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Grüss-Type Bounds for the Covariance of Transformed Random Variables
Journal of Inequalities and Applications volume 2010, Article number: 619423 (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 [2–5], 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 [2–5].
Let be an insurance or financial risk, which from the mathematical point of view is just a random variable. In this context, the expectation
is called the net premium. The insurer, wishing to remain solvent, naturally charges a premium larger than
. As demonstrated by Furman and Zitikis [2, 4], many insurance premiums can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ1_HTML.gif)
where is a nonnegative function, called the weight function, and so
is called the weighted premium. It is well known (Lehmann [12]) that if the weight function
is non-decreasing, then the inequality
holds, which is called the nonnegative loading property in insurance. (Note that when
, then
.) The weighted premium
can be written as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ2_HTML.gif)
with the ratio on the right-hand side known as the loading. The loading is a nonnegative quantity because the weight function is non-decreasing. We want to know the magnitude of the loading, given what we might know or guess about the weight function
and the random variable
. Solving this problem naturally leads to bounding the covariance
.
More generally, as noted by Furman and Zitikis [2, 4], we may wish to work with the (doubly) weighted premium
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ3_HTML.gif)
The latter premium leads to the covariance . Finally, in the more general context of capital allocations, the weighted premiums are extended into weighted capital allocations (Furman and Zitikis [3–5]), which are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ4_HTML.gif)
where the random variable can be viewed, for example, as the return on an entire portfolio and
as the return on an asset in the portfolio. In Economics,
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
and
, and also depending on the monotonicity of functions
and
. Our research in this paper is devoted to understanding the covariance
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 and
satisfy bounds
and
for all
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ5_HTML.gif)
This is known in the literature as the Grüss bound. If denotes a uniformly distributed random variable with the support
, then statement (2.1) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ6_HTML.gif)
This is a covariance bound. If we replace and
by two general random variables
and
with supports
and
, respectively, then from (2.2) we obtain the following covariance bound (Dragomir [14, 15]; also Zitikis [6]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ7_HTML.gif)
We emphasize that the random variables and
in (2.3) are not necessary uniformly distributed. They are general random variables, except that we assume
and
, and no dependence structure between
and
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 and
be two random variables with joint density function
, assuming that it exists, and denote the (marginal) densities of
and
by
and
, respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ8_HTML.gif)
Approaching the problem from a different angle, Zitikis [6] has sharpened Grüss's bound by including restrictions on the means of the random variables and
, as stated in the next proposition.
Proposition 2.2.
Let and
be two random variables. Furthermore, let
and
be intervals such that
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ9_HTML.gif)
where and
are "information coefficients'' defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ10_HTML.gif)
When there is no "useful information,'' then the two information coefficients and
are equal to
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 .
Proposition 2.3.
Let ,
and
be real functions defined on
, and let
be nonnegative and integrable. If the functions
and
are both increasing, or both decreasing, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ11_HTML.gif)
If, however, one of the two functions and
is increasing and the other one is decreasing, then inequality (2.7) is reversed.
With an appropriately defined random variable (see a note following Grüss's inequality (2.1) above), Chebyshev's integral inequality (2.7) can be rewritten in the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ12_HTML.gif)
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 in terms of the cumulative distribution functions of the random variables
and
. Among them is a result by Hoeffding [20], who proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ13_HTML.gif)
where is the joint cumulative distribution function of
, and
and
are the (marginal) cumulative distribution functions of
and
, respectively. Mardia [21], Mardia and Thompson [22] extended Hoeffding's result by showing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ14_HTML.gif)
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 and
be any real functions of bounded variation and defined, respectively, on the intervals
and
of the extended real line
. Furthermore, let
and
be any random variables such that the expectations
,
, and
are finite. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ16_HTML.gif)
which is the integrand on the right-hand side of (2.11), governs the dependence structure between the random variables and
. For example, when
for all
and
, then the random variables are independent. Hence, departure of
from
serves a measure of dependence between
and
. Depending on which side (positive or negative) the departure from
takes place, we have positive or negative dependence between the two random variables. Specifically, when
for all
and
, then
and
are called positively quadrant dependent, and when
for all
and
, 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 , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ17_HTML.gif)
irrespectively of the dependence structure between the random variables and
. Bound (3.1) can be verified as follows. First, for any event
, the probability
is the expectation
of the indicator
, which is a random variable taking on the value
if the event
happens, and
otherwise. Hence,
is equal to the covariance
. Next we use the Cauchy-Schwarz inequality to estimate the latter covariance and thus obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ18_HTML.gif)
Since is a binary random variable taking on the two values
and
with the probabilities
and
, respectively, the variance
is equal to the product of the probabilities
and
. The product does not exceed
. Likewise, the variance
does not exceed
. 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 and
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
and
are non-decreasing. Hence, we have the bound
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ19_HTML.gif)
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 and
(cf. Zitikis [6]).
Theorem 3.1.
Let and
be any random variables, and let
, which one calls the "dependence coefficient,'' be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ20_HTML.gif)
for all and
. Furthermore, let
and
be two right-continuous and non-decreasing functions defined on
and
, respectively, and let
and
be intervals such that
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ21_HTML.gif)
where and
are "information coefficients'' defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ22_HTML.gif)
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 and
inside the intervals
and
, respectively, then the two information coefficients
and
are equal to
by definition, and thus
is equal to
. Furthermore, if there is no "useful dependence information'' between
and
, then
by definition. Hence, in the presence of no "useful information'' about the means and dependence, the coefficient
reduces to the classical Grüss coefficient
.
Proof of Theorem 3.1.
Since by assumption, using (2.11) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ23_HTML.gif)
where the last equality holds because the functions and
are right-continuous and non-decreasing. Next we restart the estimation of the covariance
anew. Namely, using the Cauchy-Schwarz inequality, together with the bound
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ24_HTML.gif)
and an analogous one for , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ25_HTML.gif)
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 and
, and also assume that the random variables
and
take on values in the interval
. Grüss's bound (2.3) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ26_HTML.gif)
Assume now that the pair has a joint density function,
, and let it be equal to
for
, and
for all other
. The random variables
and
take on values in the interval
as before, but we can now calculate their means and thus apply Proposition 2.2 with appropriately specified "
-constraints.''
The joint cumulative distribution function of the pair
can be expressed by the formula
. Thus, the (marginal) cumulative distribution functions of
and
are equal to
for all
and
for all
, respectively. Using the equation
, we check that
. Likewise, we have
. Consequently, we may let the
-constraints on the means
and
be as follows:
and
. We also have
and
, because
is the support of the two random variables
and
. These notes and the definitions of
and
given in Proposition 2.2 imply that
. Consequently, bound (2.5) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ27_HTML.gif)
which is an improvement upon bound (4.1), and thus upon (4.2).
We next utilize the dependence structure between and
in order to further improve upon bound (4.2). With
and
already calculated, we next calculate
. For this, we use the above formulas for the three cumulative distribution functions and see that
. (The negative sign of
for all
reveals that the random variables
and
are negatively quadrant dependent.) Furthermore, we check that
attains its maximum at the point
. Hence, the smallest upper bound for
is
, and so we have
, which is less than
. Hence, bound (3.5) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ28_HTML.gif)
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 is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F619423/MediaObjects/13660_2009_Article_2205_Equ29_HTML.gif)
which we have calculated using the equation (cf. (2.9)) and the above given expression for
.
References
Egozcue M, Fuentes Garcia L, Wong W-K: On some covariance inequalities for monotonic and non-monotonic functions. Journal of Inequalities in Pure and Applied Mathematics 2009, 10(3, article 75):1–7.
Furman E, Zitikis R: Weighted premium calculation principles. Insurance: Mathematics and Economics 2008, 42(1):459–465. 10.1016/j.insmatheco.2007.10.006
Furman E, Zitikis R: Weighted risk capital allocations. Insurance: Mathematics and Economics 2008, 43(2):263–269. 10.1016/j.insmatheco.2008.07.003
Furman E, Zitikis R: Weighted pricing functionals with applications to insurance: an overview. North American Actuarial Journal 2009, 13: 483–496.
Furman E, Zitikis R: General Stein-type covariance decompositions with applications to insurance and Finance. to appear in ASTIN Bulletin—The Journal of the International Actuarial Association
Zitikis R: Grüss's inequality, its probabilistic interpretation, and a sharper bound. Journal of Mathematical Inequalities 2009, 3(1):15–20.
Dragomir SS: Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces. Nova Science, New York, NY, USA; 2005:viii+249.
Dragomir SS, Agarwal RP: Some inequalities and their application for estimating the moments of guessing mappings. Mathematical and Computer Modelling 2001, 34(3–4):441–468. 10.1016/S0895-7177(01)00075-9
Dragomir SS, Diamond NT: A discrete Grüss type inequality and applications for the moments of random variables and guessing mappings. In Stochastic Analysis and Applications. Volume 3. Nova Science, New York, NY, USA; 2003:21–35.
Izumino S, Pečarić JE: Some extensions of Grüss' inequality and its applications. Nihonkai Mathematical Journal 2002, 13(2):159–166.
Izumino S, Pečarić JE, Tepeš B: A Grüss-type inequality and its applications. Journal of Inequalities and Applications 2005, 2005(3):277–288. 10.1155/JIA.2005.277
Lehmann EL: Some concepts of dependence. Annals of Mathematical Statistics 1966, 37: 1137–1153. 10.1214/aoms/1177699260
Grüss G: Über das maximum des absoluten betrages von . Mathematische Zeitschrift 1935, 39(1):215–226. 10.1007/BF01201355
Dragomir SS: A generalization of Grüss's inequality in inner product spaces and applications. Journal of Mathematical Analysis and Applications 1999, 237(1):74–82. 10.1006/jmaa.1999.6452
Dragomir SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra and Its Applications 2008, 428(11–12):2750–2760. 10.1016/j.laa.2007.12.025
Anastassiou GA, Papanicolaou VG: Probabilistic inequalities and remarks. Applied Mathematics Letters 2002, 15(2):153–157. 10.1016/S0893-9659(01)00110-0
Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series). Volume 61. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.
Athey S: Monotone comparative statics under uncertainty. Quarterly Journal of Economics 2002, 117(1):187–223. 10.1162/003355302753399481
Wagener A: Chebyshev's algebraic inequality and comparative statics under uncertainty. Mathematical Social Sciences 2006, 52(2):217–221. 10.1016/j.mathsocsci.2006.05.004
Hoeffding W: Masstabinvariante korrelationstheorie. Schriften des Matematischen Instituts für Angewandte Matematik der Universität Berlin 1940, 5: 179–233.
Mardia KV: Some contributions to contingency-type bivariate distributions. Biometrika 1967, 54: 235–249.
Mardia KV, Thompson JW: Unified treatment of moment-formulae. Sankhyā Series A 1972, 34: 121–132.
Sen PK: The impact of Wassily Hoeffding's research on nonparametric. In Collected Works of Wassily Hoeffding. Edited by: Fisher NI, Sen PK. Springer, New York, NY, USA; 1994:29–55.
Cuadras CM: On the covariance between functions. Journal of Multivariate Analysis 2002, 81(1):19–27. 10.1006/jmva.2001.2000
Balakrishnan N, Lai CD: Continuous Bivariate Distributions. 2nd edition. Springer, New York, NY, USA; 2009.
Denuit M, Dhaene J, Goovaerts M, Kaas R: Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, Chichester, UK; 2005.
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.
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Egozcue, M., Fuentes García, L., Wong, WK. et al. Grüss-Type Bounds for the Covariance of Transformed Random Variables. J Inequal Appl 2010, 619423 (2010). https://doi.org/10.1155/2010/619423
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DOI: https://doi.org/10.1155/2010/619423