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**Best Bounds on Measures of Risk and Probability of Ruin for Alpha Unimodal Random Variables When There Is Limited Moment Information** ()

^{1}Department of Information, Risk and Operations Management, University of Texas, Austin, TX, USA.

^{2}Robinson College of Business, Georgia State University, Atlanta, GA, USA.

^{3}Center for Risk Management, University of Texas, Austin, TX, USA.

^{4}National Chengchi University, Taipei City.

^{5}College of Business and Technology, Morehead State University, Morehead, KY, USA.

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*Applied Mathematics*,

**7**, 765-783. doi: 10.4236/am.2016.78069.

Received 15 March 2016; accepted 17 May 2016; published 20 May 2016

1. Introduction

In financial engineering and actuarial applications, one frequently encounters situations involving a pair of random variables X and Y (with distribution functions F and G respectively) wherein it is desirable to determine if one distribution is more “dispersed”, more “variable”, or “more risky” than the other. In statistics, such situations arise, for example, in nonparametric inference when one desires to formally state a one sided alternative to the null hypothesis that F and G have the same dispersion. Other illustrations arise in queuing theory where it can be expected that as the interarrival and service times of a queue become “more variable” the waiting time should increase stochastically [1] . Still further illustrations of the importance of investigating these concepts occur in the areas of financial analysis of return distributions and in actuarial analysis of claims distributions. In these situations it is to be expected that the more “uncertain” or “disperse” random variable is a more risky financial prospect (or more dangerous risk to underwrite) and hence is less preferable, all other things being equal. To investigate these general problems, one needs to define the meaning of and quantify the notion of “more variable” or “riskier”.

Two main approaches have been used to define orderings on the space of probability distributions. The first approach attempts to order F and G according to the dispersion about some point, such as the mean, the median, or center of symmetry of the variables. Such orderings stochastically compare univariate numerical quanti-

ties such as and, or and, or some other convex functions of the quantities and where and are the appropriate central points of F and G respectively.

The variance and absolute deviation measures are particularly common measures for quantifying these concepts and obtaining a total ordering in applications, e.g., in PERT analysis.

In another direction, as a result of efforts to more generally formalize the intuitive notions of “more disperse” random variables, various partial orderings have been introduced on the space of all probability distributions. One such ordering is the dilation (which in financial applications is called the mean preserving spread) ordering. In a utility theoretic framework appropriate for decision making under uncertainty this leads to second order stochastic dominance. In this setting, a random variable Y is called a dilation of X if

(1)

for all convex functions h. In terms of utility functions, (with convex), this is the notion of second order stochastic dominance of X over Y (and Y is said to be more risky than X. [2] ).

Reflection certifies that the relationship (1) indeed yields a method for formalizing the intuitive notion that Y is more dispersed than X since, for random variable X and Y with the same means, (1) holds if and only if the mass of Y can be obtained from that of X by pushing the mass to the outside (dilating) while retaining the same center of gravity. This is the “mean preserving spread” notion used in financial analysis of return distributions [2] , the “Robin Hood transformation” used by economic researchers studying income distribution via Lorenz ordering [3] , and the “stop loss premium ordering” used by actuaries to rank order the riskiness of underwriting different hazards [4] .

In order to be able to rank distributions with differing means, it is useful to consider also the ordering defined by the inequalities

(2)

for all convex functions h for which the expectations in the above relationship (2) exist. Shaked [5] considered conditions that arise in applications which yield the inequalities (1) and (2). Rolski [6] , Whitt [1] and Brown [7] among others, studied the ordering defined by

(3)

for all non-decreasing convex functions k such that the expectations in (3) exists. Roughly speaking, if (3) holds, then Y is “more dispersed” or is stochastically larger than X. The book by Gooaverts et al. [4] characterizes these orderings (and others) and discusses their implied interrationships in an insurance context.

Two of the most common measures of dispersion for a random variable X from a pre-specified value c are and. (e.g., both are used in insurance and finance as risk measures). These two measures again have the form and can be used to define a total ordering on the space of distributions.

Unfortunately, in order to implement the above ordering criteria, it is necessary to know the entire probability distribution for the variables X and Y. Without such exact information, the expectation cannot be calculated in order to verify (1), (2) or (3). In many important practical problems, however, one only possesses partial information concerning the distribution of the variables under investigation. For example, in actuarial analysis, one may know the means (pure premium), the range of possible values for the variables (the policy limits and deductibles), and some information concerning the shape of the distributions (such as unimodality). In such situations (and with still further information such as higher moments), it is desirable to be able to assess the relative riskiness of one variable vis a vis the other. However, because the prescribed known information only incompletely determines the relevant distributions, it becomes necessary to compare the entire classes of distributions possessing the known characteristics. Accordingly, it is desirable to determine optimally tight upper and lower bounds on the expectation of the convex function of the variable under investigation where the supremum and infimum are taken over all random variables satisfying the given information constraints. This, then, produces a partial ordering on the space of probability distributions satisfying the informational constraints.

For a general function h(x) possessing nonnegative derivatives of one higher order than the number of known moments (e.g., or when only the mean is known, or with a known mean and variance), an explicit solution for the problem of obtaining the tightest possible bounds on when X is unimodal with a known mode and a know range and finite set of moments was presented

by Brockett and Cox [8] , and Brockett, Cox, and Witt [9] and used in Brockett and Kahane [10] and Brockett and Garven [11] . Their development was based on the theory of Chebychev systems of functions [12] coupled with Kemperman’s [13] “transformation of moments” technique.

This article begins by extending the arguments of Brockett and Cox [8] to a wider class of random variables (the so called alpha-unimodal or a-unimodal random variables). Then, to examine the more difficult case of

which is not covered by the previously cited theorem, we use an approach based upon the results of Kemperman [14] on the geometry of the moment problem, which does not require differentiability.

2. Bounds on E[h(X)] for Arbitrarily Bounded X

We begin by restating a result from Brockett and Cox [8] . This Lemma gives the tightest possible bounds on expectations of functions of the type referred to above. We couple this with a yet unpublished result from Chang [15] to incorporate the situation when four moments are known.

Lemma 1: 1) Let be given and let h be a twice-differentiable function on with for. Then, for any random variable X with values in the interval and mean, we have the tight bounds where.

2) Let and be given and let h be three times differentiable with for. Then, for any random variable X with values in, mean, and variance, we have the tight bounds

where

and

3) Let, , and be given, and let h be four times differentiable with for. Then, for any random variable X with values in, mean, variance, and third moment, we have the tight bounds,

where

4) Let the 4-moment vector be given, and let h be five times differentiable with for. Then, for any random variable X with values in and the given four moments, we have the tight bounds,

where

and where

Also,

where

Note that the bounds in the above theorem are optimal in the sense that there actually exist random variable and on with precisely the given set of moments for which the equality relation obtains,

namely that the distribution with the masses at the points specified within the argument of h(×) and with probability equal to the coefficient of h(×) on the sides of the two inequalities. Accordingly, the bounds cannot be improved without adding additional knowledge about the random variable X.

Before considering a-unimodal random variables, we note that a more general version of Lemma 1 can be proven in which the level of differentiability of h is decreased by one. In the case of a single moment being given, this means that we need not require h to be differentiable, but only that h be continuous and convex. This result, established for general numbers of moments by Chang [15] , is proven for the special case of convex functions in section 4, and follows from the fact that the function h can be uniformly approximated by a function with one larger derivative, and the fact that the bounding extreme measures do not depend on the actual function.

3. Bounds on E[h(X)] When X Is Known to be Alpha-Unimodal

We now turn to the problem of obtaining bounds on the expectation when more is known about the distribution than just the moments. In particular, we generalize previous results to a general notion of distributional shape known as a-unimodality originally developed by Olshen and Savage [16] as a generalization of the usual notion of unimodality.

A random variable X is said to be a-unimodal with a-mode if it satisfies either (and hence both) of the following equivalent conditions

(i) X has the same distribution as where U and Y are independent random variables with U uniformly distributed on.

(ii) is non-decreasing in for every positive bounded measurable function g.

The case corresponds to the usual notion of unimodality and, in this situation, (i) is simply L. Shepp’s reformulation of Khinchine’s [17] characterization theorem for unimodality (cf., [18] page 158). The equivalence of (1) and (2) is due to Olshen and Savage [16] . From condition (ii) it is clear that if X is a-Unimodal, then

X is also b-unimodal for any. Intuitively, in the case of an a-unimodal variable X with a-mode, this simply says that for all x.

Consider now a random variable X which is a-Unimodal on [a, b] with a-mode and which has given raw moments. By (i) we may write where U and Y are independent random variables and U is uniformly distributed on [0,I]. The moment of is, so the moment of Y is found by solving

for, which yields. The range of possible values for Y is.

In many instances, it is more convenient to work with the central moments than the raw moments. In such situations the first three central moments of Y may be easily calculated in terms of the a-mode and central moments of X as below

In the case (ordinary unimodality), the above formulae reduce to the formulae of Brockett and Cox [8] for 1, 2, 3 moments given and allow the application of Lemma 1 to the random variable Y whenever the moments of X are known.

In order to emulate Kemperman’s “transfer of moment problems” technique for mixture variables, we proceed as follows. For, consider the function g(y) obtained by calculating the expectation of h(X), conditional on. This gives

which, after the substitution, can be reduced to

,

This is valid except perhaps at. For no change of variable is required and. For, this reduces to the formulae given in Brockett and Cox [8] [19] .

Note that, so that h(X) and g(Y) have the same expectation. Accordingly, the problem of determining optimal bounds on E[h(X)] when X is a-unimodal with known moments and known a-mode can be transformed into the equivalent problem of obtaining bounds for.

When the only information about Y is its range and a known set of moments calculated from the moments of X via the above-derived formulae. Applying Lemma 1 to the variable Y and function g then produces optimal bounds for E[g(Y)] and hence E[h(X)]. This is summarized in the following theorem.

Theorem 1. Let X be an a-unimodal random variable on [a, b] with mean, variance third central moment, and a-mode. Let g denote the function

and.

1) If is given and h is twice differentiable on [a, b] with for. Then we have tight bounds

where

2) If and are given and h is three times differentiable with for. Then, we have the tight bounds

where

and

3) If, and are given and h is four times differentiable with for. Then we have the tight bounds

where

and where, and are given in terms of and according to the formulae given in the previous section.

4) Let the 4-moment vector be given, and let h be five times differentiable with for. Then, for any random variable X with values in and the given four moments, we have the tight bounds,

where

and where

Also,

where

Note that the derived function g(y) on also inherits the nonnegative derivative properties of h on [a, b]. Accordingly, Theorem 1 follows from Lemma 1 applied to the function g and the random variable Y due to the fact that. A numerical illustration of this theorem is given in Table

1 for the function, using the given support a = 0, b = 10 and the moment knowledge

and mode = 5. This is done first with only support and moment knowledge, and

Table 1. Bounds on the expectation of an alpha-unimodal random variable with different moment knowledge, alpha = 2.

then with this knowledge plus the knowledge that the random variable in question is a-unimodal with. As can be seen, at each given level of moment knowledge, the additional knowledge of a-unimodality improves the optimal bounds.

Note that in each case the permissible range of values with known unimodal situation is smaller than when unimodality is not known, and that the “indeterminacy” range decreases (sometimes dramatically) as more moments and unimodality are added.

4. Bounds on and with X Being Alpha-Unimodal

As mentioned previously, there are certain functions h which are particularly important as measures of risk in applications. One such function is on the interval [a, b]. For this function, we calculate g(y) as follows:

, and.

According to Theorem 1, the best bounds on this squared distance measure given the partial stochastic information can be explicitly obtained. We summarized the result as follows.

Theorem 2. Let X be an a-unimodal random variable on [a, b] with mean and a-mode. Then the second moment of X about c is optimally bounded as follows:

Proof: From Theorem 1 the optimal bounds are

where and g is the quadratic polynomial

The lower bound is g(E[Y]) which is calculated as follows:

The upper bound is E[g(Y)] which is

Now use the definition of p to find

which completes the proof.

Corollary 1. Let X be an a-unimodal random variable on [0, 1] with mean and a-mode. Then the variance of X is optimally bounded as follows:

Proof: This follows directly from Theorem 2 by setting a = 0, b = 1, and. (We note that the upper bound in Corollary 1 was also obtained by Dharmadhikari and Joag-Dev [20] by a completely different argument.)

We now turn to the analogue of the situation occurring in symmetric unimodal situations wherein the mean and mode coincide.

Corollary 2. Let X be a-unimodal on [a, b] with mean and a-mode. Then the second moment of X about c is optimally bounded as follows:

In particular, if X is a-unimodal on [0, 1] with, then the upper bound becomes and the variance of X is optimally bounded by. The upper bound on the variance equals in the ordinary unimodal case ().

Proof: This follows by assigning the values, and in Theorem 1.

Another function of importance in risk applications is the function. In order to apply Theorem 1 to this function we must calculate

,

except that. In the case at hand, we find that

For values of c in the range, we find that

Since is integrable and convex, g(y) is also convex.

Moreover, for any other distribution G in the expected value of h satisfies

so indeed provides the upper bound.

Proof: Using Kemperman’s results we know that

Similarly, if the contact set is an interval, then h is linear on Z(q) and hence

and g(y) is defined as follows: For c in the range,

Some simple examples follow immediately.

while in the ordinary unimodal case (),

Proof: Applying Theorem 4 with a = 0, b = 1, and, yields for. The lower bound is. The upper bound is

The special cases for and can now be obtained by substitution.

5. Application: Assessing the Probability of Ruin Using Incomplete Loss Distribution Information

where is the time of ruin, and R is the so-called adjustment coefficient which

at zero and once at a positive value. The intersections for the positive values are precisely in order, and from left to right as shown in Figure 1. Hence the corresponding adjustment coefficients must satisfy as pictured in the following chart.

From the above formula for we easily obtain the bounds on the ruin probability, namely

Figure 1. Bounds on the adjustment coefficient using partial information.

We find the following bounds on the ruin probability using partial information:

Table 2 presents the numerical results. The values of can be used to give upper bounds on, the ruin probability, because

6. Improvements in the Bounds When the Random Variables Are Known to Be Alpha-Unimodal

a-mode, then is a-unimodal about 0. According to Theorem 1,.

Table 2. Bounds upon the adjustment coefficient using only moment information.

transform the moment problems involved in the calculation of as follows:

with the line in order to find the bounds on the adjustment coefficient. This is shown graphically in Figure 2.

then calculating the bounds for. The corresponding numerical values for the adjustment coefficient R

are given in Table 3. Note that in each situation, the bounds obtained by using the unimodality assumption are

Figure 2.The best bounding curves for given unimodality, and their corresponding bounds upon the adjustment coefficient.

Table 3. Bounds on the adjustment coefficient based on moments and unimodality of the loss variable^{*}.

^{*}If the loss is known to be alpha-unimodal with alpha = 1.

Conflicts of Interest

The authors declare no conflicts of interest.

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