# Confidence interval for the difference of lognormally distributed random variables

I have two lognormally distributed random variables $$Y_i=e^{X_i}$$ where $$X_i \sim \mathcal{N}\big(\mu_i, \: \sigma_i^2 \big)$$ for $$i=1,2$$, and $$X_1$$ and $$X_2$$ are correlated by $$\rho_{12}$$. Now, Let $$Z=\alpha Y_1 - \beta Y_2$$.

Question 1

Is $$Z$$ lognormally distributed?

Question2

When $$\mu_i$$, $$\sigma_i$$, and $$\rho_{12}$$, as well as the mean and variance of $$Z$$: $$\mu_Z$$, $$\sigma_Z^2$$ are known, how can I construct a confidence interval for $$\mu_Z$$ using these known parameters?

I also know how to calculate the mu and sigma for $$Y_i$$ using these relationship.

$$\mu_{Y_i} =\mathbb{E}\big[Y_i\big] = e^{\mu_i+\frac{1}{2}\sigma_i^2}\\ \sigma_{Y_i}^2 =\mathbb{V} \big[Y_i\big] = \Big[e^{\sigma_i^2}-1 \Big]\: e^{(2\mu_i+\sigma_i^2)}.$$

I was looking at Wikipedia and this reference but struggling to understand how to adopt it for $$Z$$. I am not from mathematics background. I would appreciate if you can point me out to the right directions...

• For question 1, perhaps try the properties of the exponential functional/logartihm? For question 2, what formula for the confidence interval is given? Commented Mar 5, 2021 at 18:17
Question 1: $$Z$$ will not be lognormally distributed in general. E.g., if $$\beta>0$$ or $$\alpha<0$$, then $$P(Z<0)>0$$, and hence $$Z$$ is not lognormally distributed.
Question 2: If $$\mu_Z$$ is known, then it does not make sense to use a confidence interval for $$\mu_Z$$.
If $$\mu_Z$$ and $$\sigma_Z$$ are both unknown, then, in view of the central limit theorem, an approximate $$(1-p)$$-confidence interval for $$\mu_Z$$ based on a large iid sample $$Z_1,\dots,Z_n$$ from the distribution of $$Z$$ is $$[\bar Z-z_{p/2}S_Z/\sqrt n,\bar Z+z_{p/2}S_Z/\sqrt n], \tag{1}$$ where $$\bar Z:=\frac1n\,\sum_1^n Z_i,\quad S_Z:=\sqrt{\frac1{n-1}\,\sum_1^n(Z_i-\bar Z)^2},$$ $$z_{p/2}:=\Phi^{-1}(1-p/2)$$, and $$\Phi$$ is the standard normal cdf. Here we have also used the law of large numbers, implying that $$S_Z^2\to\sigma_Z^2$$ in probability.
If $$\mu_Z$$ is unknown but $$\sigma_Z$$ is known, then one can replace $$S_Z$$ in (1) by $$\sigma_Z$$.