7 CHI-SQUARED GOODNESS OF FIT TEST FOR GENERALIZED BIRNBAUM ...

M. S. Nikulin and X. Q. Tran – CHI-SQUARED GOODNESS-OF-FIT TETS FOR GENERALIZED BIRNBAUM-SAUNDERS MODELS FORRIGHT CENSORED DATA AND ITS RELIABILITY APPLICATIONSRT&A # 02 (29)(Vol.8) 2013, JuneX = T ∧ C ,δ = 1 { }, i = 1, 2, ⋯ , n.Defined thatf(t, θ)S(t, θ) = P θ (T > t); λ(t, θ) =S(t, θ) , Λ(t, θ) = − ln{S(t, θ)} , θ ∈ Θ ⊆ R ,be the survival, hazard rate and cumulative hazard functions, respectively. Denote by G and g arethe survival and the density function of the censoring time C , respectively. Supposing that the rightcensoring is non-informative which means that the function G does not depend on θ. So in thiscase, we obtain the following expressions for the likelihood function L(θ)L(θ) = f (X , θ)S (X , θ)g (C )G (C ).So the members with G and g do not contain θ, so they can be rejected. The likelihoodfunction is obtainedL(θ) = f (X , θ)S (X , θ) = λ (X , θ)S(X , θ) . (7)The estimator θ maximizing the likelihood function L(θ). The log-likelihood function isl(θ) = {δ ln λ(X , θ) + ln S(X , θ)}= {δ ln λ(X , θ) − Λ(X , θ)}.The maximum likelihood estimator θ satisfies the system equationsl̇θ = 0 ,where l̇(θ) are the score vectorswherel̇(θ) = l(θ) = l(θ) The Fisher information matrix is defined asI(θ) = −E l̈(θ),l̈(θ) = δ ∂ ∂θ ln λ(X , θ), l(θ), ⋯ , l(θ) . − ∂∂θ Λ(X , θ) .Supposing that θ is the true value of θ, under some regularity conditions, we haveθ → θ ; √n θ − θ = i (θ ) 1√n l̇(θ ) + O (1) , −1√n l̈θ → i(θ ),√n(θ − θ ) → 1N (0, i (θ ));√n l̇(θ ) → N (0, i(θ )),where, θ are the maximum likelihood estimation of θ and the matrixI(θi(θ ) )= lim → n .For any t ≥ 0, setN (t) = 1 { , } = 1, if t ≥ X and δ = 1,; Y0, if 0 ≤ t ≤ X . (t) = 1 { } = 1, if t ≤ X ,0, if t > X .N(t) = N (t) , Y(t) = Y (t).The process N(t) shows the number of observed failures in the interval [0, t] and the processY(t) shows the number of objects which are "at risk" just prior to time t. The sample (6) isequivalent to the sample (8)12