Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following a \(\mathrm{N}\left(\theta, \sigma^{2}ight)\) distribution and consider estimating \(\exp (\theta)\) with \(\exp \left(\bar{X}_{n}ight)\). Find an asymptotic expression for the variance of \(\exp \left(\bar{X}_{n}ight)\) using the methods detailed in Example 10.5.

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24 Example 10.5. Let {X} be a sequence of independent and identically distributed random variables following a N(0,02) distribution and consider estimating exp(0) with exp(Xn). We are not able to apply Theorem 10.1 directly to this case because the fourth derivative of exp(t), which is still exp(t), is not a bounded function. If we attempt to apply Theorem 1.13 to this problem we will end up with an error term of the form (X-0) exp($), where is a random variable that is always between 0 and Xn. This type of error term is difficult to deal with in this case because the exponential function is not bounded and hence we cannot directly bound the expectation as we did in Example 10.2 and the proof of Theorem 10.1. Instead we follow the approach of Lehmann (1999) and use the convergent Taylor series for the exponential function instead. That is, exp(Xn) = i=0 exp(0)(X-0)i i! (10.8)