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Mathematical Statistics, Winter term 2018/19 Problem sheet 14
42) Assume that Z has a t-distribution withn degrees of freedom.
Show that
P(Z ≤t) = P(−Z ≤t) ∀t ∈R.
Hint: Use the fact that, for X ∼ N(0,1), P(X ≤ u) = P(−X ≤ u) holds for all u∈R.
42) Show that the one-tailed t-test is unbiased.
43) Let X be an (n×k)-matrix withrank(X) =k.
(i) Show that XTX is a regular matrix.
(ii) Show that X(XTX)−1XT is the (unique) projection matrix onto the subspace Θ ={Xβ: β ∈Rk}.
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