^{1}

^{2}

We study a new family of random variables that each arise as the distribution of the maximum or minimum of a random number N of i.i.d. random variables X
_{1}, X
_{2},
…, X
_{N}, each distributed as a variable X with support on [0, 1]. The general scheme is first outlined, and several special cases are studied in detail. Wherever appropriate, we find estimates of the parameter θ in the one-parameter family in question.

Consider a sequence

and

have been well studied; in fact it is shown in elementary texts that _{i}’s is random, and we are instead considering the extrema

and

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 [

provided that N is independent of the sequence

The purpose of this paper is to show that the distributions in (1) and (2) can be studied in many canonical cases, even if N and _{1} and v_{2} if one of the webpages has a link to the other) are shown to be modeled by

for some constant

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

Here is our general set-up: Suppose

we see that

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

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

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 [

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

Since

Similarly, (4) gives that

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

as claimed. □

Note. Even though we take the distributions to have support on

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

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

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

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_{i} is less than θ, and we are looking for the minimum of

Specifically, let

or

In either case N has probability mass function given by

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

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

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

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

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

Proposition 3.8. Changing the variable Y of the CSUG model (in Proposition 3.1) as

Proposition 3.9. Changing the variable Z of the CSUG model (in Proposition 3.2) as

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

which is a special case, with k = 1, of the generalized half-logistic distribution [

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

Putting

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

the likelihood function is given by

The MLE of θ is a value of θ, where

Since

Suppose next that

it follows that the likelihood function is given by

As above, it now follows that

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

and

and that

UNIFORM-POISSON MODEL. Here we let

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

GEOMETRIC-BETA(2, 2) MODEL. Here

and

POISSON-BETA(2, 2) MODEL. Here

and

GEOMETRIC-ARCSINE MODEL. Here

and

POISSON-ARCSINE MODEL. Here

and

GEOMETRIC-TOPP-LEONE MODEL. Here

and

POISSON-TOPP-LEONE MODEL.

and

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.

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

Jie Hao,Anant Godbole, (2016) Distribution of the Maximum and Minimum of a Random Number of Bounded Random Variables. Open Journal of Statistics,06,274-285. doi: 10.4236/ojs.2016.62023