Chapter 3

Abstract

¹The class of all possible consistent estimates of a given parameter c is very extensive and it includes a number of particular practically important estimates thoroughly studied in many statistical texts (see, e.g., Cramér, 1946; Wilks, 1962; Kendall and Stuart, 1966; Silvey, 1970; Zacks, 1971). To compare two different unbiased consistent estimators of c, say, C ^*_N and C ^**_N , the relative efficiency e_r(C ^**_N , C ^*_N ) or C ^**_N as compared with C ^*_N is sometimes evaluated by the formula e_r(C ^**_N ,C ^*_N ) = σ²(C ^*_N )/σ²(C ^**_N ). Clearly, C ^**_N is preferable to C ^*_N if, and only if, e_r(C ^**_N C ^*_N ) > 1. Note also that under some general conditions the lower bound σ ²_N (c) to the variance σ²(C ^*_N ) of any unbiased estimator C ^*_N of a parameter c can be determined, and hence the absolute efficiency e(C ^*_N ) = σ²(C ^*_N )/σ ^*_N (c) of C ^*_N can be evaluated. The estimator C ^*_N (and the estimate c ^*_N ) is called efficient if e(C ^*_N ) = 1 and asymptotically efficient if e(C ^*_N ) → 1 as N → ∞.