The asymptotics of maximum-likelihood estimates of parameters based on a data type where the failure and the censoring time are dependent

Parameter Estimation for Bivariate Shock Models with Singular Distribution for Censored Data with Concomitant Order Statistics

“…This result is given in Theorem 4.1 and the results are used in deriving asymptotics of the MAMLE. Because the MLEθ = (λ 1 ,λ 2 ,λ 3 ) is a strongly consistent estimator of θ = (λ 1 , λ 2 , λ 3 ) (see Chen & Lu, 1998), there exists a neighbourhood n ofλ 3 such that the initial estimatorλ (0) 3 ∈ n , where n = {λ : |λ −λ 3 | ≤ δ n for δ n > 0}. Finally, use the re-expression of (3.2) as λ 3 = g(λ 3 ), where g(λ 3 ) is given by (3.4).…”

“…Note that n 1/2 (X (r) − η) d → N(0, p(1 − p) −1 γ −2 1 ). It can be shown that n 1/2 [(∂ log /∂θ )/n] d → N 3 (0, M), where M is rather complicated and is an integration of first derivatives vector products; see Chen & Lu (1998) for details. Thus, performing the following Taylor expansion of (∂ log /∂θ ) | θ=θ 0 = 0 around the true parameter value θ 0 yields n −1/2 ∂ log ∂θ…”

Distribution of random elements subjected to a flexible boundary condition

“…[8] . For random elements constrained by boundary conditions or internal processes, it can be prohibitively difficult to obtain the solution [9] . In Section 1 below, p(x) and p m (x) will be investigated separately.…”