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

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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, JuneY (θ ) = Z Σ Z, (11)where, Σ is the general inverse matrix of the covariance matrix Σ,Σ = A − C I C,Σ = A + A C G CA , G = I − CA C ,A is the diagonal k × k matrix with the elements A = on the diagonal, A is inverse matrixof A, andC = [C ] × , with C = ∑ δ ( ,): ∈ , l = 1, 2, ⋯ , m, j = 1, 2, ⋯ , k, (13)I = [ı̂ ] × , with ı̂ = 1 n δ where, ∂ ln λ(X , θ)∂θ ∂ ln λ(X , θ)∂θ (12), l, l = 1, 2, ⋯ , m. (14)From the definition of Z in (10), the test statistic Y (θ ) should be written asY θ = X + Q, (15)X = (U − e ) U , Q = W G W , W = CA Z, G = I − CA C .Under the hypothesis H , the limiting distribution of the statistics Y θ is chi-squared withr = rank(Σ ) degrees of freedom that is,lim → PY θ > x | H = P{χ > x}, for any x > 0.Statistical inference for the hypothesis H 0 : The null hypothesis H is rejected with approximatesignificance level α if Y θ > χ (r) or Y θ < χ (r) depending on an alternative, whereχ (r) and χ (r)corresponding are the upper and lower α percentage points of the χ distribution, respectively.Using the method of interval selection which is proposed by Bagdonavičius, and Nikulin [20],we used a as the random data function. DefineE = ∑ Λ(X , θ ) , E = E , j = 1, 2, ⋯ , k.Denote by X () , X () , ⋯ , X () the ordered sample from X , X , ⋯ , X . Setb = (n − i)ΛX () , θ + ΛX () , θ , i = 1, 2, ⋯ , n,if i is the smallest natural number verifying b ≤ E ≤ b then a verifying the equality(n − i + 1)Λa , θ + ΛX () , θ = E Soa = Λ E − ∑ Λ(X () , θ ), θn − i + 1 ; a = maxX () , τ , (j = 1, 2, ⋯ , k − 1). (16)where Λ is the inverse of the function Λ. We have: 0 < a < a < ⋯ < a , with this choiceof intervals, then e = , for all j.Application for GBS distributions: In particular, we shall give chi-squared tests NRR forthe hypothesis H that the data X are coming from the GBS distributions with the probability14
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).
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