Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$.

The best I can do is to take the Taylor series of both and get that the difference is the sum of the difference between two normal r.v's and two chi-squared r.v.'s in addition to the rest of the difference between the rest of the terms. Is there a more straight-forward way to get the distribution of the difference between 2 i.i.d. log-normal r.v.'s?

  • $\begingroup$ Here is a relevant paper. You will find more papers by googling! papers.ssrn.com/sol3/papers.cfm?abstract_id=2064829 $\endgroup$ – kjetil b halvorsen May 18 '15 at 15:04
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    $\begingroup$ I've taken a cursory glance at that paper, and it doesn't seem to answer my question in a satisfying way. They seem to be concerned with numerical approximations to the harder problem of finding the distribution for the sum/difference between correlated lognormal r.v.'s. I was hoping that there would be a simpler answer for the independent case. $\endgroup$ – frayedchef May 18 '15 at 15:24
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    $\begingroup$ It might be a simpler answer in the independent case, but not a simple one! The lognormal case is a famously known hard case---the moment-generating function of the lognormal distribution doesnt exist---that is, it doesnt converge on an open interval containing zero. So, you will not find an easy solution. $\endgroup$ – kjetil b halvorsen May 18 '15 at 16:42
  • $\begingroup$ I see... So would the approach I outlined above be reasonable? (i.e., if $Y_i = \log(X_i)$, $X_1 - X_2 \approx (Y_1 - Y_2) + (Y_1^2 - Y_2^2)/2 + {} ...$ Do we know anything about the higher order terms, or how to bound them? $\endgroup$ – frayedchef May 18 '15 at 17:10
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    $\begingroup$ To illustrate the difficulty---the lognormal mgf is only defined on $(-\infty,0]$. To approximate the difference distribution by saddlepoint methods, we need (K=cumulant gf) $K(s)+K(-s)$, and that sum is only defined in one point, zero. So, doesnt seem to work. Sum or average would be simpler! $\endgroup$ – kjetil b halvorsen Oct 7 '15 at 17:20

This is a difficult problem. I thought first about using (some approximation of) the moment generating function of the lognormal distribution. That doesn't work, as I will explain. But first some notation:

Let $\phi$ be the standard normal density and $\Phi$ the corresponding cumulative distribution function. We will only analyze the case lognormal distribution $lnN(0,1)$, which has density function $$ f(x)=\frac1{\sqrt{2\pi}x} e^{-\frac12 (\ln x)^2} $$ and cumulative distribution function $$ F(x) =\Phi(\ln x) $$ Suppose $X$ and $Y$ are independent random variables with the above lognormal distribution. We are interested in the distribution of $D=X-Y$, which is a symmetric distribution with mean zero. Let $M(t) = \DeclareMathOperator{\E}{E} \E e^{tX} $ be the moment generating function of $X$. It is defined only for $t\in (-\infty,0]$, so not defined in an open interval containing zero. The moment generating function for $D$ is $M_D(t)=\E e^{t(X-Y)}= \E e^{tX} \E e^{-tY}= M(t)M(-t)$. So, the moment generating function for $D$ is only defined for $t=0$, so not very useful.

That means we will need some more direct approach for finding approximations for the distribution of $D$. Suppose $t\ge 0$, calculate $$ \begin{align} P(D \le t) &= P(X-Y\le t) \\ &= \int_0^\infty P(X-y\le t | Y=y) f(y) \; dy \\ &= \int_0^\infty P(X\le t+y) f(y) \; dy \\ &= \int_0^\infty F(t+y) f(y) \; dy \end{align} $$ (and the case $t<0$ is solved by symmetry, we get $P(D\le t)=1-P(D\le |t|)$).

This expression can be used for numerical integration or as a basis for simulation. First a test:

 integrate(function(y) plnorm(y)*dlnorm(y), lower=0,  upper=+Inf)
  0.5 with absolute error < 2.3e-06

which is clearly correct. Let us wrap this up inside a function:

pDIFF  <-  function(t) {
    d  <-  t
    for (tt in seq(along=t)) {
        if (t[tt] >= 0.0) d[tt] <- integrate(function(y) plnorm(y+t[tt])*dlnorm(y),
                                         lower=0.0,  upper=+Inf)$value else
                          d[tt] <- 1-integrate(function(y) plnorm(y+abs(t[tt]))*dlnorm(y),
                                         lower=0.0, upper=+Inf)$value

> plot(pDIFF,  from=-5,  to=5)

which gives:

cumulative distribution function found by numerical integration

Then we can find the density function by differentiating under the integral sign, obtaining

dDIFF  <-  function(t) {
       d  <- t; t<- abs(t)
       for (tt in seq(along=t)) {
           d[tt]  <-  integrate(function(y) dlnorm(y+t[tt])*dlnorm(y),
                                lower=0.0,  upper=+Inf)$value

which we can test:

> integrate(dDIFF,  lower=-Inf,  upper=+Inf)
0.9999999 with absolute error < 1.3e-05

And plotting the density we get:

plot(dDIFF,  from=-5,  to=5)

density function found by numerical integration

I did also try to get some analytic approximation, but so far didn't succeed, it is not an easy problem. But numerical integration as above, programmed in R is very fast on modern hardware, so is a good alternative which probably should be used much more.

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This does not strictly answer your question, but wouldn't it be easier to look at the ratio of the $X$ and $Y$? You then simply arrive at

$$ \begin{align} \Pr\left(\frac{X}{Y} \leq t\right) &= \Pr\left(\log\left(\frac{X}{Y}\right) \leq \log(t) \right) \\ &= \Pr(\log(X) - \log(Y) \leq \log(t)) \\ &\sim \mathcal{N}(0, 2 \sigma^2) \end{align}$$

Depending on your application, this may serve your needs.

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  • 3
    $\begingroup$ But aren't we looking at X-Y instead of log(X) - log(Y) ? $\endgroup$ – Sextus Empiricus Jan 31 '18 at 12:02
  • $\begingroup$ Yes, of course. This is just in case somebody would be interested in knowing how two lognormal variables differ from each other without it necessarily needing to be a difference. That's why I also say it doesn't the answer the question. $\endgroup$ – Vincent Traag Feb 3 '18 at 21:41

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