Communications in Statistics: Theory and Methods

Smith, William B.

doi: 10.1080/03610929608831676pmid: N/A

Atilgan, Taskin

doi: 10.1080/03610929608831677pmid: N/A

When approximations of the form are used in regression or density estimation, the dimension m controls the smoothness and goodness of fit of the approximation. For this type of approximation, Akaike's Information Criterion (AIC) provides a balance between smoothness and goodness of fit, extending maximum likelihood methods from estimation of parameters for a specified dimension (model) to the selection of dimension for a given basis (ψi(x)’s). Some basis will give a smaller bias for a given dimension than others and also may suggest a parametric model for a given data. In this paper, use of AIC is first extended from selection of dimension for a given basis to selection of basis for density estimation and regression. Next, it is extended to model selection (basis and dimension) under different error distributions leading to robust model selection for regression.

Cordeiro, Gauss M.; Ferrari, Silvia L.P.

doi: 10.1080/03610929608831678pmid: N/A

Cordeiro and Ferrari (1991) have shown, quite generally, how to correct the score statistic such that the null distribution of the corrected statistic is equal to that of the asymptotic chi-squared distribution, plus terms of order n -3/2 n being the sample size. In the present paper, based on this approach, we derive improved score tests for testing the precision parameter and the full vector of regression parameters in proper dispersion models This class of models which were introduced by Barndorff-Nielsen and Jørgensen (1991), is interesting because it includes a considerable variety of important mudels, while fitting into the general framework of the generalized linear models and analysis of deviance. We also generalize some recent results by Cordeiro. Ferrari and Paula (1993).

Kim, Choongrak

doi: 10.1080/03610929608831679pmid: N/A

Cook (1986) suggested a general method for assessing the local influence of minor perturbations of a statistical model. In this paper we suggest an influence measure called replacement measure. We show that the replacement. measure is a scalar version oi an influence curve and a weighted porturba tion of a diagnostic information matrix. Comparisons between Cook's local influence and the replacement measure using the geese data and house price data (VVeisberg, 1985) are given. We apply the replacement measure to rnul-tivariate linear regression and compared it with the multivariate version of Cook's distance by Barrett and Ling (1992).

Collins, John R.; Liu, Shawn X.

doi: 10.1080/03610929608831680pmid: N/A

Following results previous obtained by DasGupta (1994), we study the problem of finding the upper and lower bounds on asymptotic relative efficiencies (ARE) among three robust estimators of location when the error distribution F lies in the class Here H is an absolutely continuous distribution; the contamination parameter ∈ is fixed, 0 < ∈ < ½; and G, the mixing distribution on the scale parameter, has fixed support [s 1,s 2], where 1 ≤ s 1 < s 2 < ∞ but is otherwise arbitrary. Our modification of DasGupta's formulation was to replace his side conditions,G[s 1, ∞) = f sdG(s) by the single side condition G[s 1,s 2]=1 The three estimators for which bounds on pairwise ARE's are found are the sample mean [Xbar], the α—trimmed mean [Xbar]α , and the sample median M. In some of the cases, the extremal problems have a simple structure and are solved explicitly by moment space methods. Numerical tabulations of the bounds on ARE's are presented.

Ferreri, Carlo

doi: 10.1080/03610929608831681pmid: N/A

In many applicatiuns count data at the event umes lead us to regard the nonhomogeneous negative binomial process as a non realistic model since it involves a time-independent process-index. In this paper a research approach implying a hyperbinomial process (HBP) in terms of a real time-function process-index α(t) involving both under- and over-Poissonian dispersion, is discussed. The estimation of the two parameter-functions specifying the HBP is dealt with mainly on the basis of the ML method by utilizing results known for the dispersion parameter of a negative binomial distribution. The instability of the estimators is further discussed and a jackknife estimator, useful for attenuating the right-tail effect of a frequency distribution, is described. A moving average procedure aimed at outlining the process-index trend reliably is then illustrated also by means of accident count data. This allows us to stress that, in the HBP context, the Poissonian reference model, to which HBP reduces at a time t 0 when α(t 0)=0, stands out as an instantaneous situation of unstable equilibrium through which an over-Poissonian dispersion behaviour appears to be the continuation of an under-Poissonian dispersion.

Naito, Kanta

doi: 10.1080/03610929608831682pmid: N/A

A modification method of the Cramér-von Mises type statistic to yield asymptotic normality proposed by Ahmad(1993) is applied to the characteristic func tion based statistic. Weak convergence theorems of the empirical characteristic process are given. Similar modification of Epps and Pulley's statistic in testing composite hypothesis for normality is also considered.

Gupta, Ramesh C.; Ma, Suchun

doi: 10.1080/03610929608831683pmid: N/A

In this paper we present several tests for testing the equality of coefficients of variation in k normal populations. Several of the known tests are reviewed and one new test is developed. An example is presented to illustrate the working of all these tests. Simulation studies are carried out for comparing the powers of these tests. Finally, conclusions are drawn from the simulation studies and some recommendations are made.

Kundu, Debasis; Mitra, Amit

doi: 10.1080/03610929608831684pmid: N/A

The consistency and asymptotic normality of the least squares estimator are derived of a particular non-linear time series model. It does not satisfy the standard sufficient conditions of Jennrich (1969) or Wu (1981). The errors are assumed to be independently and identically distributed random vaiiables each with mean zero and finite variance. Walker (1971) considered the same model and obtained the asymptotic properties of an approximate least squares estimator. It is observed that the least squares estimator and the approximate least squares estimator are asymptotically equal. Some simulations have hrcn performed to compare the two for small samples.

Lee, Andy H.; Zhao, Yuejen

doi: 10.1080/03610929608831685pmid: N/A

This paper investigates the local influence estimation of the-goodness-of-fit sia,listic in generalized linear model settings. Inspired by Cook (1986), a loral influence approach is adopted to assess model adequacy with respert to the contours uf the unperturbed generalized Pearson's statistic. Based on local perturbations to the vectors of case weights, covariates and responses, the approach can detect different aspects of influence and yield additional insight to likelihood displacement Two examples demonstrate the effectiveness of the proposed method.