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1. + ε, x ∈ {−1, 1} Support function δ(q) for q = (0, 1)T True unknown value: 0.375 n Mean Q1 Q2 Q3 100 0.374 0.360 0.375 0.390 500 0.375 0.369 0.375 0.382 1000 0.375 0.371 0.375 0.380 2500 0.375 0.372 0.375 0.378 Table 7: Percentage of rejections for the test H0 : βr ∈ B. Non-smooth set.
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2. Also, under the conditions of Proposition 9, we have: E max( z> i yi , z> i yi )2 < ∞ so that F = {fθ; θ ∈ Θ} is a Donsker class (for instance, van der Vaart, 1998, page 271). By the definition of such a class, the empirical process, √ nτn(q) = √ n( 1 n n X i=1 zqiwqi − E(zqiwqi)), converges in distribution, uniformly in Θ, to a Gaussian process with zero mean and covariance function: E(zqiwqizriwri) − E(zqiwqi)E(zriwri).
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