### Chapter 3 - Variance reduction - Exercises

Exercise 3.1 (antithetic sampling)
To a random variable $Y$, we associate $\varphi(Y)$, its antithetic. Describe carefully how to define the confidence intervals of the antithetic estimator $$I_{2,M}:=\frac{ 1 }{M }\sum_{m=1}^{M} \frac{f(Y_m)+f(\varphi(Y_m))}2,$$ where the standard deviation is computed on the sample.

Exercise 3.2 (antithetic sampling, Cauchy distribution)
Let $Y$ be a standard Cauchy variable. Write a simulation program to compare the antithetic transformation $Y\rightarrow -Y$ and the semi-antithetic transformation $Y \rightarrow 1/Y$ for the computations of $\mathbb{E}(f(Y))$ in the three cases
1. $f(y)=\sin(y)$,
2. $f(y)=\cos(y)$,
3. $f(y)=(y)_+^{1/4}$.
In each case, discuss and explain the possible variance improvement in comparison with  the standard procedure.

Exercise 3.3 (stratification, optimal allocation)
The stratification estimator for computing $\mathbb{E}(X)$ is defined by
$$I^{\rm strat.}_{M_1,\dots,M_k}=\sum_{j=1}^{k}p_j\frac 1{M_j}\sum_{m=1}^{M_j} X_{j,m}$$ where $(p_j)_{1\leq j\leq k}$ are the probabilities of strata and $(X_{j,m}:1\leq m\leq M_j)$ are independent copies of $X$ conditionally to the strata $j$.
1. Prove that the optimal allocation is indeed given by $M^*_j=M\frac{ p_j \sigma_j }{\sum_{i=1}^k p_i \sigma_i }$.
2. For such a choice, derive a central limit theorem for the estimator $I^{\rm strat., opt. alloc.}_{M_1,\dots,M_k}$ as $M\to+\infty$.
Solution.

Exercise 3.4 (stratification of Gaussian vectors)
We aim at describing in detail the generation of random variables in  steps 1 and 2 of Example 3.2.2. We assume that $Y$ is a standard $d$-dimensional Gaussian vector and that $\beta$ is normalized to 1, i.e. $|\beta|=1$.
1. What is the distribution of $Z=\beta\cdot Y$ and its parameters?
2. Assume that ${\cal S}_j=[-x_{j-1},x_j)$ with $-\infty:=x_0\leq ...\leq x_j\leq ...\leq x_k=+\infty$. Compute $p_j=\mathbb{P}(Z\in {\cal S}_j)$ as a function of the $(x_i)_i$. Derive the cumulative distribution function of $Z$ given $\{Z\in {\cal S}_j \}$. Deduce an algorithm to generate such a distribution (assuming that ${\cal N}^{-1}(\cdot)$ is known).
3. Compute explicitly the distribution of $Y-\beta Z$. Show that it is independent on $Z$.
4. Deduce a generation scheme of $Y$ given $\{Z\in {\cal S}_j\}$.
Exercise 3.5 (control variates)
In the Monte-Carlo computation of $\mathbb{E}(X)$ we use extra centered random variables $(Z_j)_j$ (the control variates) and we set $$Z(\beta):=\beta\cdot Z=\sum_{j=1}^d \beta_i Z_i.$$ Establish a central limit theorem for the control variate estimator $I^{\rm Cont.Var.}_{\beta^{*}_M,M}$ that uses the optimal empirical weights $$\beta^{*}_M=\Big[\frac 1M\sum_{m=1}^M Z_mZ_m^T\Big]^{-1}\frac 1M\sum_{m=1}^M X_m Z_m.$$ Check that the limit variance is the lowest possible variance $\mathbf{V}ar(X-Z(\beta^*))$.

Exercise 3.6 (importance sampling, Gaussian vectors)
Extend  the formula of Corollary 3.4.9 to the multidimensional case, by changing both the mean and the covariance.
Solution.

Exercise 3.7 (Esscher  transform, Gaussian and exponential distributions)
In Proposition 3.4.15, we consider a random sum $$Y=\sum_{n=1}^N Z_n$$ where $N$ has a Poisson distribution and $(Z_n)_n$ are independent, in a model defined under the probability $\mathbb{P}$. For a given function $\varphi:\mathbb{R}\mapsto \mathbb{R}$ with  linear growth at most,  $$L:=\exp\Big(\sum_{n=1}^N \varphi(Z_n)-\lambda \mathbb{E}_\mathbb{P}(e^{\varphi(Z)}-1)\Big)$$ defines the likelihood of a new probability measure $\mathbb{Q}$ equivalent to $\mathbb{P}$. Explicitly characterize the distributions of $Z$ under $\mathbb{Q}$, when we take $\varphi(z)=z$ and when under $\mathbb{P}$, $Z$ has either a Gaussian distribution or an exponential one.

Exercise 3.8 (importance sampling, Poisson distribution)
Write a simulation program for computing $\mathbb{P}(Y\geq x)$, by importance sampling, when $Y$ has a Poisson distribution with parameter 1 and $x$ is large (see Example 3.4.16).

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