4.6 Determination of Fixed-width Intervals

Khan (1969) has given a method for determining stopping rules in order to obtain fixed-width confidence intervals of prescribed coverage probability for an unknown parameter of a distribution possibly involving some unknown nuisance parameters The results are only asymptotic, and rely on the asymptotics of Chow and Robbins (1965). Below we present Khan's (1969) results.

Let p( x; ? ₁, ? ₂) denote the probability density function of a random variable X (for convenience, with respect to Lebesgue measure) with real-valued parameters ? ₁ and ? ₂ where ? ₂ is considered to be a nuisance parameter. For the sake of simplicity we assume that there is a single nuisance parameter since the case of several nuisance parameters would be analogous. We wish to determine a confidence interval of fixed-width 2 d( d>0) for ? ₁ when both ? ₁ and ? ₂ are unknown, with preassigned coverage probability 1 ? ?(0< ?<1).

Assumption We assume that all the regularity conditions of maximum likelihood estimation are satisfied. [See for instance, LeCam (1970)]. Also assume the regularity conditions of Theorem 4.5.4.

Let N denote a bona fide stopping variable (that is, N is a positive integer-valued random variable such that the stopping set {N=n} is a member of the ?-algebras of subsets generated by X ^{( n )}=( X ₁, X ₂,..., X