A Note on Comparison between Correlation Coefficients of Original and Transformed Variables
C. Rangarajan, Samprit Chatterjee · The American Statistician · 1969
_2P { 2 (V + P) ISP. up= aptp =2-1 Fjl(v + 2p)} l (8) Provided that we ignore tlhe number of terms in the summation in S2, u. is of the same form as tp7 with (v + p) written for v. Thus we have the following result: The reduced m.s.e. estimator of o-P(p > -2(71 1)) in normal samples of size it is the saine mul-tiple of SP as is the unbiased estimator of o-P inl samples of size (it + p). The case p = 2 is well-known (see e.g. [1], Vol. 2, p. 33, Exercise 17.16). Markowitz [2] gave the result for p = 1. Notice that p canl be negative, fractional or both.