Saddlepoint‐Based Bootstrap Inference for Quadratic Estimating Equations

On the Moments of Ratio of Quadratic Forms in Normal Random Variables

On the moments of ratios of quadratic forms in normal random variables

The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.

Computationally Efficient Recursions for Top-Order Invariant Polynomials with Applications

Abstract. We consider robust methods of likelihood and frequentist inference for the nonlinear parameter, sayα, in conditionally linear nonlinear regression models. We derive closed‐form expressions for robust conditional, marginal, profile and modified profile likelihood functions forαunder elliptically contoured data distributions. Next, we develop robust exact‐F confidence intervals forαand consider robust Fieller intervals for ratios of regression parameters in linear models. Several well‐known examples are considered and Monte Carlo simulation results are presented.

Evaluation of fuzzy regression models by fuzzy neural network

We develop analogs of the two classes of weighted empirical minimum distance (m.d.) estimators of the underlying parameters in linear and nonlinear regression models when covariates are observed with Berkson measurement error. One class is based on the integral of the square of symmetrized weighted empirical of residuals while the other is based on a similar integral involving a weighted empirical of residual ranks. The former class requires the regression and measurement errors to be symmetric around zero while the latter class does not need any such assumption. The first class of estimators includes the analogs of the least absolute deviation and Hodges-Lehmann estimators while the second class includes an estimator that is asymptotically more efficient than these two estimators at some error distributions when there is no measurement error. In the case of linear model, no knowledge of the measurement error distribution is needed. We also develop these estimators for nonlinear models when the measurement error distribution is known and when it is unknown but validation data is available.

The strong consistency of M estimators of the regression parameters in linear models for negatively dependent random errors under some mild conditions is established, which is an essential improvement on the relevant results in the literature on the moment conditions and dependent errors. Especially, Theorems 1 and 2 of Wu (2006) are not only extended to the case of negatively dependent random errors, but also are improved essentially on the moment conditions.

SUMMARY Residual maximum likelihood (REML) estimation is often preferred to maximum likelihood estimation as a method of estimating covariance parameters in linear models because it takes account of the loss of degrees of freedom in estimating the mean and produces unbiased estimating equations for the variance parameters. In this paper it is shown that REML has an exact conditional likelihood interpretation, where the conditioning is on an appropriate sufficient statistic to remove dependence on the nuisance parameters. This interpretation clarifies the motivation for REML and generalizes directly to non-normal models in which there is a low dimensional sufficient statistic for the fitted values. The conditional likelihood is shown to be well defined and to satisfy the properties of a likelihood function, even though this is not generally true when conditioning on statistics which depend on parameters of interest. Using the conditional likelihood representation, the concept of REML is extended to generalized linear models with varying dispersion and canonical link. Explicit calculation of the conditional likelihood is given for the one-way lay-out. A saddlepoint approximation for the conditional likelihood is also derived.

Fuzzy regression analysis is an important regression analysis method to predict uncertain information in the real world. In this paper, the input data are crisp with randomness; the output data are trapezoid fuzzy number, and three different risk preferences and chaos optimization algorithm are introduced to establish fuzzy regression model. On the basis of the principle of the minimum total spread between the observed and the estimated values, risk-neutral, risk-averse, and risk-seeking fuzzy regression model are developed to obtain the parameters of fuzzy linear regression model. Chaos optimization algorithm is used to determine the digital characteristic of random variables. The mean absolute percentage error and variance of errors are adopted to compare the modeling results. A stock rating case is used to evaluate the fuzzy regression models. The comparisons with five existing methods show that our proposed method has satisfactory performance.

Estimation in linear regression models with measurement errors subject to single-indexed distortion

In the paper some possible evaluation criteria of robustness of statistical tests are presented. The criteria have been used in evaluating robustness of the test for significance of the parameters of linear regression model in the case of outliers existence.

A design method is presented that extends least mean squared (LMS) adaptation of time-varying parameters by including general linear time-invariant filters that operate on the instantaneous gradient vector. The aim is to track time-varying parameters of linear regression models in situations where the regressors are stationary or have slowly time-varying properties. The adaptation law is optimized with respect to the steady-state parameter error covariance matrix for time-variations modeled as vector-ARIMA processes. The design method systematically uses prior information about time-varying parameters to provide filtering, prediction, or fixed lag smoothing estimates for arbitrary lags. The method is based on a transformation of the adaptation problem into a Wiener filter design problem. The filter works in open loop for slow parameter variations, whereas a time-varying closed loop has to be considered for fast variations. In the latter case, the filter design is performed iteratively. The general form of the solution at each iteration is obtained by a bilateral Diophantine polynomial matrix equation and a spectral factorization. For white gradient noise, the Diophantine equation has a closed-form solution. Further structural constraints result in very simple design equations. Under certain model assumptions, the Wiener designed adaptation laws reduce to LMS adaptation. Compared with Kalman estimators, the channel tracking performance becomes nearly the same in mobile radio applications, whereas the complexity is, in general, much lower.

In this paper we propose a modified Winsorized regression procedure for estimating parameters in linear regression models. it is shown that these estimators provide a close approximation to Tiku's MMLE. By assuming contaminated normal distributions, Monte Carlo studies indicate that the modified Winsorized estimators are more efficient than Winsorized regression estimators.

Asymptotic properties of SPS confidence regions