Power Function Algorithm for Linear Regression Weights with Weibull Data Analysis

Abstract

Weighted Linear Regression (WLR) can be used to estimate Weibull
parameters. With WLR, failure data with less variance weigh heavier. These
weights depend on the total number of test objects, which is called the sample
size n, and on the index of the ranked failure data i. The calculation of weights
can be very challenging, particularly for larger sample sizes n and for non-
integer data ranking i, which usually occurs with random censoring. There is a
demand for a light-weight computing method that is also able to deal with non-
integer ranking indices. The present paper discusses an algorithm that is both
suitable for light-weight computing as well as for non-integer ranking indices.
The development of the algorithm is based on asymptotic 3-parameter power
functions that have been successfully employed to describe the estimated
Weibull shape parameter bias and standard deviation that both monotonically
approach zero with increasing sample size n. The weight distributions for given
sample size are not monotonic functions, but there are various asymptotic
aspects that provide leads for a combination of asymptotic 3-parameter power
functions. The developed algorithm incorporates 5 power functions. The per-
formance is checked for sample sizes between 1 and 2000 for the maximum
deviation. Furthermore the weight distribution is checked for very high simi-
larity with the theoretical distribution.