From the General Affine Transform Family to a Pareto Type IV Model

Abstract

The analytical form of general affine transform families with given maximum likelihood estimators for the affine parameters is determined. In this context, the simultaneous maximum likelihood equations of the affine parameters in the generalised Pareto distribution cannot have a common solution. This pathological situation is removed by extending it to a four parameter family, called Pareto type IV model.

1. Introduction

Based on [1], the author has studied the general affine transform X of the random variable Y defined by , where and are twice differentiable monotone increasing functions, and are deterministic functions of the affine parameter vector such that . The work in [2] determines exact maximum likelihood estimators of parameters in order statistics distributions with exponential, Pareto, and Weibull parent distributions. The article [3] recovers the older result by the work in [4] that the Pareto is an exponential transform, and also notes that the latter result is not restricted to the Pareto, but applies to a lot of distributions like the truncated Cauchy, Gompertz, log-logistic, para-logistic, inverse Weibull, and log-Laplace.

A further contribution in this area is offered. Based on the method introduced in [5], we determine the analytical form that parametric models may take for specific maximum likelihood estimators of the affine parameters in a general affine transform family. Applied to the generalised Pareto distribution, of great importance in extreme value theory and its applications (e.g., [6, 7]), one observes that the simultaneous maximum likelihood equations of the affine parameters cannot have a common solution. Therefore, the highly desirable maximum likelihood method is not applicable to this distribution. Fortunately, this pathological situation can be removed by enlarging the generalised Pareto to a four-parameter family. The resulting new family, called Pareto type IV model, includes as special cases the generalised Pareto and the Beta of type II. Finally, it is worthwhile to mention the construction of alternative statistical models of Pareto type II and III in [8], and of type IV in [9]. A recent discussion of the Pareto type III is [10] and a useful monograph including Pareto type distributions is [11]. This paper is organized as follows.

Section 2 recalls the general affine transform family (GATF) and its relevance. Our main result concerns the possible form GATF models may take given specific maximum likelihood estimators (MLE) for their affine parameters and is derived in Section 3. Section 4 shows that our method does not apply to the generalised Pareto distribution and introduces the new Pareto type IV model. Section 5 concludes and gives a short outlook on further research.

2. General Affine Transform Families

Let be random variables with distribution functions and densities (provided they exist). Suppose that the distributions and densities depend on a parameter vector with values in the parameter space , where is a vector of affine parameters, is a vector of shape parameters, and . We assume that the functions and are continuous twice-differentiable monotone increasing with inverses and . Moreover, these functions do not depend on but may depend on .

Definition 2.1. The general affine transform X of Y (GATF) is the random variable defined by via a three-stage transformation. First, Y is nonlinearly transformed to , then positively linear transformed to , with , and again nonlinearly transformed to . The constants and are called location and scale parameters. A GATF family is a set of parameterised GATF X of Y whose distributions and densities satisfy the relationships In applications, very often special cases are most useful. Using [1, Table  1], the main types are summarized in [3, Table  2.1]. Some typical examples illustrate the relevance of the GATF as the generalised Pareto and the gxh-family [3, Examples  2.1 and 2.2].

3. GATF Families with Prescribed Maximum Likelihood Estimators

Consider a random sample of size, where are independent and identically distributed random variables, and denote the common random variable by X. For a real function , we define and denote the mean value of by

It is assumed that sample mean value equations like have a unique solution . Our main result characterizes GATF families by the form of the maximum likelihood estimators for their affine parameters. The proof makes use in [12, Theorem  2.2].

Theorem 3.1. Given is a GATF with support and affine parameter vector . Suppose that the distribution function of is twice differentiable, and that the MLE of the th affine parameter is solution of one of the following mean value equations. Case 1 :. with some real function .Case 2 :. with some real function .
Then there exists a twice-differentiable and monotone increasing function with derivative , and constants such that
Furthermore, for simultaneous maximum likelihood estimation of the affine parameters, the following compatibility conditions must be satisfied: Under these conditions, the distribution function has the unique representation for all .

Proof. We proceed as in [5, proof of Theorem  2.1].Case 1 (). Using (2.2) and the relations , one obtains for the negative of the random log-likelihood of X the expression Denoting partial derivatives with respect to with a lower index and making use of one obtains from (3.10) the expression for the partial derivative By assumption (3.2), one has using [12, Theorem  2.2] that for some constant . By comparison solves the second-order differential equation Setting and multiplying with this simplifies to Transform it to the equivalent system of first-order equations in [13, Chapter 19]: The second differential equation is of Bernoulli type [13, Chapter 2]. Setting , this is equivalent to the simpler system in : The second equation is linear inhomogeneous of first order and has the homogeneous solution . By variation of the constant, one sees that . On the other side, from the first equation in (3.17), one has , hence . Together, this shows the following separated differential equation: Assume momentary that has an integral such that for some . Then, has the solution . It follows that the general solution of the second differential equation in (3.17) is given by The first differential equation in (3.17) implies the separated differential equation Assume momentary that there exists a twice-differentiable function such that (). The general solution to (3.20) yields the relationship Setting and using that , one gets the random relation , which implies by (2.1) that Setting one obtains the density function The side conditions , , imply that the constants are determined by The validity of the representation (3.9) for is shown. Since has been assumed twice differentiable, so is , and as claimed in (3.4). In particular, the two momentary assumptions made above, that is, and , are fulfilled.Case 2 (). Since , one has similarly to (3.11) the relationship From (3.10), one obtains for the partial derivative of the random log-likelihood the relation By assumption (3.2) and again in [12, Theorem  2.2], one has for some constant . Through comparison, it follows that must solve Proceeding as in Case 1, one obtains a twice-differentiable function , with derivative , such that and . As in Case 1, one concludes that (3.9) for must hold.
It remains to show the compatibility conditions (3.6)–(3.8). Through differentiation of (3.9), one obtains the probability density functions
for all . Three subcases are possible.Subcase 1 (). From (3.30), one gets that with . Using (3.4), one obtains without difficulty the compatibility condition (3.6).Subcase 2 (). From (3.30), one sees that with . Using (3.4) and (3.5), one shows without difficulty condition (3.7).Subcase 3 (). From (3.30), one obtains that with . Using (3.5), one shows without difficulty condition (3.8). The proof of Theorem 3.1 is complete.

4. A Pareto Type IV Model

The generalised Pareto distribution is the GATF defined by with , Y exponential with mean one, , , , . Its probability density function is

Applying Theorem 3.1, one sees that the MLE of are determined by the real functions

According to Theorem 3.1, there are functions

and constants such that

and the compatibility condition (3.7) is fulfilled. For any random sample from this family, one observes that the simultaneous maximum likelihood equations

cannot have a common solution, hence the maximum likelihood method is not applicable.

The described pathological situation can be removed in a simple way thanks to Theorem 3.1. Our construction is motivated by the following question. What is the most general affine transform family with MLE of the affine parameter that is determined by the mean value equation ?. By Theorem 3.1, Case 1, there must exist a constant and a function such that

Using [5], formula (3.1) one obtains

A corresponding probability density function is

One notes that two well-known subfamilies are included, namely, the generalised Pareto (4.1) obtained by setting , and the Beta of type II obtained by setting . This suggests the name “generalised Pareto-Beta” but we prefer the simpler nomenclature “Pareto type IV model” for the new four-parameter family (4.8). Applying Theorem 3.1, one sees that the MLE of and are determined by

There are functions

and constants such that

and the compatibility condition (3.7), that is,

is fulfilled. For a random sample the MLE of and solves the simultaneous equations

The value of the normalising constant in (4.8) depends only on the shape vector .

Proposition 4.1. Assume that are not integers. Then the normalising constant of the Pareto type IV model (4.8) is determined by the infinite series expansion where , is a generalised binomial coefficient.

Proof. From the observation made above, one notes that To obtain convergent integrals, separate calculation in two parts and make a substitution to get The binomial expansion , valid for [14, (18.7), page 134], yields the series Under the assumption this implies without difficulty the expression (4.14).

5. Conclusions and Outlook

The proposed method is not the only way to generalize the Pareto family (4.1). The recent note [9] extends this family to the family

which looks similar to (4.8), except for the “power law” component in the second bracket, but has different statistical properties. An advantage of (5.1) is certainly the analytical closed-form expression for the survival function given by

To conclude, several advantages of (4.8) can be noted, in particular, the simple MLE estimation of the affine parameters and the inclusion of the very important generalised Pareto distribution as a submodel. From a statistical viewpoint, the interest of the extended model (4.8) is two-fold. First, it may provide a better fit of the data than any submodel. Second, it yields a simple statistical procedure to choose among submodels like the generalised Pareto and the Beta of type II. Only the model “closest” to the full model will be retained. A detailed comparison of these two four parameter Pareto families is left to further research.

Acknowledgment

The author is grateful to the referees for careful reading of the manuscript and valuable comments.

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Copyright

Copyright © 2009 Werner Hürlimann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.