10-04-2012, 03:32 PM
PARAMETER ESTIMATION FOR THE WEIBULL DISTRIBUTION
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TECHNIQUES FOR ESTIMATING WEIBULL PARAMETERS
There are, however, no correspondingly simple and statistically correct methods to calculate α ,β and γ for a Weibull distribution.
The most common technique employed in engineering to estimate α and β is a graphical method using Weibull probability paper. This method is "observer-dependent" and has resulted in some doubt as to the suitability of the Weibull distribution to represent experimental data with a satisfactory degree of accuracy .
Since 1965, extensive research has been conducted by statisticians into developing other reasonably simple methods. These include a Maximum Likelihood technique requiring a computer program, and "linear" estimation procedures involving reference to tables of constants.
Graphical Estimation
This widely accepted and convenient method involves the use of Weibull probability paper which can be homemade or obtained commercially (Figure 1). The time (or voltage) axis and the cumulative probability axis are scaled so that data drawn from a Weibull distribution, will appear as a straight line.
If experimental data cannot be represented by a reasonably straight line, the data may not belong to a Weibull distribution with γ = 0. To use Weibull probability paper, the failure times for a given voltage (or voltages) are first ordered from smallest to largest.
CONCLUS IONS
The statistical analysis of insulation breakdown can be summarized as follows:
The Weibull distribution is a general distribution which can fit a wide variety of data. Inherent in this flexibility, however, is an insensitivity to apparently large deviations in experimental data. This can result in misjudgement of various factors influencing dielectric systems.
(2) The easiest objective method to calculate the Weibull parameters α and β is the Maximum Likelihood technique. This requires only a modest computer program applicable to any number of sample breakdowns or censoring.
(3) Graphical parameter estimation will not necessarily yield statistically good estimates of α and β, but can be used to obtain first estimates for the Maximum Likelihood computation and to see if the experimental data are reasonably represented by the Weibull distribution. For uncensored experiments with large sample sizes, the graphical estimates tend to converge to those obtained analytically.