Distribution of the Maximum and Minimum of a Random Number of Bounded Random Variables

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Received 28 January 2016; accepted 23 April 2016; published 26 April 2016

1. Introduction

Consider a sequence of i.i.d. random variables with support on and having distribution function F. For any fixed n, the distributions of

and

have been well studied; in fact it is shown in elementary texts that and. But what if we have a situation where the number N of X_{i}’s is random, and we are instead considering the extrema

(1)

and

(2)

of a random number of i.i.d. random variables? Now the sum S of a random number of i.i.d. variables, defined as

satisfies, according to Wald’s Lemma [1] , the equation

provided that N is independent of the sequence and assuming that the means of X and N exist.

for some constant, where is the Riemann Zeta function

Thus if the vertices v in a large internet graph have some bounded i.i.d. property X_{i}, then the maximum and minimum values of X_{i} for the neighbors of a randomly chosen vertex can be modeled using the methods of this paper. Third, we note that N and the X_{i} may be correlated, as in the CSUG example (studied systematically in Section 3) where and follows the geometric distribution. This is an example of a situation where we might be modeling the maximum load that a device might have carried before it breaks down due to an excessive weight or current. It is also feasible in this case that the parameter θ might be unknown.

Here is our general set-up: Suppose are i.i.d. random variables following a continuous distribution on with probability density and distribution functions given by and respectively. N is a random variable following a discrete distribution on with probability mass function given by,. Let Y and Z be given by (1) and (2) respectively. Then the p.d.f.’s g of Y and Z are derived as follows: Since

we see that

and consequently, the marginal p.d.f. of Y is

(3)

In a similar fashion, the p.d.f. of Z can be shown to be

(4)

what is remarkable is that the sums in (3) and (4) will be shown to assume simple tractable forms in a variety of cases.

We want to point out that some of our distributions have been studied before but not using this motivation. For example, the Marshall-Olkin distributions [11] give a new method of adding a parameter to a distribution. Also, other distributions such as the beta and Kumaraswamy [12] distributions can be used to model continuous bounded data, but these do not apply to our set-up. See also Remark 2 in Section 3.

Our paper is organized as follows. Section 1 provided a summary and motivation for studying the distributions in the fashion we do. In Section 2, we study the case of and. We call this the Standard Uniform Geometric model. The graphs of g(y) and g(z) can be seen in Figure 1 and Figure 2 respectively. The CSUG (Correlated Standard Uniform Model) is studied in Section 3. The graphs of g(y) and g(z) in the CSUG model are plotted in Figure 3 and Figure 4 respectively. Parameter estimation is done in Section 4. Section 5 is devoted to a summary of a variety of other models.

2. Standard Uniform Geometric (SUG) Model

Since, , and for some, we have from (3) that the p.d.f. of Y in the SUG model is given by

(5)

Similarly, (4) gives that

(6)

Proposition 2.1. If the random variable Y has the “SUG maximum distribution” (5) and, then

Proof.

Figure 1. Plot of the SUG maximum density for some values of θ (see Equation (5)).

Figure 2. Plot of the SUG minimum density for some values of θ (see Equation (6)).

Figure 3. Plot of the CSUG maximum density for some values of θ.

Figure 4. Plot of CSUG minimum density for some values of θ.

as claimed. □

Note. Even though we take the distributions to have support on, this may be done by changing the survival function in [3] , where the same compounding method is used. Specifically we can use the transformation in the proofs of [3] .

Proposition 2.2. The random variable Y has mean and variance given, respectively, by

Proof. Using Proposition 2.1, we can directly compute the mean and variance by setting, and using the fact that for any random variable W. (This proof could equally well have been based on calculating the moments of and then recovering the values of and. The same is true of other proofs in the paper.) □

Proposition 2.3. If the random variable Z has the “SUG minimum distribution” and, then

Proof.

as asserted. □

Proposition 2.4. The random variable Z has mean and variance given, respectively, by

Proof. Using Proposition 2.3, it is easily to compute the mean and variance by setting k = 1, k = 2. □

The m.g.f.’s of Y, Z are easy to calculate too. Notice that the logarithmic terms above arise due to the contributions of the j = 1 and terms, and it is precisely these logarithmic terms that make, e.g., method of moments estimates for θ to be intractable in a closed (i.e., non-numerical) form. Similar difficulties arise when analyzing the likelihood function and likelihood ratios.

3. The Correlated Standard Uniform Geometric (CSUG) Model

The Correlated Standard Uniform Geometric (CSUG) model is related to the SUG model, as the name suggests, but X and N are correlated as indicated in Section 1. The CSUG problems arise in two cases. One case is that we conduct standard uniform trials until a variable X_{i} exceeds, where θ is the parameter of the correlated geometric variable, and the maximum of is what we seek. The maximum is between 0 and. The other case is where standard uniform trials are conducted until X_{i} is less than θ, and we are looking for the minimum of. The minimum is between θ and 1.

Specifically, let be a sequence of standard uniform variables and define

or

In either case N has probability mass function given by

(7)

note that this is simply a geometric random variable conditional on the success having occurred at trial 2 or later. Clearly N is dependent on the X sequence.

Proposition 3.1. Under the CSUG model, the p.d.f. of Y, defined by (1), is given by

Proof. The conditional c.d.f. of Y given that is given by

Taking the derivative, we see that the conditional density function is given by

Consequently, the p.d.f. of Y in the CSUG model is given by

This completes the proof. □

Proposition 3.2. The p.d.f. of Z under the CSUG model is given by

Proof. The conditional cumulative distribution function of Z given that is given by

Thus, the conditional density function is given by

which yields the p.d.f. of Z under the CSUG model as

which finishes the proof. □

Proposition 3.3. If the random variable Y has the “CSUG maximum distribution” and, then

Proof.

as claimed. □

Proposition 3.4. The random variable Y has mean and variance given, respectively, by

and

Proof. Using Proposition 3.3, we can directly compute the mean and variance by setting k = 1, 2. For example with k = 1 we get

Notice that the variance of Y is smaller than that of Y under the SUG model, with an identical numerator term. Also, the expected value is smaller under the CSUG model than in the SUG case. This can be best seen by the inequalities

and

valid for. □

Proposition 3.5. If the random variable Z has the “CSUG Minimum distribution” and, then

Proof. Routine, as before. □

Proposition 3.6. The random variable Z has mean and variance given, respectively, by

and

Proof. A special case of Proposition 3.3; note that as in the SUG model,. □

Remark 1. The four distributions of Y and Z under the SUG and the CSUG models can be shown to be affine transformations of the same distribution as seem by the following results (proofs omitted):

Proposition 3.7. Changing the variable Y of (5) as yields (6). Thus the SUG maximum and SUG minimum variables are related by the fact that

Proposition 3.8. Changing the variable Y of the CSUG model (in Proposition 3.1) as yields, which equals the pdf of (5). Hence

Proposition 3.9. Changing the variable Z of the CSUG model (in Proposition 3.2) as yields, which equals the pdf of (6). Thus

As a result of these affine transformations, the moment equations (Propositions from 2.1 to 2.4 and from 3.3 to 3.6) can be derived in an easier fashion, though these facts are easier to observe post facto.

Remark 2. As stated earlier the distributions of this paper are related to other distributions in the literature, but these do not exploit the extreme value connection as we do. For example, when, (5) reduces to

which is a special case, with k = 1, of the generalized half-logistic distribution [5] , eq. 23.83.

Second, the distribution of Z under the CSUG model is a special case of a truncated Pareto distribution, which, for positive a, is defined by

4. Parameter Estimation

The intermingling of polynomial and logarithmic terms makes method of moments estimation difficult in closed form, as in the SUG case. However, if θ is unknown, the maximum likelihood estimate of θ can be found in a satisfying form, both in the CGUG maximum and CSUG minimum cases. Suppose that form a random sample from the CSUG Maximum distribution with unknown θ. Since the pdf of each observation has the following form:

the likelihood function is given by

The MLE of θ is a value of θ, where for, which maximizes. Let.

Since, it follows that is a increasing function, which means the MLE is the largest possible value of θ such that for. Thus, this value should be, i.e.,.

Suppose next that form a random sample from the CSUG minimum distribution. Since the pdf of each observation has the following form:

it follows that the likelihood function is given by

As above, it now follows that. It is not too hard to write down the distribution of the MLE’s but we do not do so here.

5. A Summary of Some Other Models

The general scheme given by (3) and (4) is quite powerful. As another example, suppose (using the example from Section 1) that

and. Then it is easy to show that

and that. (The expected value of Y can also be calculated by using the identity. In this section, we collect some more results of this type, without proof:

UNIFORM-POISSON MODEL. Here we let and, so that N follows a left-truncated Poisson distribution.

Proposition 5.1. Under the Uniform-Poisson model,

In some sense, the primary motivation of this paper was to produce extreme value distributions that did not fall into the Beta family (such as for the maximum of n i.i.d. variables). A wide variety of non-Beta-based distributions may be found in [6] . Can we add extreme value distributions to that collection? In what follows, we use both the Beta families and, the arcsine distribution, and a “Beyond Beta” distribution, the Topp-Leone distribution [15] , as “input variables” to make further progress in this direction.

GEOMETRIC-BETA(2, 2) MODEL. Here and. In this case we get

and

POISSON-BETA(2, 2) MODEL. Here and, the Poisson (q) distribution left- truncated at 0. In this case we get

and

GEOMETRIC-ARCSINE MODEL. Here and. In this case we get

and

POISSON-ARCSINE MODEL. Here and. Here we have

and

GEOMETRIC-TOPP-LEONE MODEL. Here and:

and

POISSON-TOPP-LEONE MODEL. and:

and

6. Conclusion

In this paper we studied a general scheme for the distribution of the maximum or minimum of a random number of i.i.d. random variables with compact support. While some of the distributions obtained through this process have appeared before in the literature, they do not been studied using this approach. Our biggest open problem is to find data sets for which these new distributions are appropriate.

Acknowledgements

The research of AG was supported by NSF Grants 1004624 and 1040928. We thank the referees for their insightful suggestions for improvement.

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