New PDF release: An introduction to mathematical probability

By Julian Lowell Coolidge

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O ptim al estim ators. As in the preceding section we consider the problem of estimating a function g{6) from an observation ^ assuming values in {X^B) and whose distribution belongs to a parametric family {P^, 0 G 0 } . Let T = T(^) be an estimator of a function g{6) and let r(T, g) be a nonnegative loss function (a loss appears because we approximate g = g{6) by an estimator T). 14) i2(T;0) = E ,r(T (O ,^ (0)), eeQ , is called a risk function of an estimator T = T(^) if the true value of the parameter is 6.

1. ) < 00 for aZ/ i = 1 ,2 ,... ,n. 2. ESTIMATION OF A LOCATION PARAMETER 2) if the measure Pq is absolutely continuous with respect to the Lebesgue measure and its density is f{x)^ x e R ”", then the Pitman estimator is of the form / oo / /*oo \ —1 vf{x-v)d vyj f{x-v)d vj , X e R ". P r o o f .

2. Letf^^^ = ( ^ i , . . , i n ) ^ sample from the tribution. Then the sampling moments a\ and m 2 are independent. )• Proof. Consider a sample ~ (Ci dis­ Moreover = (l^i, •••,|n) where m) / z = 1 , 2 , . . , Ti. Put ai = ^ ¿ 6, i= l = 1=1 Then ai = (ai - /x)/i7 and m 2 = m 2/cr. Thus it is sufficient to prove that d\ and m 2 are independent, since C{y/nai) = 1) and C(nfh2 ) = ')^{n—\). Consider an n-dimensional vector-column h = ( 1 / n , . . , 1 /n )' and n x n matrix B = \\b* •’ b\\. It is clear that di = and nrh2 = (|(^) where A = In ~ B.

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An introduction to mathematical probability by Julian Lowell Coolidge


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