Local asymptotic normality for progressively censored likelihood ratio statistics and applications

Let Xn,1 <= Xn,2 <= ... <= Xn,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {P[theta]:[theta] [set membership, variant] [Theta]} where [Theta] is an open subset of the real line. In many practical situations the Xn,i are the observables and experimentation must be curtailed prior to Xn,n. If [tau]n is a stopping variable adapted to the [sigma]-fields {[sigma](Xn,1,...,Xn,k): 1 <= k <= n} and Pn,[theta] the projection of P[theta] onto [sigma](Xn,1,...,Xn,[tau]n), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics [Lambda]n,[tau]n = dPn,[theta]n/dPn,[theta] is established with [theta], [theta]n = [theta] + un-1/2 [set membership, variant] [Theta] and [theta], u held fixed, under certain conditions on the underlying distribution and on [tau]n. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme.