Bayesian estimation of regression parameters in elliptical measurement error models

The main object of this paper is to discuss the Bayes estimation of the regression coefficients in the elliptically distributed simple regression model with measurement errors. The posterior distribution for the line parameters is obtained in a closed form, considering the following: the ratio of the error variances is known, informative prior distribution for the error variance, and non-informative prior distributions for the regression coefficients and for the incidental parameters. We proved that the posterior distribution of the regression coefficients has at most two real modes. Situations with a single mode are more likely than those with two modes, especially in large samples. The precision of the modal estimators is studied by deriving the Hessian matrix, which although complicated can be computed numerically. The posterior mean is estimated by using the Gibbs sampling algorithm and approximations by normal distributions. The results are applied to a real data set and connections with results in the literature are reported.     

Preliminary test estimation in system regression models in view of asymmetry

In this paper, we consider the system regression model introduced by Arashi and Roozbeh (Comput Stat 30:359–376, 2015) and study the performance of the feasible preliminary test estimator (FPTE) both analytically and computationally, under the assumption that constraints may hold on the vector parameter space. The performance of the FPTE is analysed through a Monte Carlo simulation study under bounded and or asymmetric loss functions. An application of the so-called Cobb–Douglas production function in economic modelling together with the results from the simulation study shows that the bounded linear exponential (BLINEX) loss function outperforms the linear exponential loss function (LINEX) by comparing risk values.     

Combined regression models

The goal of this paper is two-fold. First, new regression models obtained by combinations of the least squares (LS), minimax (MM), and the least sum of absolute deviations (LSAD) are proposed. Second, measures for assessing the influence of observations on the fitted models are suggested. The paper is interdisciplinary because the theory behind the proposed method draws from results in the operations research area. The methods are illustrated by their application to some examples and graphical illustrations are given.     

Seemingly unrelated regression models

The cross-covariance matrix of observation vectors in two linear statistical models need not be zero matrix. In such a case the problem is to find explicit expressions for the best linear unbiased estimators of both model parameters and estimators of variance components in the simplest structure of the covariance matrix. Univariate and multivariate forms of linear models are dealt with.     

Multivariate spatial regression models

This paper describes the inference procedures required to perform Bayesian inference to some multivariate econometric models. These models have a spatial component built into commonly used multivariate models. In particular, the common component models are addressed and extended to accommodate for spatial dependence. Inference procedures are based on a variety of simulation-based schemes designed to obtain samples from the posterior distribution of model parameters. They are also used to provide a basis to forecast new observations.     

Random weightingT-statistics in linear regression models

In this paper, we have constructed a random weighting statistic to approximate the distribution of studentized least square estimator in a linear regression model with ideal accuracyo(n ^–1/2). Thus, we have provided a more practical distribution approximating method.Supported by the Doctoral Program Foundation of the Institute of Higher Education and the National Natural Science Foundation of China.     

Bayesian inference for dependent elliptical measurement error models

In this article we provide a Bayesian analysis for dependent elliptical measurement error models considering nondifferential and differential errors. In both cases we compute posterior distributions for structural parameters by using squared radial prior distributions for the precision parameters. The main result is that the posterior distribution of location parameters, for specific priors, is invariant with respect to changes in the generator function, in agreement with previous results obtained in the literature under different assumptions. Finally, although the results obtained are valid for any elliptical distribution for the error term, we illustrate those results by using the student-t distribution and a real data set.     

CLT in functional linear regression models

We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.     

Heteroscedasticity checks for regression models

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the nonparametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach may readily be extended to handle partial linear, and linear autoregressive models.     

Simple stochastic models of cracks

The stochastic Markov models describe various natural and technological processes. They are often used in rather diverse areas. The Markov models are distinguished with discrete time and relatively few states. In concrete cases, such models allow us to carry out effective calculations. In the article are considered some special models with two, four and eight states. The processes are simulated that are connected with hydraulic fracturing and wave front development.     

Semiparametric random coefficient regression models

Linear regression models with random coefficients express the idea that each individual sampled may have a different linear response function. Technically speaking, random coefficient regression encompasses a rich variety of submodels. These include deconvolution or affine-mixture models as well as certain classical linear regression models that have heteroscedastic errors, or errors-in-variables, or random effects. This paper studies minimum distance estimates for the coefficient distributions in a general, semiparametric, random coefficient regression model. The analysis yields goodness-of-fit tests for the semiparametric model, prediction regions for future responses, and confidence regions for the distribution of the random coefficients.This research was supported in part by NSF Grant DMS 9001710.     

Multiple deletion diagnostics in beta regression models

We consider the problem of identifying multiple outliers in a general class of beta regression models proposed by Ferrari and Cribari-Neto (J Appl Stat 31:799–815, 2004). The currently available single-case deletion diagnostic measures, e.g., the standardized weighted residual (SWR), the Cook-like distance (LD), etc., often fail to identify multiple outlying observations, because they suffer from the well-known problems of masking and swamping effects. In this article, we develop group deletion diagnostic measures, such as generalized SWR, generalized LD, generalized DFFITS and generalized DFBETAS, and suggest a simple procedure for identifying multiple outliers using these. The performance of the proposed methods is investigated through simulation studies and two practical examples.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

Testing in linear composite quantile regression models

Composite quantile regression (CQR) can be more efficient and sometimes arbitrarily more efficient than least squares for non-normal random errors, and almost as efficient for normal random errors. Based on CQR, we propose a test method to deal with the testing problem of the parameter in the linear regression models. The critical values of the test statistic can be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and power evaluation is possible under the local alternatives. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from a walking behavior survey.     

RANDOM WEIGHTING APPROXIMATION IN LINEAR REGRESSION MODELS

ＲＡＮＤＯＭＷＥＩＧＨＴＩＮＧＡＰＰＲＯＸＩＭＡＴＩＯＮＩＮＬＩＮＥＡＲＲＥＧＲＥＳＳＩＯＮＭＯＤＥＬＳＳＨＩＪＩＡＮ（ＤｅｐａｒｔｍｅｎｔｏｆＰｒｏｂａｂｉｌｉｔｙａｎｄＳｔａｔｉｓｔｉｃｓ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１００８...     

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

Estimation of cusp in nonregular nonlinear regression models

The asymptotic properties of the least squares estimator of the cusp in some nonlinear nonregular regression models is investigated via the study of the weak convergence of the least squares process generalizing earlier results in Prakasa Rao (Statist. Probab. Lett. 3 (1985) 15).     

Diagnostic checking for multivariate regression models

Diagnostic checking for multivariate parametric models is investigated in this article. A nonparametric Monte Carlo Test (NMCT) procedure is proposed. This Monte Carlo approximation is easy to implement and can automatically make any test procedure scale-invariant even when the test statistic is not scale-invariant. With it we do not need plug-in estimation of the asymptotic covariance matrix that is used to normalize test statistic and then the power performance can be enhanced. The consistency of NMCT approximation is proved. For comparison, we also extend the score type test to one-dimensional cases. NMCT can also be applied to diverse problems such as a classical problem for which we test whether or not certain covariables in linear model has significant impact for response. Although the Wilks lambda, a likelihood ratio test, is a proven powerful test, NMCT outperforms it especially in non-normal cases. Simulations are carried out and an application to a real data set is illustrated.