A randomly weighted estimate of the population mean

Consider the model (1.6), whereE _ij (j=1, ...,N _i,i=1,2,...) are i.i.d. with mean 0 and variance 1. Introduce a randomly weighted estimate _n defined by (1.8). Assuminge ₁₁ N(0, 1) andN _i 6, the paper gives a necessary and sufficient condition for _n to be a consistent estimate of ₀, and under some further restrictions a normal approximation fo _n is established which can be used in constructing a large sample confidence interval of ₀. Finally, in the non-normal case a theorem about the consistency of _n is proved.This work is supported by the National Natural Sciences Foundation of China.