An Exact Asymptotically Efficient Confidence Bound for Reliability in the Case of the Weibull Distribution

This paper presents a simple method for obtaining exact lower confidence bounds for reliabilities (tail probabilities) for items whose life times follow a Weibull distribution where both the “shape” and “scale” parameters are unknown. These confidence bounds are obtained both for the censored and non-censored cases and are asymptotically efficient. They are exact even for small sample sizes in that they attain the desired confidence level precisely. The case of an additional unknown “location” or “shift” parameter is also discussed in the large sample case. Tables are given of exact and asymptotic lower confidence bounds for the reliability for sample sizes of 10, 15, 20, 30, 50 and 100 for various censoring fractions.

[1]  W. Weibull,et al.  The phenomenon of rupture in solids , 1939 .

[2]  A. Basu.  Estimates of Reliability for Some Distributions Useful in Life Testing , 1964 .

[3]  R. Plackett.  Linear Estimation from Censored Data , 1958 .

[4]  J. L. Jaech.  Estimation of Weibull Distribution Shape Parameter When no More than Two Failures Occur per Lot , 1964 .

[5]  W. Weibull.  A statistical theory of the strength of materials , 1939 .

[6]  S. Dubey.  Asymptotic Properties of Several Estimators of Weibull Parameters , 1965 .

[7]  A. S. Qureishi,et al.  The Discximination Between Two Weibull Processes , 1964 .

[8]  J. D. Alanen,et al.  Sampling Inspection Plans for Discriminating Between Two Weibull Processes , 1965 .

[9]  J. Jung.  On linear estimates defined by a continuous weight function , 1956 .

[10]  On Precedence Life Testing , 1965 .

[11]  John H. K. Kao.  Computer Methods for Estimating Weibull Parameters in Reliability Studies , 1958 .

[12]  F. David,et al.  Statistical Estimates and Transformed Beta-Variables. , 1960 .

[13]  L. Weiss.  On the asymptotic distribution of an estimate of a scale parameter , 1963 .

[14]  A. Cohen,et al.  Maximum Likelihood Estimation in the Weibull Distribution Based On Complete and On Censored Samples , 1965 .

[15]  Albert H. Moore,et al.  POINT AND INTERVAL ESTIMATORS, BASED ON M ORDER STATISTICS, FOR THE SCALE PARAMETER OF A WEIBULL POPULATION WITH KNOWN SHAPE PARAMETER , 1965 .

[16]  Max Halperin,et al.  Maximum Likelihood Estimation in Truncated Samples , 1952 .

[17]  A. E. Sarhan,et al.  Contributions to order statistics , 1964 .

[18]  Albert H. Moore,et al.  Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples , 1965 .

[19]  W. Weibull.  A Statistical Distribution Function of Wide Applicability , 1951 .

[20]  J. T. Webster,et al.  A Method for Discriminating Between Failure Density Functions Used In Reliability Predictions , 1965 .