Transcription of Chapter 5 The Delta Method and Applications
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Chapter 5 The Delta Method and Applications
Chapter 5 The Delta Method and Linear approximations of functionsIn the simplest form of the central limit theorem, Theorem , we consider a sequenceX1,X2,..of independent and identically distributed (univariate) random variables withfinite variance 2. In this case, the central limit theorem states that n(Xn )d Z,( )where = EX1andZis a standard normal random this Chapter , we wish to consider the asymptotic distribution of, say, some function ofXn. In the simplest case, the answer depends on results already known: Consider a linearfunctiong(t) =at+bfor some known constantsaandb. Since EXn= , clearly Eg(Xn) =a +b=g( ) by the linearity of the expectation operator. Therefore, it is reasonable to askwhether n[g(Xn) g( )] tends to some distribution asn.
The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1,X 2,... of independent and identically distributed (univariate) random variables with finite variance σ2. In this case, the central limit theorem states that √ n(X n −µ) →d ...
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