Selecting the best one of several meifull populations

Consider k(k ≥ 2) two-parameter Meibull populations. Using type II censored data we want to select a best population. Me have proposed procedures which can be used for maximum likelihood estimators or simplified linear estimators of Che unknown parameters. The ranking of the populations is done by comparing their reliabilities at certain fixed time or by comparing their 2-eh quantiles. In selected cases, the constants needed for the procedures are tabulated using Monte Carlo methods.     

A cell means analysis of mixed linear models

The purpose of this paper is to describe a simple procedure for the estima-tion of parameters in the unbalanced mixed linear model. There are implications for hypothesis testing and interval estimation, A feature of these estimators is that they are expressed in terms of simple formulas. This has obvious advantages for computations and small sample analysis. In addition, the formulas suggest useful diagnostic procedures for assessing the quality of the data as well as possible defects in the model assumptions. The concepts are illustrated with several examples. Evidence is presented to indicate that, in cases of modest imbalance, these estimators are highly efficient and dominate AOV estimates over most of the parameter space. In cases of more extreme imbalance, the results are qualitatively the same but the estimators are less efficient than the AOV estimators for small values of the parameters. The extension of this method to factorial models with missing cells is not complete.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Robust methods for personal-income distribution models

Statistical problems in modelling personal-income distributions include estimation procedures, testing, and model choice. Typically, the parameters of a given model are estimated by classical procedures such as maximum-likelihood and least-squares estimators. Unfortunately, the classical methods are very sensitive to model deviations such as gross errors in the data, grouping effects, or model misspecifications. These deviations can ruin the values of the estimators and inequality measures and can produce false information about the distribution of the personal income in a country. In this paper we discuss the use of robust techniques for the estimation of income distributions. These methods behave like the classical procedures at the model but are less influenced by model deviations and can be applied to general estimation problems.     

Using Weight Functions in Spatial Point Pattern Analysis with Application to Plant Ecology Data

This paper presents procedures for percentile estimation in the three-parameter inverse Gaussian (IG3) and the two-parameter inverse Gaussian (IG2) distributions. All procedures require first the estimation of distribution parameters and second the computation of the desired quantile at the estimated parameters. Parameter estimation is accomplished by maximum likelihood (ML) or a mixed moments (MXM) method. A Newton-Rahpson (NR) procedure is used for inverting the CDF. Simulation and asymptotic results are given for the resulting estimators. Two sets of real data are used to illustrate the procedures.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.     

The estimation of exchangeable binomial parameters

A general class of Bayesian estimators for several Binomial parameters is proposed. The estimati on procedures are based on a logistic transformation of the parameters and a class of symmetric prior distributions.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

A Simple Diagnostic Plot Connecting Robust Estimation, Outlier Detection, and False Discovery Rates

Robust estimation of parameters, and identification of specific data points that are discordant with an assumed model, are often treated as different statistical problems. The two aims are, however, closely inter-related and in many cases the two analyses are required simultaneously. We present a simple diagnostic plot that connects existing robust estimators with simultaneous outlier detection, and uses the concept of false discovery rates to allow for the multiple comparisons induced by considering each point as a potential outlier. It is straightforward to implement, and applicable in any situation for which robust estimation procedures exist. Several examples are given.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

Parametric estimation of mixtures of two uniform distributions

In this paper, we consider a mixture of two uniform distributions and derive L-moment estimators of its parameters. Three possible ways of mixing two uniforms, namely with neither overlap nor gap, with overlap, and with gap, are studied. The performance of these L-moment estimators in terms of bias and efficiency is compared to that obtained by means of the conventional method of moments (MM), modified maximum likelihood (MML) method and the usual maximum likelihood (ML) method. These intensive simulations reveal that MML estimators are the best in most of the cases, and the L-moment estimators are less subject to bias in estimation for some mixtures and more efficient in most of the cases than the conventional MM estimators. The L-moment estimators are, in some cases, more efficient than the ML and MML estimators.     

Numerical solution of likelihood equations for a first-order response surface model for mixture paired comparisons

Summary A simple procedure for numerical solution of the likelihood equations for estimating the regression parameters of a first-order response surface model for the treatment parameters of mixture paired comparison experiments is developed. It is demonstrated that, for defined rotatable designs, those regression parameters are simple functions of the main effect parameters of a corresponding factorial model with no interactions. The maximum likelihood estimators of those main effect parameters, and hence of their corresponding regression parameters, are obtained through using procedures of treatment contrasts, factorial and iterations. A numerical example is given to illustrate applications of the procedures developed in this paper.     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Studentized robust statistics for,main effects in a two-factor manova

Multiresponse experiments in two-faoior manova are considered. StalibLical procedures of the test and estimation, based on studentized robust statistics. for location parameters in the models arc piupused. Large sample properties of their procedures as the cell sizes tend to infinity are investigated. Although Fisher's consistency is assumed in the theory ol ili-estimators, it is not needed. in this paper. For the univariate case, it is found that the asymptotic relative efficiencies (ARE's) of the proposed procedures relative to classical procedures agrees with the classical A/Sisresults of Huber's one sample Mestimator relative to the sample mean. By simulation studies, it can be seen that the proposed estimators are more efficient than the least squares estimators except for the case where the underlying distribution is normal     

Interval estimation of the largest reliability of k weibull populations

A procedure is given for obtaining a random width confidence interval for the largest reliability of k Weibull populations. The procedure does not identify the populations for which the reliability would be a maximum. The maximum likelihood estimators or the simplified linear estimators of the reliability based on type II censored data are used. The cases considered include unknown shape parameters being equal or unequal. Simultaneous confidence intervals for the k reliabilities are also obtained. Tables for the lower and upper limits in selected cases are constructed using Monte Carlo methods.     

Additive hazards regression of current status data with auxiliary covariates

This paper discusses the regression analysis of current status failure time data arising from the additive hazards model with auxiliary covariates. As often occurs in practice, it is impossible or impractical to measure the exact magnitude of covariates for all subjects in a study. To compensate the missing information, some auxiliary covariates are utilized instead. We propose two easy-to-implement procedures for estimation of regression parameters by making use of auxiliary information. The asymptotic properties of the resulting estimators are established and extensive numerical studies indicate that both procedures work well in practice.     

Bayesian inference for the parameters of Weibull distribution under progressive Type-I interval censored data with beta-binomial removals

This article considers the problem of estimating the parameters of Weibull distribution under progressive Type-I interval censoring scheme with beta-binomial removals. Classical as well as the Bayesian procedures for the estimation of unknown model parameters have been developed. The Bayes estimators are obtained under SELF and GELF using MCMC technique. The performance of the estimators, has been discussed in terms of their MSEs. Further, expression for the expected number of total failures has been obtained. A real dataset of the survival times for patients with plasma cell myeloma is used to illustrate the suitability of the proposed methodology.     

Estimation of parameters of mixed failure time distribution

The problem of estimation of parameters of a mixture of degenerate and exponential distributions is considered. A new sampling scheme is proposed and the exact bias and the mean square error (MSE) of the maximum likelihood estimators of the parameters is derived. Moment estimators, their approximate biases and the MSE are obtained. Asymptotic distributions of the estimators are also obtained for both the cases.