Linear regression parameter estimation methods for the weibull distribution

Ph.DDOCTOR OF PHILOSOPH

Reliability estimation based on operational data of manufacturing systems

Maintenance management has a direct influence on equipment reliability and safety. However, a large portion of traditional maintenance models and reliability analysis methods usually assumes that only perfect maintenance is performed on the system and the system will restore to as good as new regardless of the kind of preventive maintenance work-order that is performed. This is not practical in reality and may result in an inaccurate parametric estimation. The research objective of this paper is to develop a maximum likelihood estimation method to obtain more accurately estimated parameters based on the operational data of manufacturing systems, taking into consideration the difference between perfect and imperfect maintenance work-orders. Weibull distribution is specifically studied for this purpose. A practical case study based on industrial operational data from an automotive assembly line is performed to illustrate the implementation and efficiency of the proposed reliability estimation method. Copyright © 2008 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61212/1/959_ftp.pd

Reduced-bias estimation of some non-standard models

There is a persisting interest in methods that reduce bias in the estimation of parametric models. There is already a wide range of methods that achieve that goal, with a few of them also delivering beneficial side effects. For example, the bias-reducing adjusted scores approach of Firth (1993) has been shown to always deliver finite estimates in models like logistic regression even when the maximum likelihood (ML) estimator takes infinite values. Other proposals (e.g. reduced-bias M estimation in Kosmidis and Lunardon (2020), and indirect inference of Kuk (1995) have been shown to be able to reduce estimation bias even in cases where the model is partially specified, such as for general M-estimators. In this thesis, we examine the applicability, evaluate the performance and compare a range of bias reduction methods such as the bias-reducing adjusted score equations of Firth (1993), indirect inference and reduced-bias M estimation, in terms of their impact on estimation and inference, in well-used model classes in econometrics and statistics, which are beyond the various standard models that bias reduction methods have been used for before. In particular, we study the Heckit regression model which handles non-randomly selected samples where the observed range of the dependent variable is censored, i.e. it is only partially known whether it is above or below a fixed threshold. We also examine accelerated failure time models which are parametric survival models for censored lifetime observations. Finally, we consider two stratified models Sartori (2003) where interest lies in the estimation of a parameter in the presence of a set of nuisance parameters, whose dimension increases with the number of strata. The main challenge with these models is that even basic requirements, like consistency of the ML estimator, are not necessarily satisfied Neyman and Scott (1948). We focus on binomial matched pairs where the ML estimate of the parameter of interest may be infinite due to data separation. We propose a penalised version of the log-likelihood function based on adjusted responses which always results in a finite estimator of the log odds ratio. The probability limit of the penalised adjusted log-likelihood estimator is derived and it is shown that in certain settings the ML, conditional and modified profile log-likelihood estimators drop out as special cases of the former estimator. It is found that for the models of censored data, Firth adjustments are not available in closed form whereas indirect inference and reduced-bias M estimation are applicable and are an improvement over traditional ML estimation

Quantile-Quantile Methodology -- Detailed Results

The linear quantile-quantile relationship provides an easy-to-implement yet effective tool for transformation to and testing for normality. Its good performance is verified in this report

Monte Carlo modified profile likelihood in models for clustered data

The main focus of the analysts who deal with clustered data is usually not on the clustering variables, and hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques relying on the profile likelihood are often misleading. The use of alternative tools, such as modifications to the profile likelihood or integrated likelihoods, for making accurate inference on a parameter of interest can be complicated by the presence of nonstandard modelling and/or sampling assumptions. We show here how to employ Monte Carlo simulation in order to approximate the modified profile likelihood in some of these unconventional frameworks. The proposed solution is widely applicable and is shown to retain the usual properties of the modified profile likelihood. The approach is examined in two instances particularly relevant in applications, i.e. missing-data models and survival models with unspecified censoring distribution. The effectiveness of the proposed solution is validated via simulation studies and two clinical trial applications

Shrinkage, pretest and LASSO estimators in parametric and semiparametric linear models

The theory of pretest (Bancroft (1944)) and James-Stein (James and Stein (1961)) type shrinkage estimation has been quite well known for the last five decades though its application remains limited. In this dissertation, some contributions to different types of parametric and semiparametric linear models based on shrinkage and preliminary test estimation methods are made which improve on the maximum likelihood estimation method. The objective of this dissertation is to study the properties of improved estimators of the parameter of interest in parametric and semiparametric linear models and compare these estimators with the least absolute shrinkage and selection operator (Tibshirani (1996)) estimator. Chapter two contains a study of the properties of the shrinkage estimators of the parameters of interest in a Weibull regression model where the survival time may be subject to fixed censoring and the regression parameters are under linear restrictions. Asymptotic properties of the suggested estimators are established using the notion of asymptotic distributional risk. Bootstrapping procedures are used to develop confidence intervals. An extensive simulation study is conducted to assess the performance of the suggested estimators for moderate and large samples. In chapter three, we consider generalized linear models for binary and count data. Here, we propose James-Stein type shrinkage estimators, a pretest estimator and a Park and Hastie estimator. We demonstrate the relative performances of shrinkage and pretest estimators based on the asymptotic analysis of quadratic risk functions and it is found that the shrinkage estimators outperform the maximum likelihood estimator uniformly. On the other hand, the pretest estimator dominates the maximum likelihood estimator only in a small part of the parameter space, which is consistent with the theory. A Monte Carlo simulation study has been conducted to compare shrinkage, pretest and Park and Hastie type estimators with respect to the maximum likelihood estimator through relative efficiency. In chapter four, we consider a partial linear model where the vector of coefficients β in the linear part can be partitioned as (β 1, β2) where β1 is the coefficient vector for main effects and β2 is a vector for “nuisance” effects. In this situation, inference about β1 may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables, or from dropping the nuisance variables if there is evidence that they do not provide useful information (pre-testing). We investigate the asymptotic properties of Stein-type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein-type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein-type estimator outperforms the full model estimator uniformly. On the other hand, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty type estimator for partial linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty type estimators

Regression methods for survival and multistate models.

A common research interest in medical, biological, and engineering research is determining whether certain independent variables are correlated with the survival or failure times. Standard statistical techniques cannot usually be applied for failure-time data due to the lack of complete data or in other word, due to censoring. From a statistical perspective, the study of time to event data is even more challenging when further complexities such as high dimensionality or multivariablity is added to the model. In this dissertation, we consider the predicating patient survival from proteomic profile of patient serum using matrix-assisted laser desorption/ionization time-of-flight (MALDI-TOF) data of non-small cell lung cancer patients. Due to much larger dimension of features in a mass spectrum compared to the study sample size, traditional linear regression modeling of survival times with high number of proteomic features is not feasible. Hence, we consider latent factor and regularized/penalized methods for fitting such models in order to predict patient survival from the mass spectrometry features. Extensive numerical studies involving both simulated as well as real mass spectrometry data are used to compare four popular regression methods, namely, partial least squares (PLS), sparse partial least square (SPLS), least absolute shrinkage and selection operator (LASSO) and elastic net regularization, on processed spectra. Right censoring is handled through a residual based multiple imputation. Overall, more complex methods such as the elastic net and SPLS result in better performances provided the operational parameters are chosen carefully via cross validation. For survival time prediction, we recommend using the elastic net based on a selected set of features. As a type of multivariate survival data, multistate models have a wide range of applications. Most of the existing regression approaches to analyze such data are based on parametric and semi-parametric procedures in which one should rely on specific model structures. In this dissertation, we construct non-parametric regression estimators of a number of temporal functions in a multistate system based on a univariate continuous baseline covariate. These estimators include state occupation probabilities, state entry, exit and waiting (sojourn) times distribution functions of a general progressive (e.g. acyclic) multistate model. The data are subject to right censoring and the censoring mechanism is explainable by observable covariates that could be time dependent. The resulting estimators are valid even if the multistate process is non-Markov. The performance of the estimators is studied using a detailed simulation. We illustrate our estimators using a data set on bone marrow transplant patients. Finally, some extension of the proposed methods to more general case with multivariate covariates are presented along with plans for future developments