I. Ahmad, A., On asymptotic properties of an estimate of a functional of a probability density, Scandinavian actuarial journal , 1976(3), 1976, pp. 176-181
Bhattacharyya & Roussas (1969) proposed to estimate the functional Δ = ∫ −∞/∞ f 2(x)dx by , where is a kernel estimate of the probability density f(x). Schuster (1974) proposed, alternatively, to estimate Δ by , where F n (x) is the sample distribution function, and showed that the two estimates attain the same rate of strong convergence to Δ. In this note, two large sample properties of are presented, first strong convergence of to Δ is established under less assumptions than those of Schuster (1974), and second the asymptotic normality of established.