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.


Introduction
Let  1 ,  2 , . . .,   be  real numbers with first and second order moments   = (1/) ∑  =1    ,  = 1, 2. The Laguerre-Samuelson inequality (see Jensen and Styan [1] and Samuelson [2]) asserts that for a sample of size  no observation lies more than √  − 1 standard deviation away from the arithmetic mean; that is,       −      ≤ √  − 1 ⋅ ,  = 1, . . ., ,  =  1 ,  = √ 2 −  2  1 . ( 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   = (1/)∑  =1    ,  ≥ 3, into account.In the present note, we demonstrate that this can be done using the socalled 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  = (1/) ∑  =1 (  − ) For example, if  1 ,  2 , . . .,   is a "symmetric normal" real sequence with vanishing skewness and kurtosis  =  = 0, 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  = ( min − )/ < 0 into the first inequality and  = ( max − )/ > 0 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.Symmetric sequences from a Cauchy distribution, whose moments do not exist, generate examples for this phenomenon.

The Chebyshev-Markov Extremal Distributions by Known Skewness and Kurtosis
Given a partial order between random variables and some class of random variables, it is possible to construct extremal random variables with respect to this partial order, which provide useful information about extreme situations in probabilistic modeling.For example, the classical Chebyshev-Markov inequalities yield the extremal random variables with respect to the usual stochastic order for random variables with a given range and moments known up to a fixed order.Extremal random variables with respect to the increasing convex and other orderings are of similar general interest.
A modern account of this topic is found in Hürlimann [5], Chap.IV.
In general, to construct the stochastic ordered extremal distributions (7), it is necessary to solve the optimization problems max where  [,∞) () is the Heaviside indicator function, defined to be 0 if  <  and 1 otherwise.It belongs to the class of extremal moment problems max where () is a piecewise linear function and which have been extensively studied in Hürlimann [5][6][7].A general approach to solve these problems is the well-known polynomial majorant (minorant) method.It consists to find a polynomial () ≥ () (() ≤ ()) of degree less than or equal to  and to construct a finite atomic random variable  ∈   such that all atoms of () are simultaneously atoms of ().Indeed, suppose () and  have been For the present purpose, it suffices to restrict the attention to the construction of  (4)  st,max (),  (4)  st,min () over the space  4 of all standard random variables defined on the real line (−∞, ∞) with known skewness and kurtosis pair (, ).Recall that for  ∈  4 one has  = [ 3 ],  = [ 4 ]−3.The required main result is found in the Appendix of Hürlimann [8].First of all, the parameters must satisfy the following well-known inequality between skewness and kurtosis (e.g., Pearson [9], Wilkins [10], and Guiard [11]): Second, the extremal bounds are attained at standard triatomic random variables   ∈  4 such that Pr(  (  ) =  [,∞) (  )) = 1 for some quartic polynomial   (), and one has To describe the extremal supports of   ∈  4 and the associated probabilities one needs the following characterization result.
The Chebyshev-Markov extremal distributions over the space  4 are determined as follows.
Theorem 2. Under the assumption Δ =  −  2 + 2 ≥ 0 and in the notations of Proposition 1, the distribution functions of the Chebyshev-Markov ordered extremal random variables  (4)  ,max ,  (4)  ,min for the set  4 are described in Table 1.

A Refinement of the Laguerre-Samuelson Inequality
Let  1 ,  2 , . . .,   be  real numbers with first four order moments   where   ≤  < 0,   ≥  > 0 are both solutions of the equation () = 1/, which by ( 13) is equivalent to the quartic equation Since the probability function () is monotone decreasing, the condition   ≥  > 0 is equivalent to the inequalities Similarly, the condition   ≤  < 0 is equivalent to the inequalities Therefore, a necessary condition for the validity of ( 15) is  ≥ (1/2)(4 +  2 + || √ 4 +  2 ).The following main result has been shown.