Let X1, X2... be i.i.d. observations with mean μ and variance σ2, and suppose that given a sample of size n one wishes to estimate μ by Xn, subject to the loss function Ln= A2(Xn-μ)2+n. When σ2is unknown, the optimal sample size n0for minimizing the risk Rn= E{Ln} cannotbe found. To resolve this problem, Robbins (1959) proposed a sequential procedure whenXiis normally distributed, and then the asymptotic properties have been extensively studied. In this paper, we apply Hall's three stage procedure to the problem without specifying the distribution of Xiand derive the second order asymptotic expansions of the expected sample size and the risk.