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 . But the linearity ofg(t)allows one to write n[g(Xn) g( )]=a n(Xn ).We conclude by Theorem that n[g(Xn) g( )]d a course, the distribution on the right hand side above isN(0,a2 2).None of the preceding development is especially deep; one might even say that it is obviousthat a linear transformation of the random variableXnalters its asymptotic distribution85by a constant multiple.
a sequence of random variables, we generally refer to a nontrivial (i.e., nonconstant) distribu-tion. Thus, in the case of an independent and identically distributed sequence X 1,X ... Therefore, the delta method gives √ n(X2 n ...
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