Bayesian X̄ control limits for exponentially distributed measurements

This article deals with the construction of an X? control chart using the Bayesian perspective. We obtain new control limits for the X? chart for exponentially distributed data-generating processes through the sequential use of Bayes’ theorem and credible intervals. Construction of the control chart is illustrated using a simulated data example. The performance of the proposed, standard, tolerance interval, exponential cumulative sum (CUSUM) and exponential exponentially weighted moving average (EWMA) control limits are examined and compared via a Monte Carlo simulation study. The proposed Bayesian control limits are found to perform better than standard, tolerance interval, exponential EWMA and exponential CUSUM control limits for exponentially distributed processes.     

Parameter estimation from interval-valued data using the expectation-maximization algorithm

This paper investigates on the problem of parameter estimation in statistical model when observations are intervals assumed to be related to underlying crisp realizations of a random sample. The proposed approach relies on the extension of likelihood function in interval setting. A maximum likelihood estimate of the parameter of interest may then be defined as a crisp value maximizing the generalized likelihood function. Using the expectation-maximization (EM) to solve such maximizing problem therefore derives the so-called interval-valued EM algorithm (IEM), which makes it possible to solve a wide range of statistical problems involving interval-valued data. To show the performance of IEM, the following two classical problems are illustrated: univariate normal mean and variance estimation from interval-valued samples, and multiple linear/nonlinear regression with crisp inputs and interval output.     

Flexible parametric estimation technique for a competing risks model with unobserved heterogeneity: a Monte Carlo study

We analyse a flexible parametric estimation technique for a competing risks (CR) model with unobserved heterogeneity, by extending a local mixed proportional hazard single risk model for continuous duration time to a local mixture CR (LMCR) model for discrete duration time. The state-specific local hazard function for the LMCR model is per definition a valid density function if we have either one or two destination states. We conduct Monte Carlo experiments to compare the estimated parameters of the LMCR model, and to compare the estimated parameters of a CR model based on a Heckman–Singer-type (HS-type) technique, with the data-generating process parameters. The Monte Carlo results show that the LMCR model performs better or at least as good as the HS-type model with respect to the estimated structure parameters in most of the cases, but relatively poorer with respect to the estimated duration-dependence parameters.     

Consistent estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data

In this paper, we propose a method of estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs well compared with another prominent method of estimation in terms of bias and root mean-squared error in small-sample situations. Finally, two real data sets are used for illustrating the proposed method.     

Existence,uniqueness and consistency of estimation of life characteristics of three-parameter Weibull distribution based on Type-II right censored data

In this paper, we propose a new method of estimation for the parameters and quantiles of the three-parameter Weibull distribution based on Type-II right censored data. The method, based on a data transformation, overcomes the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators are also consistent over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method of estimation performs well compared to some prominent methods in terms of bias and root mean squared error in small-sample situations. Finally, two real data sets are used to illustrate the proposed method of estimation.     

The exponentiated Fréchet regression: an alternative model for actuarial modelling purposes

ABSTRACT

In this paper we introduce the exponentiated Fréchet regression for modelling positive responses having a long-tailed distribution in a regression model, which are common in actuarial statistics. We propose two parameterizations each of which links the regression parameters with the explanatory variables. We then discuss the maximum likelihood estimation of the parameters both theoretically and empirically. In order to meet the needs of an actuary, closed-form expressions for certain risk measures for the exponentiated Fréchet distribution are also derived. We employ the proposed model to a motorcycle claim size data set.     

Estimation of the extended Weibull parameters and acceleration factors in the step-stress accelerated life tests under an adaptive progressively hybrid censoring data

ABSTRACT

Based on the tampered failure rate model under the adaptive Type-II progressively hybrid censoring data, we discuss the maximum likelihood estimators of the unknown parameters and acceleration factors in the general step-stress accelerated life tests in this paper. We also construct the exact and unique confidence interval for the extended Weibull shape parameter. In the numerical analysis, we describe the simulation procedures to obtain the adaptive Type-II progressively hybrid censoring data in the step-stress accelerated life tests and present an experimental data to illustrate the performance of the estimators.     

Bias corrected MLEs under progressive type-II censoring scheme

ABSTRACT

Censoring frequently occurs in survival analysis but naturally observed lifetimes are not of a large size. Thus, inferences based on the popular maximum likelihood (ML) estimation which often give biased estimates should be corrected in the sense of bias. Here, we investigate the biases of ML estimates under the progressive type-II censoring scheme (pIIcs). We use a method proposed in Efron and Johnstone [Fisher's information in terms of the hazard rate. Technical Report No. 264, January 1987, Stanford University, Stanford, California; 1987] to derive general expressions for bias corrected ML estimates under the pIIcs. This requires derivation of the Fisher information matrix under the pIIcs. As an application, exact expressions are given for bias corrected ML estimates of the Weibull distribution under the pIIcs. The performance of the bias corrected ML estimates and ML estimates are compared by simulations and a real data application.     

C93. The relevance of imaginary alternatives

Some practical approaches to the problem of choosing parameters which control the smoothness of kernel-based density estimators are investigated. Fixed and variable kernels are considered, and particularly simple approaches are investigated in the latter case. The performances of a wide range of estimators are compared in a simulation study.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.