It is one of the most fundamental steps to clarify the parameters of the initial distribution of structural member's fatigue life, whenever the reliability of any structure is analized using the reliability theory. Whittaker and others reported some studies on fatigue life distributions accepting the Weibull distribution with two parameters but without the location parameter.
However, the Weibull distribution with three parameters including the location parameter should be considered as the distribution function at a stress level near the fatigue limit of a material. Since structures are designed often based on the fatigue limit of the members, it is important for the reliability analysis of the structures to clarify the fatigue life distribution at such stress levels.
From the above viewpoint, the estimation of three parameters of the Weibull distribution was made in this study by means of several methods. Before the estimation was performed, Monte Carlo technique was applied to prepare many experimental data of fatigue life governed by the Weibull distribution with three known parameters. Then the values of each parameter estimated by several methods were compared with the true values given in advance, and the applicability of these estimating methods was examined. Furthermore, the sample size necessary to estimate the parameters of the initial distribution was discussed quantitatively.

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This paper attempts to identify the best method of estimating the parameters of the two-parameter Weibull distribution using data from complete samples. Various methods of estimating the Weibull parameters have been proposed, but adequate criteria in user side to judge which method is the optimum have not been developed. In several papers comparing the biases and expected losses in the estimators of the Type I extreme value distribution has been used. However, the two parameters are simultaneously estimated from a sample, not one by one separately, and they are used in the Weibull distribution. This paper uses the concepts of the Kolmogorov-Smirnov and chi-square goodness-of-fit tests, as well as biases and expected losses of estimators in the Weibull and Type I extreme value distributions, as criteria to select an appropriate method. Monte Carlo simulation provides sets of 2000 complete samples for each combination of sample sizes n=3, 6, 10, and 20, and shape parameters β=0.5, 1, 3, and 10. The eight methods of three kinds of graphical plotting techniques, two kinds of moments methods, maximum-likelihood estimation, and two kinds of linear estimation techniques are compared. This investigation concludes the best linear unbiased estimators (BLUE) to be the best among the eight methods for general use and recommends median ranks for probability plotting.

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In reliability analysis of machines and structural members, it is important to clarify the fatigue life distribution of these engineering items, and Weibull distribution is often accepted as the original distribution of the fatigue life. Although various kinds of estimation procedures have been developed on three parameters of Weibull distribution, the statistical properties of each estimated parameter still remain unsolved for any estimation method.
The authors performed the parameter estimation by means of several methods and discussed on the distribution characteristics of estimates of Weibull parameters. In this study, three different kinds of estimation were made on each parameter of Weibull distribution, and, the estimates obtained from the respective methods, including those given in the previous paper, were compared with each other in particular attention to statistical properties such as convergency or distribution of estimates. Finally, the most preferable method for parameter estimation of Weibull distribution was selected and quantitative discussions were also performed on the sample size required to determine, experimentally, the parameter of the original distribution.

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