To use the outer measure as a probability measure for more general sets that those in F0, we have to extend the algebra F0 to a a-algebra F of events for which the outer measure is a probability measure. In this appendix it will be shown how Fcan be constructed via the following lemmas.
Lemma 1.B.1: For any sequence Bn of disjoint sets in ^, P*(U=j Bn) <
£Г=1 p "(Bn).
Proof: Given an arbitrary є > 0 it follows from (1.21) that there exists a countable sequence of sets An, j in F0 such that Bn c An, j and P *(Bn) >
ZjU P(An, j) – є2 n; hence,
CO CO CO CO CO CO
£ P*(Bn) > £ £ P(An, j) – £ £ 2-n = £ £ P(An, j) – є. n=1 n=1 j=1 n=1 n=1 j=1
Moreover, UO=1 Bn c UO=1 UO=1 An, j, where the latter is a countable union of sets in Ж0; hence, it follows from (1.21) that
и BA... Read More
Suppose you consider starting a business to sell a new product in the United States such as a European car that is not yet being imported there. To determine whether there is a market for this car in the United States, you have randomly selected n persons from the population of potential buyers of this car. Each person j in the sample is asked how much he or she would be willing to pay for this car. Let the answer be Yj. Moreover, suppose that the cost of importing this car is a fixed amount Z per car. Denote Xj = ln(Yj /Z), and assume that Xj is N(д, a2) distributed. If д > 0, then your planned car import business will be profitable; otherwise, you should forget about this idea.
To decide whether д > 0 or д < 0, you need a decision rule based on the random sample X = (X1, X2,…, Xn)T... Read More
The restricted ML estimator в can also be obtained from the first-order conditions of the Lagrange function <(в, /г) = ln(Ln (в)) – §2г, where д є Kr is a vector of Lagrange multipliers. These first-order conditions are
д<(в, г)/двТ§=1д=г = d ln(L(§))/дв*§=§ = 0,
д<(§, г)/дв2г§=вг=г = дln(L(§))/дв2§=§ – д = 0 д<(§, г)/дгт§=e^=ff = §2 = °.
/ 0 дln(L(§))/yn
V^. N д§T §=ff
Again, using the mean value theorem, we can expand this expression around the unrestricted ML estimator §, which then yields
-H 0 ) = – H-n(§ – 0) + op(1) ^d N(0, ЙAЙ),
where the last conclusion in (8.64) follows from (8.59). Hence,
дТ я(2,2 ,)д = – V, iiT) й -1
= йп(в – §)TЙл/п(в – в) + Op(1) ^d X,
where the last conclusion in ... Read More
I will now show that the conditional expectation of a random variable Y given a random variable or vectorXis the best forecasting scheme for Yin the sense that the mean-square forecast error is minimal. Let f (X) be a forecast of Y, where f is a Borel-measurable function. The mean-square forecast error (MSFE) is defined by MSFE = E[(Y — f (X))2]. The question is, For which function f is the MSFE minimal? The answer is
Theorem 3.13: If E [Y2] < to, then E [(Y — f (X))2] is minimal for f (X) = E [Y | X] ■
Proof: According to Theorem 3.10 there exists a Borel-measurable function g such that E[Y|X] = g(X) with probability 1. Let U = Y — E[Y|X] = Y — g(X)■ It follows from Theorems 3.3, 3.4, and 3.9 that
E[(7 – f (X))2|X] = E[(U + g(X) – f (X))2|X]
= E[U2|X] + 2E[(g(X) – f (X))U|X]
+ E [... Read More
The prime example of the concept of convergence in distribution is the central limit theorem, which we have seen in action in Figures 6.4—6.6:
Theorem 6.23: Let X1,Xn be i. i.d. random variables satisfying E(Xj) = X, var(Xj) = a2 < to and let X = (1/n)J2’n=1 Xj. Then *Jn(X — /г)
Proof: Without loss of generality we may assume that x = 0 and a = 1. Let p(t) be the characteristic function of Xj. The assumptions x = 0 and a = 1 imply that the first and second derivatives of p(t) at t = 0 are equal to p'(0) = 0, p"(0) = — 1, respectively; hence by Taylor’s theorem applied to Re[0(t)] and Im[0(t)] separately there exists numbers X1t, X2,t e [0, 1] such that
p(t) = p(0) + tp'(0) + 212 (Re[p"(xu ■ t)] + i ■ Im[p"(X2,, ■ t)]) = 1 — 212 + z(t )t2,
for instance, where z... Read More
Consider a square matrix A partitioned as
where A1,1 and A2 2 are submatrices of size k x k and m x m, respectively, A12 is a k x m matrix, and A2,1 is an m x k matrix. This matrix A is block-triangular if either A12 or A21 is a zero matrix, and it is block-diagonal if both A12 and A21 are zero matrices. In the latter case
where the two O blocks represent zero elements. For each block A1,1 and A2 2 we can apply Theorem I.11, that is, A1,1 = P1rL1D1U1, A2 2 = P2TL2D2U2; hence,
for instance. Then det(A) = det(P) ■ det(D) = det(P1) ■ det(P2) ■ det(D1) ■ det(D2) = det(A1,1) ■ det(A2,2). More generally, we have that
Theorem I.27: The determinant of a block-diagonal matrix is the product of the determinants of the diagonal blocks.
Next, co... Read More