Stein ’ s Paradox
نویسنده
چکیده
P erhaps the most surprising result in Statistics arises in a remarkably simple estimation problem. Let X1, ..., Xp be independent random variables, with Xi ∼ N(θi , 1) for i = 1, ..., p. Writing X = (X1, ..., Xp), suppose we want to find a good estimator θ̂ = θ̂(X) of θ = (θ1, ..., θp). To define more precisely what is meant by a good estimator, we use the language of statistical decision theory. We introduce a loss function L(θ̂, θ), which measures the loss incurred when the true value of our unknown parameter is θ, and we estimate it by θ̂. We will be particularly interested in the squared error loss function L(θ̂, θ) = �θ̂ – θ�, where � . � denotes the Euclidean norm, but other choices, such as the absolute error loss L(θ̂, θ) = ∑i=1 �θ̂i – θ i� are of course perfectly possible.
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