Chebyshev's Inequality - Unimodal Distributions

Unimodal Distributions

A distribution function F is unimodal at m if F is convex on (−∞, m) and concave on (m,∞) An empirical distribution can be tested for unimodality with the dip test.

In 1823 Gauss showed that for a unimodal distribution with a mode of zero

If the second condition holds then the second bound is always less than or equal to the first.

If the mode (ν) is not zero and the mean (μ) and standard deviation (σ) are both finite then denoting the root mean square deviation from the mode by ω, we have

and

Winkler in 1866 extended Gauss' inequality to rth moments where r > 0 and the distribution is unimodal with a mode of zero:

Gauss' bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode: see the Vysochanskiï–Petunin inequality for details.

The Vysochanskiï–Petunin inequality has been extended by Dharmadhikari and Joag-Dev

where s is a constant satisfying both s > r + 1 and s( s - r - 1 ) = rr and r > 0.

It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions.