By William Feller

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1, f(v(X), v(Y)) is the MLE of g(8). 82) implies that the MLE of R is R(2L, Y) = Ri>r)(f(v(X), v(Y)). 86). 8 Let T^v = T^v(^, 77) be a sufficient statistic for r based on (£,77) and let there exist an UMVUE ^/(T^^) of R based on observations (£,77)- Then, TX,Y = T^tn{v(X_),v(Y_)) is a sufficient statistic for 6 based Transformation Methods 41 on the sample Q£, Y_) and the UMVUE R of R based on X_ and Y_ is given by ( 2 - 87 ) R = i>(Tx,Y)- Moreover, if ^fi(T^tV) is the UMVUE of the variance of the unbiased estimator ^f(T^tV), then the UMVUE of the variance of R is of the form Var(R) = * i ( T x , r ) .

G. Lehmann and Casella (1998)) 1(9) is the matrix with (i, j)-th element Yet another possibility - in the absence of specific information about 9 is to match the Bayesian solution with the frequentist solution of the problem. A prior which satisfies this condition is called a matching prior. It is derived by requiring the classical frequentist coverage probability of the posterior region of a real-valued parametric function to match the nominal level with a remainder of the order of O(n~^2). g.

TI\T) = r))At)Av), (2-80) g(v(x),v(y)\g(e))v'(x)v'(y). 77) is monotonically increasing, P(£ < rf) = P(u(£) < u(r])) — P(X < Y), so that for this model R remains invariant. Therefore, if R^tT)(r) and RXty{9) are two parametric expressions for R in terms of r and 9, respectively, and RX,Y(6) = RUQW), Ri,v(7) = RX,Y{"

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