Computational Methods in Statistics and Econometrics - download pdf or read online

By Thomas R. Cundari

ISBN-10: 0824704789

ISBN-13: 9780824704780

Highlights Monte Carlo and nonparametric statistical tools for types, simulations, analyses, and interpretations of statistical and econometric facts. good points useful purposes.

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Extra resources for Computational Methods in Statistics and Econometrics

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4. Theorem: Let tj>\(6), $2(6), • • •, n(ff) bg the moment-generating functions of X, , X 2 , - - •, Xn, which are mutually independently distributed random variables. Define Y = X^ + X2 + • • • + Xn. , 0,(0) = E(e8Y) = 0, (d)2(9) • • • ^,,(0), where 4>y(ff) represents the moment-generating function of Y. 26 CHAPTER 1. , 0V(0), is rewritten as: 0},(0) = E(eeY) = E(e^+x^-+^) = E(eex<)E(eex>) • • -E(eex") = 0,(0)0 2 (0)---0,,(0). The third equality holds because X), Xi, • • • , Xn are mutually independently distributed random variables.

Since A n Ac = 0, we have the fact that A and Ac are mutually exclusive. 2: Cast a coin three times. In this case, we have the following eight sample points: tol = (H,H,H), o>5 = (T,H,H), o>2 = (H,H,T), o>6 = (T,H,T), 0)3 = (H,T,H), o>7 = (T,T,H), o>4 = (H,T,T), o>8 = (T,T,T), where H represents head while T indicates tail. For example, (H,T,H) means that the first flip lands head, the second flip is tail and the third one is head. Therefore, the sample space of this experiment can be written as: il = {0)1,012,0)3,014,0)5,0)6,0)7,018}.

Theorem: When X], X2, ••-, Xn are mutually independently and identically distributed and the moment-generating function of X, is given by (ff) for all /', the moment-generating function of Y is represented by (0(0)) , where Y X,+X2 + --- + Xn. Proof: Using the above theorem, we have the following: 0/0) = 0, (0)02(0) • • • 0n(0) = 0(0)0(0) - • • 0(6) = (0(0))". Note that 0,{0) = 0(0) for all z. 6. Theorem: When X\, X2, • •-, Xn are mutually independently and identically distributed and the moment-generating function of Xt is given by 0(0) for all z, the moment-generating function of X is represented by (0(-)) \ n i , where X = (l/«)I7=iXi.

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Computational Methods in Statistics and Econometrics by Thomas R. Cundari


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