What is the distribution of the quotient of two consecutive order statistics from a lognormal distribution.

Question

While I know that in principle the distribution of the quotient of two random variables, including order statistics, can be derived from the Mellin transform, I was hoping to find a paper or book that does it? Any advice?

Answer 1

A good reference:

Malik, Henrick J., and Roger Trudel. "Probability density function of quotient of order statistics from the pareto, power and weibull distributions." Communications in Statistics-Theory and Methods 11.7 (1982): 801-814.

It is also worth to keep in mind that if all you are interested is the expected value of the log ratio or ratio then it can be calculated as $\int_0^\infty \int_{X_{i:n}}^\infty(ln[X_{j:n}-X_{i:n}]*g(X_{j:n},X_{i:n})dX_{j:n}dX_{i:n}$ where $g(X_{j:n},X_{i:n})$ is the joint distribution of the two order statistics $X_{i:n}$ $X_{j:n}$ with $i<j\leq n$. Here I am assuming the sample comes from distribution with $x>0$ and unbounded from above, e.g., lognormal.

So you can get away without actually deriving the distribution of the quotient.