fitgrouped1 | R Documentation |
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,…,m. So, the likelihood function is given by
L(Θ)=\frac{n!}{f_{1}!f_{2}!… f_{m}!}∏_{i=1}^{m}\Bigl[F\bigl(r_{i}\big|Θ\bigr)-F\bigl(r_{i-1}\big|Θ\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=∑_{i=1}^{m}f_{i}. The cdf of a three-parameter BS, GE, and Weibull distributions are given by
F(x;Θ)=\biggl(1-\exp \bigl\{-β(x-μ)\bigr\} \biggr)^{α},
F(x;Θ)=Φ\Biggl(\frac{√{\frac{x}{β}}-√{\frac{β}{x}}}{α}\Biggr),
and
F(x;Θ)=1- \exp \Bigl\{-≤ft(\frac{x-μ}{β} \right)^{α} \Bigr\},
where Θ=(α,β,μ)^T.
fitgrouped1(r, f, family, method1, starts, method2)
r |
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,…,m. The lower bound of the first group and upper bound of the m-th group are chosen arbitrarily. |
f |
A numeric vector of length m containing the group's frequency. |
family |
Can be either |
method1 |
A character string determining the method of estimation. It can be one of |
""aml"
(for method of approximated maximum likelihood (aml)),
""em"
(for method of expectation maximization (em)), and
""ml"
(for method of maximum likelihood (ml)).
starts |
A numeric vector of the initial values for the shape, scale, and location parameters, respectively. |
method2 |
The method for optimizing the log-likelihood function. It invovles one of |
If the method is "em"
, then the initial values ("starts"
) and the log-likelihood optimizing method ("method2"
) are ignored.
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.
Mahdi Teimouri
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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