Modified maximum likelihood estimator under the Jones and Faddy's skew t-error distribution for censored regression model

It is well-known that classical Tobit estimator of the parameters of the censored regression (CR) model is inefficient in case of non-normal error terms. In this paper, we propose to use the modified maximum likelihood (MML) estimator under the Jones and Faddy''s skew t-error distribution, which covers a wide range of skew and symmetric distributions, for the CR model. The MML estimators, providing an alternative to the Tobit estimator, are explicitly expressed and they are asymptotically equivalent to the maximum likelihood estimator. A simulation study is conducted to compare the efficiencies of the MML estimators with the classical estimators such as the ordinary least squares, Tobit, censored least absolute deviations and symmetrically trimmed least squares estimators. The results of the simulation study show that the MML estimators work well among the others with respect to the root mean square error criterion for the CR model. A real life example is also provided to show the suitability of the MML methodology.     

Robust pairwise multiple comparisons under short-tailed symmetric distributions

In one-way ANOVA, most of the pairwise multiple comparison procedures depend on normality assumption of errors. In practice, errors have non-normal distributions so frequently. Therefore, it is very important to develop robust estimators of location and the associated variance under non-normality. In this paper, we consider the estimation of one-way ANOVA model parameters to make pairwise multiple comparisons under short-tailed symmetric (STS) distribution. The classical least squares method is neither efficient nor robust and maximum likelihood estimation technique is problematic in this situation. Modified maximum likelihood (MML) estimation technique gives the opportunity to estimate model parameters in closed forms under non-normal distributions. Hence, the use of MML estimators in the test statistic is proposed for pairwise multiple comparisons under STS distribution. The efficiency and power comparisons of the test statistic based on sample mean, trimmed mean, wave and MML estimators are given and the robustness of the test obtained using these estimators under plausible alternatives and inlier model are examined. It is demonstrated that the test statistic based on MML estimators is efficient and robust and the corresponding test is more powerful and having smallest Type I error.     

Effects of non-normality and mild heteroscedasticity on estimators in regression

Several estimators are examined for the simple linear regression model under a controlled, experimental situation with multiple observations at each design point. The model is examined under normal and non-normal error distributions and mild heterogeneity of variances across the chosen design points. We consider the ordinary, generalized, and estimated generalized least squares estimators and several examples of M estimators. The asymptotic properties of the M estimator using the Huber ψ are presented under these conditions for the multiple regression model. A simulation study is also presented which indicates that the M estimator possesses strong robustness properties under the presence of both non-normality and mild heteroscedasticity o£ errors. Finally, the M estimates are compared to the least squares estimates in two examples.     

Hypothesis testing in mixture regression models

Summary.  We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of n ^−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p -value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results.     

A Comparison of Partially Adaptive and Reweighted Least Squares Estimation

The small sample performance of least median of squares, reweighted least squares, least squares, least absolute deviations, and three partially adaptive estimators are compared using Monte Carlo simulations. Two data problems are addressed in the paper: (1) data generated from non-normal error distributions and (2) contaminated data. Breakdown plots are used to investigate the sensitivity of partially adaptive estimators to data contamination relative to RLS. One partially adaptive estimator performs especially well when the errors are skewed, while another partially adaptive estimator and RLS perform particularly well when the errors are extremely leptokur-totic. In comparison with RLS, partially adaptive estimators are only moderately effective in resisting data contamination; however, they outperform least squares and least absolute deviation estimators.     

A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

Parameter estimation for a new distribution for the strength of brittle fibers: a simulation study

Parameter estimates of a new distribution for the strength of brittle fibers and composite materials are considered. An algorithm for generating random numbers from the distribution is suggested. Two parameter estimation methods, one based on a simple least squares procedure and the other based on the maximum likelihood principle, are studied using Monte Carlo simulation. In most cases, the maximum likelihood estimators were found to have somewhat smaller root mean squared error and bias than the least squares estimators. However, the least squares estimates are generally good and provide useful initial values for the numerical iteration used to find the maximum likelihood estimates.     

A new estimator for the unit root

We compare the ordinary least squares, weighted symmetric, modified weighted symmetric (MWS), maximum likelihood, and our new modification for least squares (MLS) estimator for first-order autoregressive in the case of unit root using Monte Carlo method. The Monte Carlo study sheds some light on how well the estimators and the predictors perform on different samples sizes. We found that MLS estimator is less biased and has less mean squared error (MSE) than any other estimators, and MWS predictor error performs well, in the sense of MSE, than any other predictors’ methods. The sample percentiles for the distribution of the τ statistic for the first, second, and third periods in the future, for alternative estimators, are reported to know if it agrees with those of normal distribution or not.     

Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using Kernel Density Method

In this article, we propose a nonparametric estimator for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method. The properties of the proposed kernel estimator are investigated and compared with well-known maximum likelihood and ordinary least squares estimators via a simulation technique. The mean squared error and the length of the bootstrap confidence interval are used as the basis criteria of the comparisons. The simulation study shows that the performance of the kernel estimator is acceptable as a general estimator. When the distribution of the data is assumed to be known, the maximum likelihood and ordinary least squares estimators perform better than the kernel estimator, while the kernel estimator is superior when the assumption of our knowledge of the data distribution is violated. A comparison among different estimators is achieved using a real data set.     

Adaptive Estimation Using Weighted Least Squares

This paper proposes an adaptive estimator that is more precise than the ordinary least squares estimator if the distribution of random errors is skewed or has long tails. The adaptive estimates are computed using a weighted least squares approach with weights based on the lengths of the tails of the distribution of residuals. Smaller weights are assigned to those observations that have residuals in the tails of long-tailed distributions and larger weights are assigned to observations having residuals in the tails of short-tailed distributions. Monte Carlo methods are used to compare the performance of the proposed estimator and the performance of the ordinary least squares estimator. The estimates that were studied in this simulation include the difference between the means of two populations, the mean of a symmetric distribution, and the slope of a regression line. The adaptive estimators are shown to have lower mean squared errors than those for the ordinary least squares estimators for short-tailed, long-tailed, and skewed distributions, provided the sample size is at least 20. The ordinary least squares estimator has slightly lower mean squared error for normally distributed errors. The adaptive estimator is recommended for general use for studies having sample sizes of at least 20 observations unless the random errors are known to be normally distributed.     

A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

Mixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second-order least squares estimator and a simulation-based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model.     

Multiple Linear Regression Model Under Nonnormality

Abstract

We consider multiple linear regression models under nonnormality. We derive modified maximum likelihood estimators (MMLEs) of the parameters and show that they are efficient and robust. We show that the least squares esimators are considerably less efficient. We compare the efficiencies of the MMLEs and the M estimators for symmetric distributions and show that, for plausible alternatives to an assumed distribution, the former are more efficient. We provide real-life examples.     

Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

When there are outliers in the carriers:the univariate case

Robust regression estimators studied to date are robust against non-normal distributions of the errors only If the carriers ‘Independent variables’ do not also contain outliers. Several alternative estimators that are robust even 1f there are outliers in the carriers are studied. Two estimators seem to be preferable, but even these can be very Inefficient ‘relative to least squares’ If the errors are normally distributed.     

Robust Estimation and Hypothesis Testing of Linear Contrasts in Analysis of Covariance with Stochastic Covariates

Estimators of parameters are derived by using the method of modified maximum likelihood (MML) estimation when the distribution of covariate X and the error e are both non-normal in a simple analysis of covariance (ANCOVA) model. We show that our estimators are efficient. We also develop a test statistic for testing a linear contrast and show that it is robust. We give a real life example.     

Statistical inference for partially time-varying coefficient errors-in-variables models

This paper studies the partially time-varying coefficient models where some covariates are measured with additive errors. In order to overcome the bias of the usual profile least squares estimation when measurement errors are ignored, we propose a modified profile least squares estimator of the regression parameter and construct estimators of the nonlinear coefficient function and error variance. The proposed three estimators are proved to be asymptotically normal under mild conditions. In addition, we introduce the profile likelihood ratio test and then demonstrate that it follows an asymptotically χ²${\chi }^2$ distribution under the null hypothesis. Finite sample behavior of the estimators is investigated via simulations too.     

Robust inference in an heteroscedastic measurement error model

In this paper we deal with robust inference in heteroscedastic measurement error models. Rather than the normal distribution, we postulate a Student t distribution for the observed variables. Maximum likelihood estimates are computed numerically. Consistent estimation of the asymptotic covariance matrices of the maximum likelihood and generalized least squares estimators is also discussed. Three test statistics are proposed for testing hypotheses of interest with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of simulations and an application to a real data set are also reported.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.