15. (i) If XI" '" Xn is a sample from a Poisson distribution with mean E(X;) = ).., then {n (IX - IX) tends in law to N(O,~) as n -> 00 . (ii) If X has the binomial distribution b(p, n), then {n[arcsinJX/n - arcsin{i] tends in law to N(O,nas n -> 00 . (iii) If (XI' YI ) , . . . ,(Xn , y") is a sample from a bivariate normal distribution, then as n -> 00 (in the notation of Chapter 5, Section 15) [ l+R l+ P] {n log-- - log-- -> N(O,4).

Certain refinements of these transformations arc discussed by Anscombe (1948), Freeman and Tukey (1950), and Hotelling (1953). Transformations of data to achieve approximately a normal linear model are considered by Box and Cox (1964); for later developments stemming from this work see Bickel and Doksum (1981), Box and Cox (1982), and Hinkley and Runger (1984).