fitWeibull {ForestFit}

Estimating parameters of the Weibull distribution through classical methods

Description

Estimates the parameters of the two- and three-parameter Weibull model with pdf and cdf given by

f(x;\alpha,\beta,\theta)=\frac{\alpha}{\beta} \left(\frac{x-\theta}{\beta }\right)^{\alpha -1} \exp \biggl\{-\left(\frac{x-\theta}{\beta } \right)^{\alpha } \biggr\},

and

F(x;\alpha,\beta,\theta)=1- \exp \biggl\{-\left(\frac{x-\theta}{\beta } \right)^{\alpha } \biggr\},

where x>\theta, \alpha > 0, \beta >0 and -\infty<\theta<\infty. Here, the parameters \alpha, \beta, and \theta are known in the literature as the shape, scale, and location, respectively. If \theta=0, then f(x;\alpha,\beta) and F(x;\alpha,\beta) in above are the pdf and cdf of a two-parameter Weibull distribution, respectively.

Usage

fitWeibull(data, location, method, starts)

Arguments

data

Vector of observations

starts

Initial values for starting the iterative procedures such as Newton-Raphson.

location

Either TRUE or FALSE. If location=TRUE, then shift parameter will be considered; otherwise the shift parameter omitted.

method

Used method for estimating the parameters. In the two-parameter case, methods are "greg1" (for the method of generalized regression type 1), "greg2" (for the method of generalized regression type 2), "lm" (for the method of L-moment), "ml" (for the method of maximum likelihood (ML)), "mlm" (for the method of logarithmic moment), "moment" (for the method of moment), "pm" (for the method of percentile), "rank" (for the method of rank correlation), "reg" (for the method of least square), "ustat" (for the method of U-statistic), "wml" (for the method of weighted ML), and "wreg" (for the method of weighted least square). In three-parameter case the methods are "mle" (for the method of ML), "mm1" (for the method of modified moment (MM) type 1), "mm2" (for the method of MM type 2), "mm3" (for the method of MM type 3), "mml1" (for the method of modified ML type 1), "mml2" (for the method of modified ML type 2), "mml3" (for the method of modified ML type 3), "mml4" (for the method of modified ML type 4), "moment" (for the method of moment), "mps" (for the method of maximum product spacing), "tlm" (for the method of T-L moment), and "wml" (for the method of weighted ML).

Details

For the method wml, all weights have been provided for sample size less that or equal to 100. This means that both methods ml and wml give the same estimates for samples of size larger than 100.

Value

A list of objects in two parts given by the following:

1. Estimated parameters for two- or three-parameter Weibull distribution.

2. A sequence of goodness-of-fit measures consist of Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling (AD), Cram\'eer-von Misses (CVM), Kolmogorov-Smirnov (KS), and log-likelihood (log-likelihood) statistics.

Author(s)

Mahdi Teimouri

References

R. C. H. Cheng and M. A. Stephens, 1989. A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76(2), 385-392.

C. A. Clifford and B. Whitten, 1982. Modified maximum likelihood and modified moment estimators for the three-parameter Weibull distribution, Communication in Statistics-Theory and Methods, 11(23), 2631-2656.

D. Cousineau, 2009. Nearly unbiased estimators for the three-parameter Weibull distribution with greater efficiency than the iterative likelihood method, British Journal of Mathematical and Statistical Psychology, 62, 167-191.

G. Cran, 1988. Moment estimators for the 3-parameter Weibull distribution, IEEE Transactions on Reliability, 37(4), 360-363.

J. R. Hosking, 1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics, Journal of the Royal Statistical Society. Series B (Methodological), 52(1), 105-124.

Y. M. Kantar, 2015. Generalized least squares and weighted least squares estimation methods for distributional parameters, REVSTAT-Statistical Journal, 13(3), 263-282.

M. Teimouri and S. Nadarajah, 2012. A simple estimator for the Weibull shape parameter, International Journal of Structural Stability and Dynamics, 12(2), 2395-402.

M. Teimouri, S. M. Hoseini, and S. Nadarajah, 2013. Comparison of estimation methods for the Weibull distribution, Statistics, 47(1), 93-109.

F. Wang and J. B. Keats, 1995. Improved percentile estimation for the two-parameter Weibull distribution, Microelectronics Reliability, 35(6), 883-892.

L. Zhang, M. Xie, and L. Tang, 2008. On Weighted Least Squares Estimation for the Parameters of Weibull Distribution. In: Pham H. (eds) Recent Advances in Reliability and Quality in Design. Springer Series in Reliability Engineering. Springer, London.

Examples

n<-100
alpha<-2
beta<-2
theta<-3
data<-rweibull(n,shape=alpha,scale=beta)+theta
starts<-c(2,2,3)
fitWeibull(data, TRUE, "mps", starts)
fitWeibull(data, TRUE, "wml", starts)
fitWeibull(data, FALSE, "mlm", starts)
fitWeibull(data, FALSE, "ustat", starts)