The method of antithetic variates is a technique for reducing the variance of simulation estimators. Antithetic variates are negatively correlated random variables with an expected mean and variance. The variance of the average of two antithetic variates is smaller than the variance of the average of two i.i.d. variables. In this exercise, we shall see how to use antithetic variates for importance sampling, but the method is very general. Suppose that we wish to compute \(\smallint \,g\left( x \right)\,\,dx\), and we wish to use the importance function f. Suppose that we generate pseudo-random variables with the p.d.f. f using the integral probability transformation. For \(\,{\bf{i = 1,2,}}...{\bf{,\nu ,}}\,\)let \({{\bf{X}}^{\left( {\bf{i}} \right)}}{\bf{ = }}{{\bf{F}}^{{\bf{ - 1}}}}\left( {{\bf{1 - }}{{\bf{U}}^{\left( {\bf{i}} \right)}}} \right)\), where \({{\bf{U}}^{\left( {\bf{i}} \right)}}\)has the uniform distribution on the interval (0, 1) and F is the c.d.f. Corresponding to the p.d.f. f . For each \(\,{\bf{i = 1,2,}}...{\bf{,\nu ,}}\,\) define
\(\begin{aligned}{l}{{\bf{T}}^{\left( {\bf{i}} \right)}}{\bf{ = }}{{\bf{F}}^{ - {\bf{1}}}}\left( {{\bf{1}} - {{\bf{U}}^{\left( {\bf{i}} \right)}}} \right)\,\,{\bf{.}}\\{{\bf{W}}^{\left( {\bf{i}} \right)}}{\bf{ = }}\frac{{{\bf{g}}\left( {{{\bf{X}}^{\left( {\bf{i}} \right)}}} \right)}}{{{\bf{f}}\left( {{{\bf{X}}^{\left( {\bf{i}} \right)}}} \right)}}\\{{\bf{V}}^{\left( {\bf{i}} \right)}}{\bf{ = }}\frac{{{\bf{g}}\left( {{{\bf{T}}^{\left( {\bf{i}} \right)}}} \right)}}{{{\bf{f}}\left( {{{\bf{T}}^{\left( {\bf{i}} \right)}}} \right)}}\\{{\bf{Y}}^{\left( {\bf{i}} \right)}}{\bf{ = 0}}{\bf{.5}}\left( {{{\bf{W}}^{\left( {\bf{i}} \right)}}{\bf{ + k}}{{\bf{V}}^{\left( {\bf{i}} \right)}}} \right){\bf{.}}\end{aligned}\)
Our estimator of\(\smallint \,{\bf{g}}\left( {\bf{x}} \right)\,\,{\bf{dx}}\)is then\({\bf{Z = }}\frac{{\bf{I}}}{{\bf{\nu }}}\sum\nolimits_{{\bf{i = 1}}}^{\bf{\nu }} {{{\bf{Y}}^{\left( {\bf{i}} \right)}}{\bf{.}}} \).
a. Prove that\({T^{\left( i \right)}}\)has the same distribution as\({X^{\left( i \right)}}\).
b. Prove that\({\bf{E}}\left( {\bf{Z}} \right){\bf{ = }}\smallint \,\,{\bf{g}}\left( {\bf{x}} \right)\,\,{\bf{dx}}\).
c. If\({\bf{g}}\left( {\bf{x}} \right)\,{\bf{/f}}\left( {\bf{x}} \right)\)it is a monotone function, explain why we expect it \({{\bf{V}}^{\left( {\bf{i}} \right)}}\)to be negatively correlated.
d. If \({{\bf{W}}^{\left( {\bf{i}} \right)}}\) and \({{\bf{V}}^{\left( {\bf{i}} \right)}}\)are negatively correlated, show that Var(Z) is less than the variance one would get with 2v simulations without antithetic variates.
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