Yuri K Belyaev

Computer Intensive Methods Based on Resampling in Analysis of Reliability and Survival Data

Computer Intensive Methods Based on Resampling in Analysis of Reliability and Survival Data

A class of non-parametric tests in the competing risks model for comparing two samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D1, obtained when the process is under control. Our main goal is to test if the failure rates in D1 are equal to or less than the failure rates in another data set D2, against “undesirable” neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests

Approach to Analysis of Self-Selected Interval Data

We analyze an approach to quantitative information elicitation in surveys that includes many currently popular variants as special cases. Rather than asking the individual to state a point estimate or select between given brackets, the individual can self-select any interval of choice. We propose a new estimator for such interval censored data. It can be viewed as an extension of Turnbulls estimator (Turnbull(1976)) for interval censored data. A detailed empirical example is provided, using a survey on the valuation of a public good. We estimate survival functions based on a Weibull and a mixed Weibull/exponential distribution and prove that a consistent maximum likelihood estimator exists and that its accuracy can be consistently estimated by re-sampling methods in these two families of distributions.

Asymptotical properties of nonparametric point estimators based on complexly structured reliability data with right-censoring

This paper presents a general approach to nonparametric estimation of unknown distribution functions and related characteristics such as cumulative hazard functions. It is based on the notion of portions of statistical data and on the property of discertely separated distributions of statistical data General assumptions are given under which the corresponding generalized maximum likelihood estimators are consistent and their deviations have asymptotically normal distributions, if the number of portions increases to indinity.

Two-Step Approach to Self-Selected Interval Data in Elicitation Surveys

We propose a novel two-step approach to elicitation in surveys and provide supporting statistical theory for the models suggested. The essential idea is to combine self-selected intervals in a first step and then employ brackets generated from the intervals in a second step. In this way we combine the advantages of selfselected intervals, mainly related to the fact that individuals often fi nd it difficult to report a precise point-estimate of a quantity of interest, with the documented usefulness of brackets. Because the brackets are generated from the first sample we sidestep the thorny problem of the optimal design of brackets and additional assumptions on dependency between the self-selected intervals and their points of interest. Our set-up necessitates development of new statistical models. First, we propose a stopping rule for sampling in the first step. Second, Theorem 1 proves that the proposed non-parametric ML-estimator of the underlying distribution function is consistent. Third, a special recursion for quick estimation of the ML-estimators is suggested. Theorem 2 shows that the accuracy of the estimator can be consistently estimated by resampling. Fourth, we have developed an R-package for efficient application of the method. We illustrate the approach using the problem of eliciting willingness-to-pay for a public good.

Methoden der Wahrscheinlichkeitsrechnung und Statistik bei der Analise von Zuverlassigkeitsdaten

Methoden der Wahrscheinlichkeitsrechnung und Statistik bei der Analise von Zuverlassigkeitsdaten

A moment-distance hybrid method for estimating a mixture of two symmetric densities

In clustering of high-dimensional data a variable selection is commonly applied to obtain an accurate grouping of the samples. For two-class problems this selection may be carried out by fitting a ...

The HRD-Algorithm: A General Method for Parametric Estimation of Two-Component Mixture Models

We introduce a novel approach to estimate the parameters of a mixture of two distributions. The method combines a grid approach with the method of moments and can be applied to a wide range of two- ...

Assessing Accuracy of Statistical Inferences by Resamplings

Suppose that a list of explanatory variables and corresponding random responses was obtained during a series of regression experiments. The characteristic of interest is the mean value of responses considered as a regression function of corresponding values of explanatory variables. For example, if responses are failure times of tested elements, then the conditional mean value of life time given the value of explanatory variable is one of the important reliability characteristics of the tested elements. The analysis of this type of data can be realized in the framework of linear heteroscedastic regression models. Here, one of the central problems is a consistent estimation of the unknown regression function when the size of data grows unboundedly. The problems related to analysis of regression data attracted many researches, see Wu [Ann. Statist. 14, 1261–1350 (1986)]. We give an approach to consistent solution of the problems under the assumption that values of explanatory variables are real numbers and the regression function is a polynomial with unknown degree and coefficients. The selection of regression function is based on resamplings from terms in the sum of the residuals estimated by the ordinary least squares method with various values of polynomial degree. In a similar way, resamplings from the weighted estimated residuals are used for consistent estimation of the deviations distributions of estimated coefficients from their true unknown values. The consistency of applied resamplings methods holds under certain assumptions, e.g. it is assumed that the residuals distributions have uniformly integrable second moments (assumption AW 2). Given in Appendix a variant of the Central Limit Resampling Theorem is used in the proofs of Theorems 1 and 2.

Asymptotic Properties of Estimators in a Model of Life Data with Warnings

Abstract We consider a model where elements of a single type are life tested. All elements are observed up to the time of their failures or censorings. Three types of events are possible to observe during life testing for each element: failure, censoring, and warning, where a warning can only be observed before a failure or before censoring has occurred. It is essential to know if warnings influence subsequent failures. Two subsets of data are simultaneously considered: the first consisting of only the times of the first occurrences of failure, censoring, or warning, and the second consisting of the times for those elements where warnings occurred before failures or censorings. The first subset belongs to the competing risks model, and the second consists of left-truncated data. Estimators of the cumulative hazard function before and after warnings are derived and proved to be consistent, with asymptotic normal distributions. A null hypothesis where the cumulative hazard functions before and after warnings are proportional and a corresponding alternative hypothesis that they are not proportional are defined. Under this null hypothesis an estimator for the constant of proportionality is derived and showed to be strongly consistent. Martingale techniques are used and numerical examples are provided.