Open Access

Grüss-Type Bounds for the Covariance of Transformed Random Variables

  • Martín Egozcue1, 2,
  • Luis Fuentes García3,
  • Wing-Keung Wong4 and
  • Ričardas Zitikis5Email author
Journal of Inequalities and Applications20102010:619423

DOI: 10.1155/2010/619423

Received: 9 November 2009

Accepted: 16 March 2010

Published: 28 March 2010

Abstract

A number of problems in Economics, Finance, Information Theory, Insurance, and generally in decision making under uncertainty rely on estimates of the covariance between (transformed) random variables, which can, for example, be losses, risks, incomes, financial returns, and so forth. Several avenues relying on inequalities for analyzing the covariance are available in the literature, bearing the names of Chebyshev, Grüss, Hoeffding, Kantorovich, and others. In the present paper we sharpen the upper bound of a Grüss-type covariance inequality by incorporating a notion of quadrant dependence between random variables and also utilizing the idea of constraining the means of the random variables.

1. Introduction

Analyzing and estimating covariances between random variables is an important and interesting problem with manifold applications to Economics, Finance, Actuarial Science, Engineering, Statistics, and other areas (see, e.g., Egozcue et al. [1], Furman and Zitikis [25], Zitikis [6], and references therein). Well-known covariance inequalities include those of Chebyshev and Grüss (see, e.g., Dragomir [7] and references therein). There are many interesting applications of Grüss's inequality in areas such as Computer Science, Engineering, and Information Theory. In particular, the inequality has been actively investigated in the context of Guessing Theory, and we refer to Dragomir and Agarwal [8], Dragomir and Diamond [9], Izumino and Pečarić [10], Izumino et al. [11], and references therein.

Motivated by an open problem posed by Zitikis [6] concerning Grüss's bound in the context of dependent random variables, in the present paper we offer a tighter Grüss-type bound for the covariance of two transformed random variables by incorporating a notion of quadrant dependence and also utilizing the idea of constraining the means of the random variables. To see how this problem arises in the context of insurance and financial pricing, we next present an illustrative example. For further details and references on the topic, we refer to Furman and Zitikis [25].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq1_HTML.gif be an insurance or financial risk, which from the mathematical point of view is just a random variable. In this context, the expectation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq2_HTML.gif is called the net premium. The insurer, wishing to remain solvent, naturally charges a premium larger than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq3_HTML.gif . As demonstrated by Furman and Zitikis [2, 4], many insurance premiums can be written in the form

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq4_HTML.gif is a nonnegative function, called the weight function, and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq5_HTML.gif is called the weighted premium. It is well known (Lehmann [12]) that if the weight function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq6_HTML.gif is non-decreasing, then the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq7_HTML.gif holds, which is called the nonnegative loading property in insurance. (Note that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq8_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq9_HTML.gif .) The weighted premium https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq10_HTML.gif can be written as follows:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ2_HTML.gif
(1.2)

with the ratio on the right-hand side known as the loading. The loading is a nonnegative quantity because the weight function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq11_HTML.gif is non-decreasing. We want to know the magnitude of the loading, given what we might know or guess about the weight function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq12_HTML.gif and the random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq13_HTML.gif . Solving this problem naturally leads to bounding the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq14_HTML.gif .

More generally, as noted by Furman and Zitikis [2, 4], we may wish to work with the (doubly) weighted premium

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ3_HTML.gif
(1.3)

The latter premium leads to the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq15_HTML.gif . Finally, in the more general context of capital allocations, the weighted premiums are extended into weighted capital allocations (Furman and Zitikis [35]), which are

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ4_HTML.gif
(1.4)

where the random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq16_HTML.gif can be viewed, for example, as the return on an entire portfolio and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq17_HTML.gif as the return on an asset in the portfolio. In Economics, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq18_HTML.gif is known as the expected utility, or the expected valuation, depending on a context. The `loading' ratio on the right-hand side of (1.4) can be negative, zero, or positive, depending on the dependence structure between the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq20_HTML.gif , and also depending on the monotonicity of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq22_HTML.gif . Our research in this paper is devoted to understanding the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq23_HTML.gif and especially its magnitude, depending on the information that might be available to the researcher and/or decision maker.

The rest of the paper is organized as follows. In Section 2 we discuss a number of known results, which we call propositions throughout the section. Those propositions lead naturally to our main result, which is formulated in Section 3 as Theorem 3.1. In Section 4 we give an illustrative example that demonstrates the sharpness of the newly established Grüss-type bound.

2. A Discussion of Known Results

Grüss [13] proved that if two functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq25_HTML.gif satisfy bounds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq27_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq28_HTML.gif , then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ5_HTML.gif
(2.1)

This is known in the literature as the Grüss bound. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq29_HTML.gif denotes a uniformly distributed random variable with the support https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq30_HTML.gif , then statement (2.1) can be rewritten as

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ6_HTML.gif
(2.2)

This is a covariance bound. If we replace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq32_HTML.gif by two general random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq34_HTML.gif with supports https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq36_HTML.gif , respectively, then from (2.2) we obtain the following covariance bound (Dragomir [14, 15]; also Zitikis [6]):

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ7_HTML.gif
(2.3)

We emphasize that the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq38_HTML.gif in (2.3) are not necessary uniformly distributed. They are general random variables, except that we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq40_HTML.gif , and no dependence structure between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq42_HTML.gif is assumed.

There are many results sharpening Grüss's bound under various bits of additional information (see, e.g., Dragomir [14, 15], and references therein). For example, Anastassiou and Papanicolaou [16] have established the following bound.

Proposition 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq44_HTML.gif be two random variables with joint density function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq45_HTML.gif , assuming that it exists, and denote the (marginal) densities of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq47_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq49_HTML.gif , respectively. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ8_HTML.gif
(2.4)

Approaching the problem from a different angle, Zitikis [6] has sharpened Grüss's bound by including restrictions on the means of the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq51_HTML.gif , as stated in the next proposition.

Proposition 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq53_HTML.gif be two random variables. Furthermore, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq55_HTML.gif be intervals such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq57_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ9_HTML.gif
(2.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq59_HTML.gif are "information coefficients'' defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ10_HTML.gif
(2.6)

When there is no "useful information,'' then the two information coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq61_HTML.gif are equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq62_HTML.gif by definition (Zitikis [6]), and thus bound (2.5) reduces to the classical Grüss bound.

Mitrinović et al. [17] have in detail discussed Chebyshev's integral inequality, formulated next as a proposition, which gives an insight into Grüss's inequality and especially into the sign of the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq63_HTML.gif .

Proposition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq66_HTML.gif be real functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq67_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq68_HTML.gif be nonnegative and integrable. If the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq70_HTML.gif are both increasing, or both decreasing, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ11_HTML.gif
(2.7)

If, however, one of the two functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq72_HTML.gif is increasing and the other one is decreasing, then inequality (2.7) is reversed.

With an appropriately defined random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq73_HTML.gif (see a note following Grüss's inequality (2.1) above), Chebyshev's integral inequality (2.7) can be rewritten in the following form:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ12_HTML.gif
(2.8)

As we will see in a moment, inequality (2.8) is also implied by the notion of positive quadrant dependence (Lehmann [12]). For details on economic applications of Chebyshev's integral inequality (2.8), we refer to Athey [18], Wagener [19], and references therein.

There have been many attempts to express the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq74_HTML.gif in terms of the cumulative distribution functions of the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq76_HTML.gif . Among them is a result by Hoeffding [20], who proved that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ13_HTML.gif
(2.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq77_HTML.gif is the joint cumulative distribution function of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq78_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq80_HTML.gif are the (marginal) cumulative distribution functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq82_HTML.gif , respectively. Mardia [21], Mardia and Thompson [22] extended Hoeffding's result by showing that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ14_HTML.gif
(2.10)

For further extensions of these results, we refer to Sen [23] and Lehmann [12]. Cuadras [24] has generalized these works by establishing the following result.

Proposition 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq84_HTML.gif be any real functions of bounded variation and defined, respectively, on the intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq86_HTML.gif of the extended real line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq87_HTML.gif . Furthermore, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq89_HTML.gif be any random variables such that the expectations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq91_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq92_HTML.gif are finite. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ15_HTML.gif
(2.11)

Equation (2.11) plays a crucial role in establishing our main result, which is Theorem 3.1 in the next section. To facilitate easier intuitive understanding of that section, we note that the function

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ16_HTML.gif
(2.12)

which is the integrand on the right-hand side of (2.11), governs the dependence structure between the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq94_HTML.gif . For example, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq95_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq97_HTML.gif , then the random variables are independent. Hence, departure of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq98_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq99_HTML.gif serves a measure of dependence between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq101_HTML.gif . Depending on which side (positive or negative) the departure from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq102_HTML.gif takes place, we have positive or negative dependence between the two random variables. Specifically, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq103_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq105_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq107_HTML.gif are called positively quadrant dependent, and when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq108_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq110_HTML.gif , then the random variables are negatively quadrant dependent. For applications of these notions of dependence and also for further references, we refer to the monographs by Balakrishnan and Lai [25], Denuit et al. [26].

3. A New Grüss-Type Bound

We start this section with a bound that plays a fundamental role in our subsequent considerations. Namely, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq111_HTML.gif , we have that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ17_HTML.gif
(3.1)

irrespectively of the dependence structure between the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq113_HTML.gif . Bound (3.1) can be verified as follows. First, for any event https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq114_HTML.gif , the probability https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq115_HTML.gif is the expectation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq116_HTML.gif of the indicator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq117_HTML.gif , which is a random variable taking on the value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq118_HTML.gif if the event https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq119_HTML.gif happens, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq120_HTML.gif otherwise. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq121_HTML.gif is equal to the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq122_HTML.gif . Next we use the Cauchy-Schwarz inequality to estimate the latter covariance and thus obtain that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ18_HTML.gif
(3.2)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq123_HTML.gif is a binary random variable taking on the two values https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq125_HTML.gif with the probabilities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq127_HTML.gif , respectively, the variance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq128_HTML.gif is equal to the product of the probabilities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq130_HTML.gif . The product does not exceed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq131_HTML.gif . Likewise, the variance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq132_HTML.gif does not exceed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq133_HTML.gif . From bound (3.2) we thus have bound (3.1).

To see how bound (3.1) is related to Grüss's bound, we apply it on the right-hand side of (2.11). We also assume that the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq135_HTML.gif are right-continuous and monotonic. Note that, without loss of generality in our context, the latter monotonicity assumption can be replaced by the assumption that the two functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq137_HTML.gif are non-decreasing. Hence, we have the bound

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ19_HTML.gif
(3.3)

which is Grüss's bound written in a somewhat different form than that in (2.2).

The following theorem sharpens the upper bound of Grüss's covariance inequality (3.3) by utilizing the notion of quadrant dependence (cf. Lehmann [12]) and incorporating constrains on the means of random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq139_HTML.gif (cf. Zitikis [6]).

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq141_HTML.gif be any random variables, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq142_HTML.gif , which one calls the "dependence coefficient,'' be such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ20_HTML.gif
(3.4)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq144_HTML.gif . Furthermore, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq146_HTML.gif be two right-continuous and non-decreasing functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq148_HTML.gif , respectively, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq150_HTML.gif be intervals such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq152_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ21_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq154_HTML.gif are "information coefficients'' defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ22_HTML.gif
(3.6)

Before proving the theorem, a few clarifying notes follow. If there is no "useful information'' (see Zitikis [6] for the meaning) about the location of the means https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq156_HTML.gif inside the intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq158_HTML.gif , respectively, then the two information coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq160_HTML.gif are equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq161_HTML.gif by definition, and thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq162_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq163_HTML.gif . Furthermore, if there is no "useful dependence information'' between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq165_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq166_HTML.gif by definition. Hence, in the presence of no "useful information'' about the means and dependence, the coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq167_HTML.gif reduces to the classical Grüss coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq168_HTML.gif .

Proof of Theorem 3.1.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq169_HTML.gif by assumption, using (2.11) we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ23_HTML.gif
(3.7)
where the last equality holds because the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq171_HTML.gif are right-continuous and non-decreasing. Next we restart the estimation of the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq172_HTML.gif anew. Namely, using the Cauchy-Schwarz inequality, together with the bound
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ24_HTML.gif
(3.8)
and an analogous one for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq173_HTML.gif , we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ25_HTML.gif
(3.9)

Combining bounds (3.7) and (3.9), we arrive at bound (3.5), thus completing the proof of Theorem 3.1.

4. An Example

Here we present an example that helps to compare the bounds of Grüss [13], Zitikis [6], and the one of Theorem 3.1.

To make our considerations as simple as possible, yet meaningful, we choose to work with the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq175_HTML.gif , and also assume that the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq177_HTML.gif take on values in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq178_HTML.gif . Grüss's bound (2.3) implies that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ26_HTML.gif
(4.1)

Assume now that the pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq179_HTML.gif has a joint density function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq180_HTML.gif , and let it be equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq181_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq182_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq183_HTML.gif for all other https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq184_HTML.gif . The random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq185_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq186_HTML.gif take on values in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq187_HTML.gif as before, but we can now calculate their means and thus apply Proposition 2.2 with appropriately specified " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq188_HTML.gif -constraints.''

The joint cumulative distribution function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq189_HTML.gif of the pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq190_HTML.gif can be expressed by the formula https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq191_HTML.gif . Thus, the (marginal) cumulative distribution functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq193_HTML.gif are equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq194_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq195_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq196_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq197_HTML.gif , respectively. Using the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq198_HTML.gif , we check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq199_HTML.gif . Likewise, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq200_HTML.gif . Consequently, we may let the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq201_HTML.gif -constraints on the means https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq203_HTML.gif be as follows: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq205_HTML.gif . We also have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq207_HTML.gif , because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq208_HTML.gif is the support of the two random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq210_HTML.gif . These notes and the definitions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq212_HTML.gif given in Proposition 2.2 imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq213_HTML.gif . Consequently, bound (2.5) implies that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ27_HTML.gif
(4.2)

which is an improvement upon bound (4.1), and thus upon (4.2).

We next utilize the dependence structure between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq215_HTML.gif in order to further improve upon bound (4.2). With https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq217_HTML.gif already calculated, we next calculate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq218_HTML.gif . For this, we use the above formulas for the three cumulative distribution functions and see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq219_HTML.gif . (The negative sign of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq220_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq221_HTML.gif reveals that the random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq222_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq223_HTML.gif are negatively quadrant dependent.) Furthermore, we check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq224_HTML.gif attains its maximum at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq225_HTML.gif . Hence, the smallest upper bound for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq226_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq227_HTML.gif , and so we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq228_HTML.gif , which is less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq229_HTML.gif . Hence, bound (3.5) implies that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ28_HTML.gif
(4.3)

which is a considerable improvement upon bounds (4.1) and (4.2).

We conclude this example by noting that the true value of the covariance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq230_HTML.gif is

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ29_HTML.gif
(4.4)

which we have calculated using the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq231_HTML.gif (cf. (2.9)) and the above given expression for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_IEq232_HTML.gif .

Declarations

Acknowledgments

The authors are indebted to four anonymous referees, the editor in charge of the manuscript, Soo Hak Sung, and the Editor-in-Chief, Ravi P. Agarwal, for their constructive criticism and numerous suggestions that have resulted in a considerable improvement of the paper. The third author would also like to thank Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. The research has been partially supported by grants from the University of Montevideo, University of Coruña, Hong Kong Baptist University, and the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Authors’ Affiliations

(1)
Department of Economics, University of Montevideo
(2)
Accounting and Finance Department, Norte Construcciones
(3)
Departamento de Métodos Matemáticos e de Representación, Escola Técnica Superior de Enxeñeiros de Camiños, Canais e Portos, Universidade da Coruña
(4)
Department of Economics, Institute for Computational Mathematics, Hong Kong Baptist University
(5)
Department of Statistical and Actuarial Sciences, University of Western Ontario

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© Martín Egozcue et al. 2010

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