How should I use importance sampling in this case?

Question

Consider these two cases:

• Case 1: $T_1\sim\operatorname{exp}(\lambda_1)$, $T_2\sim\operatorname{exp}(\lambda_2),T_3\sim\operatorname{exp}(\lambda_3)$,
• Case 2: $T_1\sim\operatorname{exp}(\theta_1)$, $T_2\sim\operatorname{exp}(\theta_2),T_3\sim\operatorname{exp}(\theta_3)$.

We define $T_4 = \max(T_1 + T_2, T_3)$. Suppose that we want to estimate $\mu_2 = E[T_4]$, where $T_4$ is distributed according to case $2$. How can we do so using importance sampling based on a sample $T_1, \ldots, T_n$ generated from case $1$?

I want something like $$\mu_2 \approx \dfrac{1}{n}\sum\limits_{i = 1}^nT_{4i}\dfrac{f(T_{4i})}{q(T_{4i})}$$ where the $T_{4i}$ are generated based on the distributions from case $1$ and where $f$ is the density function of $T_4$ for case $2$ and $q$ is the density function of $T_4$ for case $1$.

Question: I know how to generate $T_{4i}$ based on case $1$, but I don't know how to figure out $f$ and $g$. How should I proceed?

Edit: As pointed out in the comments; I know that I can just use regular Monte Carlo to estimate $\mu_2$. I'm just interested in how I would use importance sampling if I wouldn't be able to simulate directly from case 2.

Answer 1

Importance sampling is based on the identity $$\mathbb{E}_f[h(X)]=\int h(x) f(x)\text{d}x =\int h(x) \frac{f(x)}{g(x)}g(x)\text{d}x $$where both $f$ and $g$ are probability densities. This means one can use a sample generated from $g$ to approximate an expectation under $f$.

In the current problem, the density of$$T_4=\max\{T_1+T_2,T_3\}$$is not provided and while available within a few steps, it does not need to be derived. Another approach is indeed to consider $T_4$ as a function of $(T_1,T_2,T_3)$ and integrate under the distribution of this triplet: \begin{align*} \mathbb{E}_\theta[T_4]&=\mathbb{E}_\theta[\max\{T_1+T_2,T_3\}]\\ &=\int \max\{t_1+t_2,t_3\} f_\theta(t_1,t_2,t_3)\,\text{d}t_1\text{d}t_2\text{d}t_3\\ &=\int \max\{t_1+t_2,t_3\} \dfrac{f_\theta(t_1,t_2,t_3)}{f_\lambda(t_1,t_2,t_3)}f_\lambda(t_1,t_2,t_3)\,\text{d}t_1\text{d}t_2\text{d}t_3\\ &=\mathbb{E}_\lambda\left[T_4 \dfrac{f_\theta(T_1,T_2,T_3)}{f_\lambda(T_1,T_2,T_3)}\right] \end{align*}