# Estimating covariance based on Delta approach

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?

• 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.
– whuber
Jan 29, 2023 at 18:45
• @whuber Thanks for your comment. How then can I analytically derive the covariance in this case? Jan 29, 2023 at 20:44
• 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.
– whuber
Jan 30, 2023 at 14:29