Intelligent Constructing Exact Tolerance Limits for Prediction of Future Outcomes Under Parametric Uncertainty

2021 ◽  
pp. 701-729
Author(s):  
Nicholas A. Nechval
Keyword(s):  
Parametric Uncertainty ◽  
Statistical Problem ◽  
Tolerance Limits ◽  
Novel Technique ◽  
Simulation Monte Carlo ◽  
Statistical Tolerance ◽  
Future Outcomes ◽  
Tolerance Factors ◽  
Two Parameter ◽  
Future Sample

The problem of constructing one-sided exact statistical tolerance limits on the kth order statistic in a future sample of m observations from a distribution of log-location-scale family on the basis of an observed sample from the same distribution is considered. The new technique proposed here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. The exact tolerance limits on order statistics associated with sampling from underlying distributions can be found easily and quickly making tables, simulation, Monte Carlo estimated percentiles, special computer programs, and approximation unnecessary. Finally, numerical examples are given, where the tolerance limits obtained by using the known methods are compared with the results obtained through the proposed novel technique, which is illustrated in terms of the extreme-value and two-parameter Weibull distributions.

Novel Approaches to Prediction of a Future Number of Failures Based on Previous In-Service Inspections

2018 ◽  
pp. 225-248
Author(s):  
Nicholas A. Nechval ◽  
Konstantin N. Nechval
Keyword(s):  
Performance Index ◽  
Failure Time ◽  
Parametric Uncertainty ◽  
Statistical Problem ◽  
Unknown Parameters ◽  
Invariant Embedding ◽  
Prediction Limits ◽  
Novel Approaches ◽  
Special Case ◽  
Two Parameter

In this chapter, we present novel approaches to predictions of the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the previous in-service inspections of the same sample. The failure-time of such units is modeled with a distribution from a two-parameter Weibull distribution. The different cases of parametric uncertainty are considered. The pivotal quantity averaging approach proposed here for constructing point prediction and simple prediction limits emphasizes pivotal quantities relevant for eliminating unknown parameters from the problems and represents a special case of the method of invariant embedding of sample statistics into a performance index applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space. For illustration, a numerical example is given.

scholarly journals A Novel Intelligent Technique of Invariant Statistical Embedding and Averaging via Pivotal Quantities for Optimization or Improvement of Statistical Decision Rules under Parametric Uncertainty

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Probability Distributions ◽  
Decision Rules ◽  
Parametric Uncertainty ◽  
Decision Criterion ◽  
Unknown Parameters ◽  
Statistical Decision ◽  
Intelligent Technique ◽  
Statistical Decisions ◽  
Statistical Tolerance ◽  
Future Outcomes

In the present paper, for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a decision criterion and averaging this criterion over pivots’ probability distributions is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, illustrative examples of intelligent constructing exact statistical tolerance limits for prediction of future outcomes coming from log-location-scale distributions under parametric uncertainty are given

One-Sided β-Content Tolerance Factors forthe Two Parameter Exponential Distribution

Technometrics ◽  
1976 ◽  
Vol 18 (3) ◽  
pp. 333-340 ◽  
Author(s):  
William C. Guenther ◽  
S. A. Patil ◽  
V. R.R. Uppuluri
Keyword(s):  
Exponential Distribution ◽  
Tolerance Factors ◽  
Two Parameter

A New Technique for Optimization of Product Acceptance Process in Terms of Misclassification Probability

2018 ◽  
pp. 1-32
Author(s):  
Nicholas A. Nechval ◽  
Konstantin N. Nechval
Keyword(s):  
Weibull Distribution ◽  
Optimization Technique ◽  
Parametric Uncertainty ◽  
Lifetime Distribution ◽  
Statistical Quality Control ◽  
Misclassification Probability ◽  
Distribution Parameters ◽  
A New Technique ◽  
Two Parameter

A product acceptance process is an inspecting one in statistical quality control or reliability tests, which are used to make decisions about accepting or rejecting lots of products to be submitted. This process is important for industrial and business purposes of quality management. To determine the optimal parameters of the product acceptance process under parametric uncertainty of underlying lifetime models (in terms of misclassification probability), a new optimization technique is proposed. The most popular lifetime distribution used in the field of product acceptance is a two-parameter Weibull distribution, with the assumption that the shape parameter is known. Such oversimplified assumptions can facilitate the follow-up analyses, but may overlook the fact that the lifetime distribution can significantly affect the estimation of the failure rate of a product. Therefore, the situations are also considered when both Weibull distribution parameters are unknown. An illustrative numerical example is given.

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