighted" r.v. is given and the Inverse Gaussian distribution is characterized through a distributional property the "weighted" r.v. observed under 1ength biased sampling.     

A Note on Whittle's Likelihood

The approximate likelihood function introduced by Whittle has been used to estimate the spectral density and certain parameters of a variety of time series models. In this note we attempt to empirically quantify the loss of efficiency of Whittle's method in nonstandard settings. A recently developed representation of some first-order non-Gaussian stationary autoregressive process allows a direct comparison of the true likelihood function with that of Whittle. The conclusion is that Whittle's likelihood can produce unreliable estimates in the non-Gaussian case, even for moderate sample sizes. Moreover, for small samples, and if the autocorrelation of the process is high, Whittle's approximation is not efficient even in the Gaussian case. While these facts are known to some extent, the present study sheds more light on the degree of efficiency loss incurred by using Whittle's likelihood, in both Gaussian and non-Gaussian cases.     

A Note on Model-Based Stratification

A slight improvement on the stratification methodology in Godfrey et al. (1984) is proposed and an example is advanced to demonstrate its usefulness.     

A Note on Stagewise Regression

Suppose one estimates the coefficient β₂ in E[Y] = β₀ + β₁ X ₁ + β₂ X ₂ by stagewise regression. That is, first the model E[Y] ≌ β₀ + β₁ X ₁ is fit using simple linear regression followed by a simple linear regression of the residuals from this model on X ₂ to yield the estimator β₂. The ratio of the squared t statistic for the estimate b ₂ from multiple regression to the squared t statistic for β₂ is greater than or equal to 1.0 and is shown to be a convenient function of correlation coefficients among Y, X ₁, and X ₂. Examination of stagewise regression can provide useful insights when introducing concepts of multiple regression.     

A Note on Estimating Covariance Components

A well-known formula for expressing a covariance in terms of variances is shown to hold true for estimating components of covariance.     

A Note on Screening Regression Equations

Consider developing a regression model in a context where substantive theory is weak. To focus on an extreme case, suppose that in fact there is no relationship between the dependent variable and the explanatory variables. Even so, if there are many explanatory variables, the R ² will be high. If explanatory variables with small t statistics are dropped and the equation refitted, the R ² will stay high and the overall F will become highly significant. This is demonstrated by simulation and by asymptotic calculation.     

关于随机DEA模型的一个注记

文章证明了参考集为有限集时的综合DEA模型的最优值的分布的无缺陷性.     

A Note On Beta Inverse-Weibull Distribution

The inverse Weibull distribution is one of the widely applied distribution for problems in reliability theory. In this article, we introduce a generalization—referred to as the Beta Inverse-Weibull distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the Beta Inverse-Weibull distribution. The shapes of the corresponding probability density function and the hazard rate function have been obtained and graphical illustrations have been given. The distribution is found to be unimodal. Results for the non central moments are obtained. The relationship between the parameters and the mean, variance, skewness, and kurtosis are provided. The method of maximum likelihood is proposed for estimating the model parameters. We hope that this generalization will attract wider applicability to the problems in reliability theory and mechanical engineering.     

A Note on Dirichlet One-Armed Bandits

One of the two independent stochastic processes (or ‘arms’) is selected and observed sequentially at each of n(≤ ∝) stages. Arm 1 yields observations identically distributed with unknown probability measure P with a Dirichlet process prior whereas observations from arm 2 have known probability measure Q. Future observations are discounted and at stage m, the payoff is a _m(≥0) times the observation Z _m at that stage. The objective is to maximize the total expected payoff. Clayton and Berry (1985) consider this problem when a _m equals 1 for m ≤ n and 0 for m > n(< ∝) In this paper, the Clayton and Berry (1985) results are extended to the case of regular discount sequences of horizon n, which may also be infinite. The results are illustrated with numerical examples. In case of geometric discounting, the results apply to a bandit with many independent unknown Dirichlet arms.     

A Note on the Kuks-Olman Estimator

In the linear regression model the unknown parameter vector θ is supposed to vary in a known ellipsoid. Under this parameter constraint Kuks and Olman derived an estimator by demanding a minimax property. Since sometimes the Kuks-Olman estimator takes values outside of the ellipsoid a modification is proposed in the paper. It is shown that this modified variant is a least squares estimator in the restricted model.     

Rangefinder Box Plots: A Note

This note introduces the rangefinder box plot, an extension of the familiar box plot.