An Improved Laguerre-Samuelson Inequality of Chebyshev-Markov Type

The Chebyshev-Markov extremal distributions by known moments to order four are used to improve the Laguerre-Samuelson inequality for finite real sequences. In general, the refined bound depends not only on the sample size but also on the sample skewness and kurtosis. Numerical illustrations suggest that the refined inequality can almost be attained for randomly distributed completely symmetric sequences from a Cauchy distribution. 1. Introduction Let be real numbers with first and second order moments , . The Laguerre-Samuelson inequality (see Jensen and Styan [1] and Samuelson [2]) asserts that for a sample of size no observation lies more than standard deviation away from the arithmetic mean; that is, Experiments with random samples generated from various distributions on the real line suggest that there is considerable room for improvement if one takes higher order moments , , into account. In the present note, we demonstrate that this can be done using the so-called Chebyshev-Markov extremal distributions based on the moments of order three and four or equivalently on the (sample) knowledge of the skewness and (excess) kurtosis of a real sequence. The latter quantities are denoted and defined by For example, if is a “symmetric normal” real sequence with vanishing skewness and kurtosis , then the following improvement holds (see Example 5): Since the bounds in (1) can be attained, one might argue that (3) is not a genuine improvement because it depends on the property of a sequence to be “symmetric normal.” However, this objection cannot be made if one states improved general bounds of the type with some analytical function depending on all feasible values of and the sample size . According to Arnold and Balakrishnan [3] the idea of using probability inequalities to derive (1) goes back (at least) to Smith [4] (see Jensen and Styan [1], Section 2.7). The derivation is very simple. Indeed, consider the discrete uniform random variable defined by Clearly, is a standard random variable, which therefore satisfies the Chebyshev-Markov inequalities (also called Cantelli inequalities): Substituting into the first inequality and into the second inequality, one gets through combination the Laguerre-Samuelson bound (1). Along the same line of proof, we derive in Section 3 a refinement of the type (4) by considering the generalized Chebyshev-Markov inequalities by known skewness and kurtosis, which is recalled in preliminary Section 2. The result is illustrated for some sequences of symmetric type. We observe that the new bounds are sometimes rather tight.