Average Run Lengths of an Optimal Method of Detecting a Change in Distribution

Suppose one is able to observe sequentially a series of independent observations X₁, X₂, …, such that X₁, X₂, …, Xν - 1 are i.i.d. with known density f₀ and X_ν, Xν + 1, … are i.i.d. with density f_θ where ν is unknown. Define R(n, θ) = ∑ⁿk = 1 ∏ⁿi = k f_θ(Xᵢ)/f₀(Xᵢ). It is known that rules, which call for stopping and raising an alarm the first time n that R(n, θ) or a mixture thereof exceeds a prespecified level A, are optimal methods of detecting that the density of the observations is not f₀ any more. Practical applications of such stopping rules require knowledge of their operating characteristics, whose exact evaluation is difficult. Here are presented asymptotic (A → ∞) expressions for the expected stopping times of such stopping rules (a) when ν = ∞ and (b) when ν = 1. We assume that the densities f_θ form an exponential family and that the distribution of log(f_ν(X₁)/f₀(X₁)) is (strongly) nonlattice. Monte Carlo studies indicate that the asymptotic expressions are very good approximations, even when the expected sample sizes are small.