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).
→d N(0,σ2) by the central limit theorem, which implies that nX n →d σ2χ2 1. Example 5.4 Estimating binomial variance: Suppose X n ∼ binomial(n,p). Because X n/n is the maximum likelihood estimator for p, the maximum likelihood esti-mator for p(1−p) is δ n = X n(n−X n)/n2. The central limit theorem tells us that √ n(X n/n−p)
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