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, JuneDistribution Notation c g(z ), z ∈ RPearson VIIKotz typePVII(q, r)KT(q, r, s)() √1 + , q > , ν > 0 () ⁄ z () e , q > , r > 0, s > 0Table 1: Kernel g(∙) and normalization constants c for some indicated distributions.Following Díaz-García et al. the random variable T in (1) allows the GBS distributions,denoted by T ≈ GBS(α, β ; g),T = β α z 4 + α z 2 + 1 ≈ GBS(α, β, g), α > 0, β > 0,iff the random variable Z which is given by the expressionZ = − ≈ EC(0, 1; g).So, the probability density function of T can be written asf (t, α, β) = + g − , t > 0, α > 0, β > 0, (3)the cumulative distribution function of T ≈ GBS(α, β ; g) is expressed byF (t, α, β) = F , t > 0, α > 0, β > 0, (4)the GBS hazard rate, survival and cumulative hazard functions areλ (t, α, β) = f a (α, β)A (α, β)1 − F a (α, β) ,S (t, α, β) = 1 − F a (α, β) , and Λ (t, α, β) = −ln{S (t, α, β)},(5)respectively, wherea (α, β) = − ; A (α, β) = + . It is clear that the properties of GBS distributions depends on the kernel function g(∙) and theunknown parameter θ = (α, β) . The statistical theory and methodology of the GBS distributions,also some results for this flexible family of distributions mainly related to transformations, thehazard failure and censored data type II which can be found in the works of Sanhueza, Leiva etal.[36].Table 2 below shown some probability density function of T ≈ GBS(α, β ; g),corresponding the specific symmetric distribution EC(0, 1; g) in Table 1.The Figure 1, 2, 3 and 4 below illustrates some curve of the probability densities and hazardrate functions of T ≈ GBS(α, β ; g), allows with the kernel indicative.Kernel g(∙)N(0, 1)DistributionGBS-Normal(BS)Probability density function of T ≈ GBS(α, β ; g),f(t, α, β; g), (t > 0, α > 0, β > 0)1 31 2 2 1 t exp 2222t t t 9