By Finkenstadt B. F.

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**Additional info for A stochastic model for extinction and recurrence of epidemics estimation and inference for measles o**

**Sample text**

18) Thus E(RSS) = (T − 2)σ 2 , so an unbiased estimator of σ 2 is s2 = RSS . 19) Note that RSS has only (T − 2) degrees of freedom. The two degrees of freedom are lost due to estimation of b0 and b1 . 21) respectively. T It is observed that t=1 ˆt = 0. In the light of this property, ˆt can be regarded as an estimate of unknown t (t = 1, . . , T ) and helps in verifying the diﬀerent model assumptions in the given sample. The methods to verify the model assumptions are discussed in chapter 7. T T T T Further, note that t=1 xt ˆt = 0, t=1 yˆt ˆt = 0, t=1 yt = t=1 yˆt and the ﬁtted line always passes through (¯ x, y¯).

Putting I − C = M , the problem is that of ﬁnding min M M subject to M X = X . 54) 46 3. The Multiple Linear Regression Model and Its Extensions Since PX and Z span the whole RT , we can write M = PX A + ZB for some A and B , giving X =XM MM = = X A, A PX A + B Z ZB = = A X(X X)− X A + B Z ZB X(X X)− X + B Z ZB ≥ PX with equality when B = 0. Then M = PX A = X(X X)− X A = X(X X)− X , and the best predictor of is ˆ = C y = (I − M )y = (I − PX )y . 55) s2 = T −r T −r since (with rank (X) = r) E(s2 ) = = 1 1 E[y (I − PX )y] = tr(I − PX ) D(y) T −r T −r σ2 T −r tr(I − PX ) = σ 2 = σ2 .

14) where P = X(X X)− X is the orthogonal projection operator on R(X). ˆ ˆ = 0. Proof: Let θˆ ∈ R(X) be such that (y − θ)⊥R(X), that is, X (y − θ) Then y−θ 2 = = (y − θˆ + θˆ − θ) (y − θˆ + θˆ − θ) ˆ (y − θ) ˆ + (θˆ − θ) (θˆ − θ) ≥ y − θˆ (y − θ) 2 ˆ (θˆ − θ) vanishes using the orthogonality condition. since the term (y − θ) ˆ Writing θˆ = X β, ˆ the orthogonality The minimum is attained when θ = θ. ˆ ˆ condition implies X (y − X β) = 0, that is, X X β = X y. 79. This shows that θˆ exists. Let (X X)− be any g-inverse of X X.