Write a program in \(\mathrm{R}\) that simulates 100 samples of size \(n\) from distributions that are specified below. Let \(T(F)\) be the variance functional described in Example 9.8. For each sample compute \(T\left(\hat{F}_{n}ight)-T(F)\) and \(\Delta_{1} T\left(F, \hat{F}_{n}-ight.\) \(F)\) where the differential was found in Example 9.8. For each sample size and distribution, construct a scatterplot the 100 values \(T\left(\hat{F}_{n}ight)-T(F)\) and \(\Delta_{1} T\left(F, \hat{F}_{n}-Fight)\). Describe the behavior observed in the plots with relation to the expansion given in Equation (9.1). Use \(n=5,10,25,50\), and 100 .

a. \(\mathrm{N}(0,1)\)

b. Exponential(1)

c. \(\operatorname{UNIFORm}(0,1)\)

d. \(\operatorname{Cauchy}(0,1)\)

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Example 9.8. Consider the variance functional given by F) = [ {t - LtdF (t)]" dF(t), T(F) where FF, a collection of distribution functions with mean and finite variance . In order to take advantage of the result given in Theorem 9.1 we must first write this functional as a multiple integral of a symmetric function. To find such a function we can note that if X1 and X2 are independent random variables both following the distribution F, then E[(X1X2)] = E(X + X - 2X1X2) = 2 + 20 - 2 = 20, so that E[(X1-X2)] = 0. Therefore, it follows that T(F) can equivalently be written as T(F) = LL h(x1, x2)dF(x1)dF(x2),