The wikipedia article about Chebyshev's inequality also lists a one-sided sharpened variant, based on upper/lower semivariances. It is additionally mentioned there there that "The inequality with the lower semivariance has been found to be of use in estimating downside risk in finance and agriculture", but I have troubles accessing the referenced papers (they all seem to require a registration, though the first page of the "Using the Semivariance to Estimate Safety-First Rules" paper by Peter Berck and Jairus M. Hihn appears to be available for free and already contains enough information).

There is also a rather informative question "Does a sample version of the one-sided Chebyshev inequality exist?" about a similar topic already answered here. My current understanding is that specifically looking at the upper and lower semivariances separately may make sense for substantially asymmetric distributions. It can be probably interpreted as just throwing away the samples above/below the population mean and replacing them with the mirror-reflected samples from the other tail. Then the ordinary Chebyshev's inequality is getting applied to this modified data set.

The question is about how applicable is this lower/upper semivariance variant of Chebyshev's inequality for dealing with sampled data. Is it theoretically sound? Or is it more like some kind of heuristics invented by the finance/agriculture folks?


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