Nonparametric Estimation in the Presence of Length Bias

We derive the nonparametric maximum likelihood estimate, F̂ say, of a lifetime distribution F on the basis of two independent samples, one a sample of size m from F and the other a sample of size n from the length-biased distribution of F, i.e. from GF(x) = ∫ˣ₀ u dF(u)/μ, μ = ∫^∞₀ x dF(x). We further show that (m + n)1/2(F̂ - F) converges weakly to a pinned Gaussian process with a simple covariance function, when m + n → ∞ and m/n → constant. Potential applications are described.