Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following a \(\operatorname{ChiSquared}(\theta)\) distribution. In Example 7.12 we derived a saddlepoint expansion that approximates the density of \(n \bar{X}_{n}\) at a point \(x\) with relative error \(O\left(n^{-1}ight)\) as \(n ightarrow \infty\), given by

\(f_{n}(x)=\left[4 \pi n^{-1} x^{2} \thetaight]^{-1 / 2} \exp \left[-\frac{1}{2} n \theta \log \left(n \theta x^{-1}ight)-\frac{1}{2}\left(1-n x^{-1} \thetaight)ight]\left[1+O\left(n^{-1}ight)ight]\).

Prove that an application of Theorem 1.8 (Stirling) implies that the leading term of this expansion has the same form as a GAMma distribution.

Transcribed Image Text:

Example 7.12. Let {X} be a sequence of independent and identically distributed random variables following a CHISQUARED (0) distribution. We wish to derive a saddlepoint expansion that approximates the density of nXn at a point r with relative error O(n-1) as n o. The cumulant generating function of a CHISQUARED (9) distribution is given by c(t) = -log(1-2t) for t