Statistical inference for the extreme value distribution under adaptive Type-II progressive censoring schemes

Adaptive Type-II progressive censoring schemes have been shown to be useful in striking a balance between statistical estimation efficiency and the time spent on a life-testing experiment. In this article, some general statistical properties of an adaptive Type-II progressive censoring scheme are first investigated. A bias correction procedure is proposed to reduce the bias of the maximum likelihood estimators (MLEs). We then focus on the extreme value distributed lifetimes and derive the Fisher information matrix for the MLEs based on these properties. Four different approaches are proposed to construct confidence intervals for the parameters of the extreme value distribution. Performance of these methods is compared through an extensive Monte Carlo simulation.     

Nested coordinate descent algorithms for empirical likelihood

Empirical likelihood (EL) as a nonparametric approach has been demonstrated to have many desirable merits. While it has intensive development in methodological research, its practical application is less explored due to the requirements of intensive optimizations. Effective and stable algorithms therefore are highly desired for practical implementation of EL. This paper bears the effort to narrow the gap between methodological research and practical application of EL. We try to tackle the computation problems, which are considered difficult by practitioners, by introducing a nested coordinate descent algorithm and one modified version to EL. Coordinate descent as a class of convenient and robust algorithms has been shown in the existing literature to be effective in optimizations. We show that the nested coordinate descent algorithms can be conveniently and stably applied in general EL problems. The combination of nested coordinate descent with the MM algorithm further simplifies the computation. The nested coordinate descent algorithms are a natural and perfect match with inferences based on profile estimation and variable selection in high-dimensional data. Extensive examples are conducted to demonstrate the performance of the nested coordinate descent algorithms in the context of EL.     

Assessing circular error probable when the errors are elliptical normal

An important problem of continuing interest to engineers is the need to assess the circular error probable (CEP), a measure of the impact accuracy of a projectile or a measure of GPS point positioning accuracy. One of the challenges in addressing this problem is to construct some accurate confidence bounds or intervals for CEP in the small sample settings, where certain amount of systematic biases exist in testing experiments. Currently there is no general method available to deal with this challenge due to the intractability of the distributions of the existing CEP estimators. In this paper, in order to meet this challenge, several new approximate formulas are derived for calculating CEP, which are more accurate than the existing ones but still simple to use. Both exact and empirical expressions for the derivatives of CEP with respect to the population means and variances are also given. Using these formulas, three kinds of confidence bounds or intervals for CEP are proposed, which are based on the parametric bootstrap, the asymptotic distribution, and the Cornish–Fisher expansion, respectively. Moreover, a bias-corrected estimator of CEP is provided. The performances of these procedures are evaluated based on some Monte Carlo simulation studies. Both the theoretical and simulation results show that the Cornish–Fisher expansion-based procedure performs slightly better than the other two procedures when the downrange and cross-range variances are assumed the same. However, when these two variances are different, the simulation demonstrates that the bootstrap approach can be superior to the Cornish–Fisher for the small samples (say n=10), and vice versa for the moderate samples (say n=20).     

Linear bootstrap methods for vector autoregressive moving-average models

We provide the theoretical justification of bootstrapping stationary invertible echelon vector autoregressive moving-average (VARMA) models using linear methods. The asymptotic validity of the bootstrap is established with strong white noise under parametric and nonparametric assumptions. Our methods are practical and useful for building reliable simulation-based inference and forecasting without implementing nonlinear estimation techniques such as ML which is usually burdensome, time demanding or impractical, particularly in big or highly persistent systems. The relevance of our procedures is more pronounced in the context of dynamic simulation-based techniques such as maximized Monte Carlo (MMC) tests [see Dufour J-M. Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and nonstandard asymptotics in econometrics. J Econom. 2006;133(2):443–477 and Dufour J-M, Jouini T. Finite-sample simulation-based tests in VAR models with applications to Granger causality testing. J Econom. 2006;135(1–2):229–254 for the VAR case]. Simulation evidence shows that, compared with conventional asymptotics, our bootstrap methods have good finite-sample properties in approximating the actual distribution of the studentized echelon VARMA parameter estimates, and in providing echelon parameter confidence sets with satisfactory coverage.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

The effectiveness of bootstrap methods in evaluating skewed auditing populations: a simulation study

This article describes a comparison among four bootstrap methods: the percentile, reflective, bootstrap-t, and variance stabilized bootstrap-t using a simple new stabilization procedure. The four methods are employed in constructing upper confidence bounds for the mean error in a wide variety of audit populations. The simulation results indicate that the variance stabilized bootstrap-t bound is to be preferred. It exhibits reliable coverage while maintaining reasonable tightness.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

Confidence Intervals for Variance Components in Measurement System Capability Studies

In Measurement System Analysis a relevant issue is how to find confidence intervals for the parameters used to evaluate the capability of a gauge. In literature approximate solutions are available but they produce so wide intervals that they are often not effective in the decision process. In this article we introduce a new approach and, with particular reference to the parameter γ_R, i.e., the ratio of the variance due to the process and the variance due to the instrument, we show that, under quite realistic assumptions, we obtain confidence intervals narrower than other methods. An application to a real microelectronic case study is reported.     

A note on effective sample size for constructing confidence intervals for the difference of two proportions

Proportion differences are often used to estimate and test treatment effects in clinical trials with binary outcomes. In order to adjust for other covariates or intra-subject correlation among repeated measures, logistic regression or longitudinal data analysis models such as generalized estimating equation or generalized linear mixed models may be used for the analyses. However, these analysis models are often based on the logit link which results in parameter estimates and comparisons in the log-odds ratio scale rather than in the proportion difference scale. A two-step method is proposed in the literature to approximate the calculation of confidence intervals for the proportion difference using a concept of effective sample sizes. However, the performance of this two-step method has not been investigated in their paper. On this note, we examine the properties of the two-step method and propose an adjustment to the effective sample size formula based on Bayesian information theory. Simulations are conducted to evaluate the performance and to show that the modified effective sample size improves the coverage property of the confidence intervals.