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I have a bivariate normal distribution $\left(X_1, X_2\right)$ with mean vector $\left(\mu_1, \mu_2\right)$ and some VCV matrix as $\Sigma$.

Now I want to estimate the covariance between some continuous functions of the bivariate normal distribution as $COV\left(Y_1,Y_2\right), Y_1=2{X_1}^2, Y_2=e^{X_2}$.

According to https://www.jepusto.com/multivariate-delta-method/, such covariance can be estimated using delta approach as

$COV\left(Y_1,Y_2\right)= \sum_{i=1}^{2} \sum_{j=1}^{2} \frac{\delta f}{\delta \mu_i} \frac{\delta g}{\delta \mu_j} \sigma_{i,j}$.

In the present case, $f = 2x^2, g =e^y$.

Therefore, $COV\left(Y_1,Y_2\right)= \left(2 \times 2 \times \mu_1 \right) \times \left(e^{\mu_2} \right) \times \sigma_{1,2}$.

However when checked above result using simulation, I get fairly different results as follows using R,

Setting up parameters

set.seed(1)
mu1 = rnorm(1, 10); mu2 = rnorm(1, -10); sigma1 = runif(1, 0, 1) * 10; sigma2 = runif(1, 0, 1) * 3; corr = 0.90; 
X = mvtnorm::rmvnorm(10000, mean = c(mu1, mu2), sigma = matrix(c(sigma1^2, corr*sigma1*sigma2, corr*sigma1*sigma2, sigma2^2), 2,2))

Simulation result

print(cov(data.frame(2*X[, 1]^2, exp(X[, 2]))))[1,2] ## 0.3849776

Based on above formula

print(4*mu1*exp(mu2)*(corr*sigma1*sigma2)). ## 0.01000622

Could you please help me to understand why I am getting such different result?

My goal is to analytically derive the covariance between these 2 random variables. Approximation should also be fine. However @whuber commented that use of delata method is not suitable here as underlying distributions are non-normal. In such case, what other methodology can I use to estimate the covariance analytically?

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  • $\begingroup$ Make a scatterplot of $(Y_1,Y_2).$ Since it doesn't look approximately Normal, you're in trouble: the delta method will not give a useful result. $\endgroup$
    – whuber
    Commented Jan 29, 2023 at 18:45
  • $\begingroup$ @whuber Thanks for your comment. How then can I analytically derive the covariance in this case? $\endgroup$ Commented Jan 29, 2023 at 20:44
  • $\begingroup$ This one requires only basic techniques of integration. A convenient approach is to express $(Y_1,Y_2)=(X+Y_2,Y_2)$ where $(X,Y_2)$ has a bivariate Normal distribution with zero correlation. The calculations reduce to finding the expectations of $Y_2\exp(Y_2)$ and $Y_2^2\exp(Y_2),$ which are no different than computing the mgf of a Normal distribution. $\endgroup$
    – whuber
    Commented Jan 30, 2023 at 14:29

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