THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.     

THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.     

THE EXACT DISTRIBUTION OF THE BELL-DOKSUM TEST OF INDEPENDENCE

An expression for the exact distribution of the Bell-Doksum test of independence based upon the use of order statistics from normally distributed random samples is derived and a table of critical values is provided. Two extensions of the theory are considered and possible applications to Poisson processes and queueing theory are noted.     

ON THE DISTRIBUTION OF THE REGRESSION COEFFICIENT MATRIX OF A NORMAL DISTRIBUTION

In this paper we generalize a result of Kshirsagar's (1960, pp. 83–84) on the distribution of the regression coefficient matrix for a multivariate normal population.     

THE DISTRIBUTION OF STOCHASTIC SHRINKAGE PARAMETERS IN RIDGE REGRESSION

ABSTRACT

In this article we derive the density and distribution functions of the stochastic shrinkage parameters of three well known operational Ridge Regression (RR) estimators by assuming normality. The stochastic behavior of these parameters is likely to affect the properties of the resulting RR estimator, therefore such knowledge can be useful in the selection of the shrinkage rule. Some numerical calculations are carried out to illustrate the behavior of these distributions, throwing light on the performance of the different RR estimators.     

TESTING THE STABILITY OF REGRESSION COEFFICIENTS USING GENERALIZED RECURSIVE RESIDUALS

Simple but flexible methods to detect deviations from the assumption of constant coefficients in linear regression are presented. Based on recursive residuals a runs test is developed as an alternative to CUSUM- and MOSUM-techniques. Finally a simulation study gives insight into the new method.     

BAYESIAN ANALYSIS OF THE LINEAR REGRESSION MODEL WITH NON-NORMAL DISTURBANCES

This paper considers the Bayesian analysis of a linear regression model with identically independently distributed non-normal disturbances. The distribution of disturbances is approximated by an Edgeworth series distribution with cumulants, of order higher than fourth, negligible. The posterior distribution of the regression coefficients vector is obtained under the assumption of a g-prior distribution for the parameters of the model. The Bayes estimator and its Bayes risk of the estimator are derived under a quadratic loss structure.     

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator.

     

AN ANALYTIC FORM FOR THE CUMULATIVE DISTRIBUTION FUNCTION OF THE BELL-DOKSUM STATISTIC

A closed form analytic expression for the Bell-Doksum Statistic is developed. The use of the generalized hypergeometric function to evaluate the expression is demonstrated, and a table of critical values for 2 n 25 is presented.     

A CLOSED-FORM EXPRESSION FOR THE PEARSON TYPE IV DISTRIBUTION FUNCTION

A closed-form expression is presented for the probability integral of the Pearson Type IV distribution, and a corresponding method of evaluation is given. This analysis addresses a long-standing gap in the theory of the Pearson system of distributions. In addition, a simple derivation is given of an expression for the normalizing constant in the Type IV integral.     

ON ESTIMATING THE SPREAD OF A DISTRIBUTION ABOUT ITS REGRESSION SLOPES1

Nonparametric tolerance regions are used to estimate the spread of a distribution about known regression slopes. When the slopes are not known their sample estimates are employed.     

ON TESTING THE GOODNESS-OF-FIT OF NONLINEAR HETEROSCEDASTIC REGRESSION MODELS

For testing the adequacy of a parametric model in regression, various test statistics can be constructed on the basis of a marked empirical process of residuals. By using a discretized version of the decomposition of the corresponding Gaussian limiting process into its principal components, we obtain a test statistic with an asymptotic chi-squared distribution under the null hypothesis. We investigate the consistency of this test statistic and of the estimators needed to compute it. Numerical experiments indicate that the distributional approximations already work for small to moderate sample sizes and reveal that the test has good power properties against a variety of alternatives. The test has a simple implementation. We present an application to a real-data example for testing the adequacy of a possible heteroscedastic exponential model.     

ON CHARACTERIZING DISTRIBUTIONS VIA LINEARITY OF REGRESSION FOR ORDER STATISTICS

Let X₁X_n be a random sample from an absolutely continuous distribution with the corresponding order statistics X_1:n≤X_2:n≤X_n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X_k+2:n with respect to X_k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.     

ESTIMATING THE ROOT OF A NONPARAMETRIC REGRESSION FUNCTION IN A ROBUST FASHION

For a nonparametric regression model y = m(x)+e with n independent observations, we analyze a robust method of finding the root of m(x) based on an M-estimation first discussed by Härdle & Gasser (1984). It is shown here that the robustness properties (minimaxity and breakdown function) of such an estimate are quite analogous to those of an M -estimator in the simple location model, but the rate of convergence is somewhat limited due to the nonparametric nature of the problem.     

ESTIMATION OF REALIZED SIGNAL TO NOISE RATIO FOR A PAIR OF MULTIVARIATE SIGNALS

Consider the case of classifying an incoming message as one of two known p-dimension signals or as a pure noise. Let the noise co-variance matrix (assumed to be same in all the three cases) be unknown. We consider the problem of estimation of “realized signal to noise ratio matrix”, which is an index of discriminatory power, under various loss functions. Optimum estimators are obtained under these loss functions. Finally, an attempt is made to provide a lower confidence bound for the realized signal to noise ratio matrix. In the process, the probability distribution of the smaller eigenvalue of a 2 × 2 confluent hypergeometric random matrix is obtained.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

BINARY REGRESSION WITH A CLASS OF SKEWED t LINK MODELS

ABSTRACT

In this paper we propose a class of skewed t link models for analyzing binary response data with covariates. It is a class of asymmetric link models designed to improve the overall fit when commonly used symmetric links, such as the logit and probit links, do not provide the best fit available for a given binary response dataset. Introducing a skewed t distribution for the underlying latent variable, we develop the class of models. For the analysis of the models, a Bayesian and non-Bayesian methods are pursued using a Markov chain Monte Carlo (MCMC) sampling based approach. Necessary theories involved in modelling and computation are provided. Finally, a simulation study and a real data example are used to illustrate the proposed methodology.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

THE q-BETA-GEOMETRIC DISTRIBUTION AS A MODEL FOR FECUNDABILITY

The number of sterile couples in a retrospective study of the number of cycles to conception is necessarily zero; this is not so for a prospective study. The paper puts forward a modification of Weinberg and Gladen's beta geometric model for cycles to conception that is suitable for both types of investigation. The probability that a couple achieves conception at the xth cycle, but not earlier, is assumed to take the form R_x = (1 ? ρ)/(1 ? m ^x?1 ρ/u), instead of μ/(1 ? θ + θx). The set of parameter restraints (0 < m < 1, 0< ρ < 1, 1 < u) is appropriate for retrospective data, whilst the alternative set of restraints (1 < m, 1 < ρ, 0 < u < 1) is appropriate for prospective data. The decrease in R_x over time can be interpreted not only as a time effect, but also as a heterogeneity effect by replacing Weinberg and Gladen's beta mixture of geometric distributions by a q-beta mixture.     

A REGRESSION APPROACH TO THE ESTIMATION OF THE FINITE POPULATION MEAN IN THE PRESENCE OF NON‐RESPONSE

This paper presents various estimators for estimating the population mean of the study variable y using information on the auxiliary variable x in the presence of non‐response. Properties of the suggested estimators are studied and compared with those of existing estimators. It is shown that the estimators suggested in this paper are among the best of all the estimators considered. An empirical study is carried out to demonstrate the performance of the suggested estimators and of others, and it is found that the empirical results support the theoretical study.