Suppose a sample of \(n\) independent observations each follows a three-parameter BS, GE, or Weibull distributions have been divided into \(m\) separate groups of the form \((r_{i-1},r_i]\), for \(i=1,\dots,m\). So, the likelihood function is given by $$ L(\Theta)=\frac{n!}{f_{1}!f_{2}!\dots f_{m}!}\prod_{i=1}^{m}\Bigl[F\bigl(r_{i}\big|\Theta\bigr)-F\bigl(r_{i-1}\big|\Theta\bigr)\Bigr]^{f_i},$$ where the \(r_0\) is the lower bound of the first group, \(r_m\) is the upper bound of the last group, and \(f_i\) is the frequency of observations within \(i\)-th group provided that \(n=\sum_{i=1}^{m}f_{i}\). The cdf of a three-parameter BS, GE, and Weibull distributions are given by $$F(x;\Theta)=\biggl(1-\exp \bigl\{-\beta(x-\mu)\bigr\} \biggr)^{\alpha},$$ $$F(x;\Theta)=\Phi\Biggl(\frac{\sqrt{\frac{x}{\beta}}-\sqrt{\frac{\beta}{x}}}{\alpha}\Biggr),$$ and $$F(x;\Theta)=1- \exp \Bigl\{-\left(\frac{x-\mu}{\beta} \right)^{\alpha} \Bigr\},$$
where \(\Theta=(\alpha,\beta,\mu)^T\).
fitgrouped1(r, f, family, method1, starts, method2)A two-part list of objects given by the following:
Estimated parameters of the three-parameter GE, Birnbaum-Saunders, or Weibull distribution fitted to the gropued data.
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), Chi-square (Chi-square),
Cram\'eer-von Misses (CVM), Kolmogorov-Smirnov (KS), and log-likelihood (log-likelihood) statistics.
A numeric vector of length \(m+1\). The first element of \(r\) is lower bound of the first group and other \(m\) elements are upper bound of the \(m\) groups. We note that upper bound of the \((i-1)\)-th group is the lower bound of the \(i\)-th group, for \(i=2,\dots,m\). The lower bound of the first group and upper bound of the \(m\)-th group are chosen arbitrarily.
A numeric vector of length \(m\) containing the group's frequency.
Can be either "birnbaum-saunders", "ge", or "weibull".
A character string determining the method of estimation. It can be one of "aml", "em" and "ml". The short forms "aml", "em", and "ml" are described as follows.
""aml" (for method of approximated maximum likelihood (aml)),
""em" (for method of expectation maximization (em)), and
""ml" (for method of maximum likelihood (ml)).
A numeric vector of the initial values for the shape, scale, and location parameters, respectively.
The method for optimizing the log-likelihood function. It invovles one of "BFGS", "Nelder-Mead", "CG", "L-BFGS-B" or "SANN".
Mahdi Teimouri
If the method is "em", then the initial values ("starts") and the log-likelihood optimizing method ("method2") are ignored.
G. J. McLachlan and T. Krishnan, 2007. The EM Algorithm and Extensions, John Wiley & Sons.
A. P. Dempster, N. M. Laird, and D. B. Rubin, 1977. Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society, Series B (methodological), 1-38.
M. Teimouri and A. K. Gupta, 2012. Estimation Methods for the Gompertz–Makeham Distribution Under Progressively Type-I Interval Censoring Scheme, National Academy Science Letters, 35(3).
r<-c(0,1,2,3,4,10)
f<-c(2,8,12,15,4)
starts<-c(2,2,0)
fitgrouped1(r,f,"birnbaum-saunders","em")
fitgrouped1(r,f,"weibull","ml",starts,"CG")
fitgrouped1(r,f,"ge","em")
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