New PDF release: A Course in Statistical Theory

By Olive D.J.

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Example text

It will be useful to partition X, µ, and Σ. Let X 1 and µ1 be q × 1 vectors, let X 2 and µ2 be (p − q) × 1 vectors, let Σ11 be a q × q matrix, let Σ12 be a q × (p − q) matrix, let Σ21 be a (p − q) × q matrix, and let Σ22 be a (p − q) × (p − q) matrix. Then CHAPTER 2. MULTIVARIATE DISTRIBUTIONS X= X1 X2 , µ= µ1 µ2 58 Σ11 Σ12 Σ21 Σ22 , and Σ = . , Xkq )T ˜ where µ ˜ ij = Cov(Xk , Xk ). In particular, ˜ Σ) ˜ i = E(Xki ) and Σ ∼ Nq (µ, i j X 1 ∼ Nq (µ1 , Σ11 ) and X 2 ∼ Np−q (µ2, Σ22). b) If X 1 and X 2 are independent, then Cov(X 1 , X 2) = 0, a q × (p − q) matrix of zeroes.

Y1 ! · · · yn ! 27) is a pmf. , Yn−1 are important and the nth outcome means that none of the n − 1 important outcomes occurred. , Yik occurred. Then Wk = k−1 k−1 k m − j=1 Yij and P (Wk ) = 1 − j=1 ρij . , ik occurred,” an outcome with probability kj=1 ρij . Hence kj=1 Yij ∼ BIN(m, kj=1 ρij ). Now consider conditional distributions. , Yik . The conditional probabilities of Yi remains proportional to ρi , but the conditional probabilities must sum to one. Hence the conditional distribution is again multinomial.

Otherwise the expectation does not exist. The expected value is ··· E[h(Y )] = h(y)f(y) = y1 h(y)f(y) = y∈ yn if f is a joint pmf and if does not exist. 11) y ∈Y |h(y)|f(y) exists. Otherwise the expectation The following theorem is useful since multiple integrals with smaller dimension are easier to compute than those with higher dimension. 4. , yn). ,Yik . , Yik )] exists. , yik ) yik CHAPTER 2. ,Yi k if f is a pmf. Proof. The proof for a joint pdf is given below. For a joint pmf, replace the integrals by appropriate sums.

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A Course in Statistical Theory by Olive D.J.


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