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On Equivalence of Likelihood-Based Confidence Bands for Fatigue-Life and Fatigue-Strength Distributions

Peng Liu, Yili Hong, Luis A. Escobar, William Q. Meeker

Abstract

Fatigue data arise in many research and applied areas and there have been statistical methods developed to model and analyze such data. The distributions of fatigue life and fatigue strength are often of interest to engineers designing products that might fail due to fatigue from cyclic-stress loading. Based on a specified statistical model and the maximum likelihood method, the cumulative distribution function (cdf) and quantile function (qf) can be estimated for the fatigue-life and fatigue-strength distributions. Likelihood-based confidence bands then can be obtained for the cdf and qf. This paper provides equivalence results for confidence bands for fatigue-life and fatigue-strength models. These results are useful for data analysis and computing implementation. We show (a) the equivalence of the confidence bands for the fatigue-life cdf and the fatigue-life qf, (b) the equivalence of confidence bands for the fatigue-strength cdf and the fatigue-strength qf, and (c) the equivalence of confidence bands for the fatigue-life qf and the fatigue-strength qf. Then we illustrate the usefulness of those equivalence results with two examples using experimental fatigue data.

On Equivalence of Likelihood-Based Confidence Bands for Fatigue-Life and Fatigue-Strength Distributions

Abstract

Fatigue data arise in many research and applied areas and there have been statistical methods developed to model and analyze such data. The distributions of fatigue life and fatigue strength are often of interest to engineers designing products that might fail due to fatigue from cyclic-stress loading. Based on a specified statistical model and the maximum likelihood method, the cumulative distribution function (cdf) and quantile function (qf) can be estimated for the fatigue-life and fatigue-strength distributions. Likelihood-based confidence bands then can be obtained for the cdf and qf. This paper provides equivalence results for confidence bands for fatigue-life and fatigue-strength models. These results are useful for data analysis and computing implementation. We show (a) the equivalence of the confidence bands for the fatigue-life cdf and the fatigue-life qf, (b) the equivalence of confidence bands for the fatigue-strength cdf and the fatigue-strength qf, and (c) the equivalence of confidence bands for the fatigue-life qf and the fatigue-strength qf. Then we illustrate the usefulness of those equivalence results with two examples using experimental fatigue data.
Paper Structure (26 sections, 28 equations, 4 figures)

This paper contains 26 sections, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Fatigue-Life and Fatigue-Strength Random Variables
  4. Related Literature and Contributions of This Work
  5. Overview
  6. Data, Model, and Statistical Inference with a Specified Fatigue-Life Model
  7. Data and Model
  8. Maximum Likelihood Estimation
  9. Likelihood-Ratio-Based Confidence Intervals
  10. Likelihood-ratio-based confidence intervals for a single parameter
  11. Likelihood-ratio-based confidence intervals for a scalar function of the parameters
  12. Likelihood-Ratio-Based Confidence Bands
  13. Likelihood confidence bands for fatigue-life qfs and cdfs when the fatigue-life model is specified
  14. Likelihood confidence bands for an induced fatigue-strength qf and an induced fatigue-strength cdf when the fatigue-life model is given
  15. Equivalence Results when the Fatigue-Life Model is Specified
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Plot showing fatigue-life and fatigue-strength lognormal distributions (a); Plot showing fatigue-life and fatigue-strength Weibull distributions (b).
  • Figure 2: Plot showing LR confidence bands for $\xi(w)$ as a collection of individual confidence intervals of $\xi(w)$ at individual $w$ (a); Plot showing a transposed LR confidence bands for $\xi^{-1}(v)$ as a collection of individual confidence intervals of $\xi^{-1}(v)$ at individual $v$ (b).
  • Figure 3: Fatigue-life cdf estimates at 500,000 cycles as a function of Stroke (a); Fatigue-life 0.10 quantile estimates as a function of Stroke (b); the dashed lines are pointwise 90% confidence bands.
  • Figure 4: Fraction failing as a function of percent strain for time 600 million cycles (a); Strength quantile as a function of percent strain for time 600 million cycles (b); the dashed lines are pointwise 90% confidence bands.