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.
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. , Furman and Zitikis [2–5], Zitikis , and references therein). Well-known covariance inequalities include those of Chebyshev and Grüss (see, e.g., Dragomir  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 , Dragomir and Diamond , Izumino and Pečarić , Izumino et al. , and references therein.
Motivated by an open problem posed by Zitikis  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
where is a nonnegative function, called the weight function, and so is called the weighted premium. It is well known (Lehmann ) 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:
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 .
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
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  proved that if two functions and satisfy bounds and for all , then
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
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 ):
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  have established the following bound.
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
Approaching the problem from a different angle, Zitikis  has sharpened Grüss's bound by including restrictions on the means of the random variables and , as stated in the next proposition.
Let and be two random variables. Furthermore, let and be intervals such that and . Then
where and are "information coefficients'' defined by
When there is no "useful information,'' then the two information coefficients and are equal to by definition (Zitikis ), and thus bound (2.5) reduces to the classical Grüss bound.
Mitrinović et al.  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 .
Let , and be real functions defined on , and let be nonnegative and integrable. If the functions and are both increasing, or both decreasing, then
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:
As we will see in a moment, inequality (2.8) is also implied by the notion of positive quadrant dependence (Lehmann ). For details on economic applications of Chebyshev's integral inequality (2.8), we refer to Athey , Wagener , 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 , who proved that
where is the joint cumulative distribution function of , and and are the (marginal) cumulative distribution functions of and , respectively. Mardia , Mardia and Thompson  extended Hoeffding's result by showing that
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
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
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 , Denuit et al. .
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
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
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
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 ) and incorporating constrains on the means of random variables and (cf. Zitikis ).
Let and be any random variables, and let , which one calls the "dependence coefficient,'' be such that
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
where and are "information coefficients'' defined by
Before proving the theorem, a few clarifying notes follow. If there is no "useful information'' (see Zitikis  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
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
and an analogous one for , we obtain that
Combining bounds (3.7) and (3.9), we arrive at bound (3.5), thus completing the proof of Theorem 3.1.
4. An Example
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
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
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
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
which we have calculated using the equation (cf. (2.9)) and the above given expression for .
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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