Weibull distribution is used to find the failure rates, and reliability analysis. In 2-parameters analysis, there are 2 parameters under this distribution, one is beta which is the shape parameter or the slope of the regression function of the data, and other is eta, which is the scale parameter in the distribution.
There are 2 ways with which we can estimate the parameters’ values of the Weibull failure distribution, that are, maximum likelihood method and the medium rank regression method. Both give the different values. According to Evans, Kretschmann and Green (2019), for the small data set medium rank method is the best method to estimate the parameters’ values.
Under the medium rank regression values, the following steps will be taken,
Step 1: Find independent and dependent variables, where independent variable = ln(rank of the cycles), and dependent variable = ln[ln(1-pi)], where pi = (rank - 0.3) / (number of sample size + 0.4), and pi = medium rank, and i = time or rank of the cycle (CFI, 2020).
Step 2: Then regressing ln(cycles) on dependent variable, and get the regression line:
Y = -13.245 + 3.013 ln(cycles) (Antonitsin, 2012)
From the regression line, the intercept value = eta’s estimate = -13.245, and the slope value = beta’s parameter = 3.013.
Beta represents the shape parameter of the Weibull distribution function or the slope of the function which tells the change in y due to change in x, in other words, it tells the change in probability or the rate of the failure due to change in cycles of the failures.
In the given situation, the beta value is positive and approximately equals to 3 and greater than 1, which tells the following things:-
Using the formula for the reliability that is,
R(t) = exp [-(time / scale)^ shape] (HBM Prenscia Inc., 2020)
Therefore, the reliability for 90000 cycles will be as follows:
R(90) = exp [-(90 / -13.245)^3.013] = 2.028 e^(-140)
We have taken 90 as the cycles are here, taken in units of 000’s.
From the regression analysis, we get the confidence intervals for ln(cycles) for 90% that is, (2.623, 3.402). Therefore, for cycles, with 90% reliability, the confidence interval will be (13.777, 30.024). Therefore, the cycles should start at 14000 cycles approximately to have reliability of 90% in the failure distribution of data.
HBM Prenscia Inc. (2020). Characteristics of the Weibull distribution. Weibull. Retrieved from https://www.weibull.com/hotwire/issue14/relbasics14.htm
Antonitsin, A. (2012). Statistical methods in reliability testing. Simon Fraser University. Retrieved from: https://www.stat.sfu.ca/content/dam/sfu/stat/alumnitheses/2012/Thesis_Alexey_full_4-26-2012.pdf
Evans, J.W., Kretschmann, D.E. & Green, D.W. (2019). United States Department of Agriculture. Retrieved from: https://www.fpl.fs.fed.us/documnts/fplgtr/fpl_gtr264.pdf
CFI. (2020). Excel Weibull distribution. Retrieved from: https://corporatefinanceinstitute.com/resources/excel/functions/weibull-dist-excel-weibull-distribution/
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