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, Junedensity, cumulative distribution, hazard rate, survival and cumulative hazard functions give informulas (3), (4) and (5), respectively.The GBS log-likelihood functions l(θ), (θ = (α, β) ) isl(θ) = −δ ln α − δ ln β + δ ln β β + X X + δ ln{1 − F (a (α, β))}15+ δ lngK (α, β)Let θ = α, β be maximum likelihood estimations which are solutions of the non-linearsystem equationsl̇ (θ), l̇ (θ) = 0 .Using the formula (13) – (14), the elements ı̂ , (l, l = 1, 2) of the Fisher information matrixI = [ı̂ ] × are1ı̂ =nα δ −1 + K α, βv K α, β − A (α, β)f (A (α, β))1 − F (A (α, β)) ,ı̂ =ı̂ =1nβ δ −1 + 1 21 + 3 1 + + 1 2 A α, βB α, βv K α, β − 1 21nαβ δ −1 + K α, βv K α, β − A (α, β)f (A (α, β))1 − F (A (α, β)) ×× −1 + 1 1 + 3 2 1 + + 1 2 A α, βB α, βv K α, β − 1 2 and the matrix C = C ×given by1C =C =where,nα δ −1 + K α, βv K α, β − A (α, β)f (A (α, β))1 − F (A (α, β)) B (α, β)f (A (α, β))1 − F (A (α, β)) ,∶ ∈ ,B (α, β)f (A (α, β))1 − F (A (α, β)) ,1 δ −1 + 1 1 + 3 nβ2 1 + + 1 2 A α, βB α, βv K α, β − 1 B (α, β)f (A (α, β))2 1 − F (A (α, β)) .∶ ∈ A α, β = 1 α X β − β ,X B α, β = 1 α X β + β ,X K α, β = 1 α X β + β − 2 , i = 1, 2, ⋯ , n.X and f (u) = c g(u ), F (∙) are the probability density function and cumulative function of therandom variable Z ≈ EC(0, 1; g) which follows a standard symmetrical distribution in R with thekernel g(∙), respectively, andv(u) = −2w(u); w(u) = g (u), u > 0,g(u)are the transformations functions of kernel function g(u), and w (u) is the derivative of w(u)([15], [16]). Table 3 below shown some transformations functions w(u) and its derivativew (u), (u > 0) corresponding with kernel g(u) of indicated Elliptic distributions EC(0, 1; g).