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Only few failure data can be collected because of increasing costs and reliability. Therefore, improving the estimation accuracy of Weibull distribution with small samples has become an urgent problem. In this study, a method for estimating the Weibull parameter and analyzing reliability is proposed based on the successive approximation method. In this method, the effect of sample size on the location parameter is considered to establish an iterative model of the location parameter. The iterative equations of scale and shape parameters are constructed based on the location parameter and failure data. Then, an extensive Monte-Carlo simulation study is conducted to compare the performance of the proposed method with the existing methods. Simulation results show that the proposed method is superior for estimating Weibull parameters and reliability in terms of precision, robustness, and efficiency. Finally, two real examples are analyzed to illustrate the application of the proposed method. [ABSTRACT FROM AUTHOR]

The most applied principles for parameter estimation are maximum likelihood and least square error. This paper presents a new principle with regard to the parameter estimation of the three-parameter Weibull distribution. By transforming the cumulative distribution function, constructed is a mapping from the value of the random variable and its corresponding cumulative distribution probability to the scale parameter. The scale parameter estimated by such a mapping is the random variable value and the corresponding cumulative distribution probability dependent when the shape parameter and/ or location parameter applied in the mapping is subject to error. Given a set of random variable values or a set of sample values, a larger error in the shape/location parameter brings about larger differences between the scale parameter values obtained with the individual random variable values or sample values, respectively. Based on such a causal relationship between the discrepancy and the shape/ location parameter value applied in the mapping relation, a new parameter estimation method is proposed. For the Weibull distribution parameter estimation according to a set of sample values, the right shape parameter and location parameter are those minimizing the discrepancy between the scale parameter values obtained with the individual sample values, respectively. Case studies demonstrate that the proposed method outperforms the maximum likelihood method and the Weibull plot-based least squares method. [ABSTRACT FROM AUTHOR]

The three-parameter Weibull distribution is popular in reliability analysis. However, in the context of a small sample size, the issue of parameter estimation of the three-parameter Weibull distribution is difficult to address. If only a small sample size is available, a reasonable method is to empirically determine the shape parameter, and the emphasis is placed on location parameter estimation. This paper presents a theoretical model to establish the relationship between the location parameter, the minimal sample value, and the sample size. Moreover, an iterative method is proposed to estimate the Weibull location parameter and scale parameter with an empirical value for the shape parameter. Compared with the maximum likelihood method (MLE), the Weibull plot with the sample correlation coefficient and TL-moment, the proposed method can more effectively estimate the parameters of the three-parameter Weibull distribution. [ABSTRACT FROM AUTHOR]