Consider lognormal random variables $X_1$ and $X_2$ with correlation coefficient $ρ$ and a partial observation sample of them of length N, the sample being partial because it only contains occurrences of ($X_1$ , $X_2$) when $X_1 > X_2$.

Is there a way to estimate the variance and covariance of $X_1$ and $X_2$ only from this partial sample?

Correlation of log-normal random variables gives the formula for the covariance in the case of $X_1$ and $X_2$ beeing lognormals (notations differ). Can I write a conditionnal covariance like:

$cov(X_1 , X_2 | Z>0)$ with $Z=X_1 - X_2$ and derive a formula for this (similar to the one in Correlation of log-normal random variables)? Then, is it possible to find from this the unconditional covariance?

Is this possible for gaussian variables? Or for any distribution?

  • $\begingroup$ Are the unconditional variances known, unknown, or unknown but assumed equal? Are the unconditional means known, unknown, or unknown but assumed equal? Is the fraction of unconditional variables with $X_1>X_2$ known or unknown? Those questions give many plausible versions of the main question even for the easier-to-analyze case of Gaussian instead of lognornal variables. So an intended application of the question would clarify a lot. $\endgroup$
    – Matt F.
    Sep 23, 2022 at 7:56


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