Journal of Probability and Statistics

Journal of Probability and Statistics / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 364901 | https://doi.org/10.1155/2009/364901

Werner Hรผrlimann, "From the General Affine Transform Family to a Pareto Type IV Model", Journal of Probability and Statistics, vol. 2009, Article ID 364901, 10 pages, 2009. https://doi.org/10.1155/2009/364901

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

Academic Editor: Josรฉ Marรญa Sarabia
Received30 Mar 2009
Revised30 Sep 2009
Accepted15 Oct 2009
Published15 Nov 2009

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 ๐‘‹=๐‘ˆ[๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ)], where ๐œ“(๐‘ฅ) and ๐‘ˆ(๐‘ฅ) are twice differentiable monotone increasing functions, and ๐ด(๐›ผ),๐ต(๐›ผ) are deterministic functions of the affine parameter vector ๐›ผ such that ๐ต(๐›ผ)>0. 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 ๐‘‹,๐‘Œ be random variables with distribution functions ๐น๐‘‹,๐น๐‘Œ and densities ๐‘“๐‘‹,๐‘“๐‘Œ (provided they exist). Suppose that the distributions and densities depend on a parameter vector ๐œƒ=(๐›ผ,๐›พ) with values in the parameter space ฮ˜โŠ‚๐‘…๐‘š, where ๐›ผ=(๐›ผ1,โ€ฆ,๐›ผ๐‘Ÿ) is a vector of affine parameters, ๐›พ=(๐›พ1,โ€ฆ,๐›พ๐‘ ) is a vector of shape parameters, and ๐‘š=๐‘Ÿ+๐‘ . We assume that the functions ๐œ“(๐‘ฅ) and ๐‘ˆ(๐‘ฅ) are continuous twice-differentiable monotone increasing with inverses ๐œ‘(๐‘ฅ)=๐œ“โˆ’1(๐‘ฅ) and ๐‘‡(๐‘ฅ)=๐‘ˆโˆ’1(๐‘ฅ). Moreover, these functions do not depend on ๐›ผ but may depend on ๐›พ.

Definition 2.1. The general affine transform X of Y (GATF) is the random variable defined by ๐‘‹=๐‘ˆ[๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ)] via a three-stage transformation. First, Y is nonlinearly transformed to ๐œ“(๐‘Œ), then positively linear transformed to ๐‘‡(๐‘Œ)=๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ), with ๐ต(๐›ผ)>0, and again nonlinearly transformed to ๐‘‹=๐‘ˆ[๐‘‡(๐‘Œ)]. The constants ๐ด(๐›ผ) and ๐ต(๐›ผ) are called location and scale parameters. A GATF family ๐น{๐‘Œ}={๐‘‹=๐‘ˆ[๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ)]โˆผ๐น๐‘‹(๐‘ฅ;๐œƒ),๐œƒ=(๐›ผ,๐›พ)โˆˆฮ˜} is a set of parameterised GATF X of Y whose distributions and densities satisfy the relationships ๐น๐‘‹(๐‘ฅ)=๐น๐‘Œ๎‚ป๐œ‘๎‚ธ๐‘‡(๐‘ฅ)โˆ’๐ด(๐›ผ)๐‘“๐ต(๐›ผ)๎‚น๎‚ผ,(2.1)๐‘‹1(๐‘ฅ)=๐ต(๐›ผ)โ‹…๐‘‡๎…ž(๐‘ฅ)โ‹…๐œ‘๎…ž๎‚ธ๐‘‡(๐‘ฅ)โˆ’๐ด(๐›ผ)๎‚น๐ต(๐›ผ)โ‹…๐‘“๐‘Œ๎‚ป๐œ‘๎‚ธ๐‘‡(๐‘ฅ)โˆ’๐ด(๐›ผ)๐ต(๐›ผ)๎‚น๎‚ผ.(2.2) In applications, very often special cases are most useful. Using [1, Table โ€‰1], the main types are summarized in [3, Table โ€‰2.1]. Some typical examples illustrate the relevance of the GATF as the generalised Pareto and the gxh-family [3, Examples โ€‰2.1 and 2.2].

3. GATF Families with Prescribed Maximum Likelihood Estimators

Consider a random sample ๐œ‰=(๐‘‹1,โ€ฆ,๐‘‹๐‘›) of size๐‘›, where ๐‘‹๐‘– are independent and identically distributed random variables, and denote the common random variable by X. For a real function ๐ป(๐‘ฅ), we define and denote the mean value of ๐ป(๐œ‰) by

1๐ป(๐œ‰)=๐‘›๐‘›๎“๐‘–=1๐ป๎€ท๐‘‹๐‘–๎€ธ.(3.1) It is assumed that sample mean value equations like ๐ป(๐œ‰/๎๐›ผ)=1 have a unique solution ๎๐›ผ=๎๐›ผ(๐œ‰,๐ป). 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 โ€‰2.2].

Theorem 3.1. Given is a GATF ๐‘‹=๐‘ˆ[๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ)] with support ๐ผ๐‘‹=[๐‘Ž๐‘‹,๐‘๐‘‹] and affine parameter vector ๐›ผ=(๐›ผ1,โ€ฆ,๐›ผ๐‘Ÿ)โˆˆฮ˜โŠ‚๐‘…๐‘Ÿ. Suppose that the distribution function ๐น๐‘‹(๐‘ฅ) of ๐‘‹ is twice differentiable, and that the MLE of the ๐‘˜th affine parameter ๐›ผ๐‘˜ is solution of one of the following mean value equations. Case 1 :. ๐ต๐‘˜=๐œ•๐ต(๐›ผ)๐œ•๐›ผ๐‘˜โ‰ 0,๐ด๐‘˜=๐œ•๐ด(๐›ผ)๐œ•๐›ผ๐‘˜๎€ฝarbitrary,๐‘˜โˆˆ1,โ€ฆ,๐‘Ÿ1๎€พ,๐‘†๐‘˜๎‚ต๐‘‡(๐œ‰)โˆ’๐ด(๎๐›ผ)+๐ด๐ต(๎๐›ผ)๐‘˜๐ต๐‘˜๎‚ถ=1,(3.2) with some real function ๐‘†๐‘˜(๐‘ฅ).Case 2 :. ๐ต๐‘˜=๐œ•๐ต(๐›ผ)๐œ•๐›ผ๐‘˜โ‰ก0,๐ด๐‘˜=๐œ•๐ด(๐›ผ)๐œ•๐›ผ๐‘˜๎€ฝ๐‘Ÿโ‰ 0,๐‘˜โˆˆ1๎€พ,+1,โ€ฆ,๐‘Ÿ๐ฟ๐‘˜๎‚ต๐‘‡(๐œ‰)โˆ’๐ด(๎๐›ผ)๎‚ถ๐ต(๎๐›ผ)=0,(3.3) with some real function ๐ฟ๐‘˜(๐‘ฅ).
Then there exists a twice-differentiable and monotone increasing function ๐‘„๐‘˜(๐‘ฅ) with derivative ๐‘ž๐‘˜(๐‘ฅ)=๐‘„๎…ž๐‘˜(๐‘ฅ), and constants ๐‘๐‘˜,๐‘‘๐‘˜โ‰ 0 such that
๐‘๐‘˜๐‘†๐‘˜(๐‘ฅ)+1โˆ’๐‘๐‘˜๐‘‘=โˆ’๐‘ฅโ‹…๎€ฝ๐‘ž๐‘‘๐‘ฅln๐‘˜๎€พ๐‘‘(๐‘ฅ),inCase1,(3.4)๐‘˜๐ฟ๐‘˜๐‘‘(๐‘ฅ)=โˆ’๎€ฝ๐‘ž๐‘‘๐‘ฅln๐‘˜๎€พ(๐‘ฅ),inCase2.(3.5) Furthermore, for simultaneous maximum likelihood estimation of the affine parameters, the following compatibility conditions must be satisfied: ๎‚ต๐ด๐‘ฅ+๐‘—๐ต๐‘—๎‚ถโ‹…๎‚ต๐‘๐‘–๐‘†๐‘–๎‚ต๐ด๐‘ฅ+๐‘–๐ต๐‘–๎‚ถ+1โˆ’๐‘๐‘–๎‚ถ=๎‚ต๐ด๐‘ฅ+๐‘–๐ต๐‘–๎‚ถโ‹…๎‚ต๐‘๐‘—๐‘†๐‘—๎‚ต๐ด๐‘ฅ+๐‘—๐ต๐‘—๎‚ถ+1โˆ’๐‘๐‘—๎‚ถ,๎€ฝ๐‘–,๐‘—โˆˆ1,โ€ฆ,๐‘Ÿ1๎€พ,๐‘(3.6)๐‘–๐‘†๐‘–๎‚ต๐ด๐‘ฅ+๐‘–๐ต๐‘–๎‚ถ+1โˆ’๐‘๐‘–=๎‚ต๐ด๐‘ฅ+๐‘–๐ต๐‘–๎‚ถโ‹…๐‘‘๐‘—๐ฟ๐‘—๎€ฝ(๐‘ฅ),๐‘–โˆˆ1,โ€ฆ,๐‘Ÿ1๎€พ๎€ฝ๐‘Ÿ,๐‘—โˆˆ1๎€พ๐‘‘+1,โ€ฆ,๐‘Ÿ,(3.7)๐‘–๐ฟ๐‘–(๐‘ฅ)=๐‘‘๐‘—๐ฟ๐‘—๎€ฝ๐‘Ÿ(๐‘ฅ),๐‘–,๐‘—โˆˆ1๎€พ+1,โ€ฆ,๐‘Ÿ.(3.8) Under these conditions, the distribution function has the unique representation ๐น๐‘‹๐‘„(๐‘ฅ)=๐‘–๎€ท๎€ท๐ด(๐‘‡(๐‘ฅ)โˆ’๐ด)/๐ต+๐‘–/๐ต๐‘–๎€ธ๎€ธโˆ’๐‘„๐‘–๐‘‡๎€ท๐‘Ž๎€ท๎€ท๐‘‹๎€ธ๎€ธ๎€ท๐ดโˆ’๐ด/๐ต+๐‘–/๐ต๐‘–๎€ธ๎€ธ๐‘„๐‘–๐‘‡๎€ท๐‘๎€ท๎€ท๐‘‹๎€ธ๎€ธ๎€ท๐ดโˆ’๐ด/๐ต+๐‘–/๐ต๐‘–๎€ธ๎€ธโˆ’๐‘„๐‘–๐‘‡๎€ท๐‘Ž๎€ท๎€ท๐‘‹๎€ธ๎€ธ๎€ท๐ดโˆ’๐ด/๐ต+๐‘–/๐ต๐‘–=๐‘„๎€ธ๎€ธ๐‘—((๐‘‡(๐‘ฅ)โˆ’๐ด)/๐ต)โˆ’๐‘„๐‘—๎€ท๐‘‡๎€ท๐‘Ž๐‘‹๎€ธ๎€ธโˆ’๐ด/๐ต๐‘„๐‘—๎€ท๐‘‡๎€ท๐‘๐‘‹๎€ธ๎€ธโˆ’๐ด/๐ตโˆ’๐‘„๐‘—๎€ท๐‘‡๎€ท๐‘Ž๐‘‹๎€ธ๎€ธ,โˆ’๐ด/๐ต(3.9) for all ๐‘ฅโˆˆ๐ผ๐‘‹=[๐‘Ž๐‘‹,๐‘๐‘‹],๐‘–โˆˆ{1,โ€ฆ,๐‘Ÿ1},๐‘—โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ}.

Proof. We proceed as in [5, proof of Theorem โ€‰2.1].Case 1 (๐‘˜โˆˆ{1,โ€ฆ,๐‘Ÿ1}). Using (2.2) and the relations ๐‘Œ=๐œ‘((๐‘‡(๐‘‹)โˆ’๐ด)/๐ต),๐œ‘๎…ž[๐œ“(๐‘Œ)]=๐œ“๎…ž(๐‘Œ)โˆ’1, one obtains for the negative of the random log-likelihood of X the expression โˆ’โ„“(๐‘‹)=ln๐ต(๐›ผ)โˆ’ln๐‘‡๎…ž(๐‘‹)+ln๐œ“๎…ž(๐‘Œ)โˆ’ln๐‘“๐‘Œ(๐‘Œ).(3.10) Denoting partial derivatives with respect to ๐›ผ๐‘˜ with a lower index ๐‘˜ and making use of ๐‘Œ๐‘˜=๐œ‘๎…ž๎‚ต๐‘‡(๐‘‹)โˆ’๐ด๐ต๎‚ถโ‹…โˆ’๐ด๐‘˜๐ตโˆ’(๐‘‡(๐‘‹)โˆ’๐ด)๐ต๐‘˜๐ต2๐ด=โˆ’๐‘˜+๐ต๐‘˜๐œ“(๐‘Œ)๐ต๐œ“๎…ž(๐‘Œ),(3.11) one obtains from (3.10) the expression for the partial derivative โˆ’๐ต๐ต๐‘˜โ‹…โ„“๐‘˜๎ƒฉ๎€ท๐ด(๐‘‹)=1โˆ’๐œ“(๐‘Œ)+๐‘˜/๐ต๐‘˜๎€ธ๎ƒชโ‹…๎‚ต๐œ“๐œ“โ€ฒ(๐‘Œ)๎…ž๎…ž(๐‘Œ)๐œ“๎…žโˆ’๐‘‘(๐‘Œ)๎€ฝ๐‘“๐‘‘๐‘Œln๐‘Œ๎€พ๎‚ถ(๐‘Œ).(3.12) By assumption (3.2), one has using [12, Theorem โ€‰2.2] that โˆ’๐ต๐ต๐‘˜โ‹…โ„“๐‘˜(๐‘‹)=๐‘๐‘˜โ‹…๎‚ป1โˆ’๐‘†๐‘˜๎‚ต๐ด๐œ“(๐‘Œ)+๐‘˜๐ต๐‘˜๎‚ถ๎‚ผ(3.13) for some constant ๐‘๐‘˜โ‰ 0. By comparison ๐‘ฆ(๐‘ฅ)=๐œ“(๐‘ฅ)+(๐ด๐‘˜/๐ต๐‘˜) solves the second-order differential equation ๐‘ฆ๎…ž๎…ž๐‘ฆ๎…žโˆ’๎€ฝ๐‘๐‘˜๐‘†๐‘˜(๐‘ฆ)+1โˆ’๐‘๐‘˜๎€พโ‹…๐‘ฆ๎…ž๐‘ฆ=๐‘‘๎€ฝ๐‘“๐‘‘๐‘ฅln๐‘Œ๎€พ(๐‘ฅ).(3.14) Setting ๐‘”๐‘˜(๐‘ฅ)=(๐‘๐‘˜๐‘†๐‘˜(๐‘ฅ)+(1โˆ’๐‘๐‘˜))/๐‘ฅ and multiplying with ๐‘ฆ๎…ž, this simplifies to ๐‘ฆ๎…ž๎…žโˆ’๐‘‘๎€ฝ๐‘“๐‘‘๐‘ฅln๐‘Œ๎€พ(๐‘ฅ)โ‹…๐‘ฆ๎…žโˆ’๐‘”๐‘˜(๐‘ฆ)โ‹…๐‘ฆ๎…ž2=0.(3.15) Transform it to the equivalent system of first-order equations in (๐‘ฆ1=๐‘ฆ,๐‘ฆ2) [13, Chapter 19]: ๐‘ฆ๎…ž1=๐‘ฆ2,๐‘ฆ๎…ž2=๐‘‘๎€ฝ๐‘“๐‘‘๐‘ฅln๐‘Œ๎€พ(๐‘ฅ)โ‹…๐‘ฆ2+๐‘”๐‘˜๎€ท๐‘ฆ1๎€ธโ‹…๐‘ฆ22.(3.16) The second differential equation is of Bernoulli type [13, Chapter 2]. Setting ๐‘ฆ2=๐‘ง2โˆ’1, this is equivalent to the simpler system in (๐‘ฆ1,๐‘ง2): ๐‘ฆ๎…ž1=๐‘ง2โˆ’1,๐‘ง๎…ž2๐‘‘=โˆ’๎€ฝ๐‘“๐‘‘๐‘ฅln๐‘Œ๎€พ(๐‘ฅ)โ‹…๐‘ง2+๐‘”๐‘˜๎€ท๐‘ฆ1๎€ธ.(3.17) The second equation is linear inhomogeneous of first order and has the homogeneous solution ๐‘ง2=๐ถ๐‘˜โ‹…๐‘“๐‘Œ(๐‘ฅ)โˆ’1. By variation of the constant, one sees that ๐ถ๎…ž๐‘˜(๐‘ฅ)=โˆ’๐‘”๐‘˜(๐‘ฆ1)โ‹…๐‘“๐‘Œ(๐‘ฅ). On the other side, from the first equation in (3.17), one has ๐‘ฆ๎…ž=๐‘ฆ๎…ž1=๐‘ง2โˆ’1=๐ถ๐‘˜(๐‘ฅ)โˆ’1โ‹…๐‘“๐‘Œ(๐‘ฅ), hence ๐‘“๐‘Œ(๐‘ฅ)=๐‘ฆ๎…ž1โ‹…๐ถ๐‘˜(๐‘ฅ). Together, this shows the following separated differential equation: ๐‘‘๎€ฝ๐ถ๐‘‘๐‘ฅln๐‘˜๎€พ(๐‘ฅ)=โˆ’๐‘”๐‘˜(๐‘ฆ)โ‹…๐‘ฆ๎…ž.(3.18) Assume momentary that ๐‘”๐‘˜(๐‘ฅ) has an integral such that ๐บ๎…ž๐‘˜(๐‘ฅ)=๐‘”๐‘˜(๐‘ฅ) for some ๐บ๐‘˜(๐‘ฅ). Then, (๐‘‘/๐‘‘๐‘ฅ)ln{๐ถ๐‘˜(๐‘ฅ)}=โˆ’(๐‘‘/๐‘‘๐‘ฅ)๐บ๐‘˜(๐‘ฆ) has the solution ๐ถ๐‘˜(๐‘ฅ)=๐ถ๐‘˜โˆ’1โ‹…exp{โˆ’๐บ(๐‘ฆ)},๐ถ๐‘˜>0. It follows that the general solution of the second differential equation in (3.17) is given by ๐‘ง2=๎€ฝexpโˆ’๐บ๐‘˜๎€พ(๐‘ฆ)๐ถ๐‘˜๐‘“๐‘Œ(๐‘ฅ).(3.19) The first differential equation in (3.17) implies the separated differential equation ๐‘ฆ๎…ž๎€ฝโ‹…expโˆ’๐บ๐‘˜๎€พ(๐‘ฆ)=๐ถ๐‘˜โ‹…๐‘“๐‘Œ(๐‘ฅ).(3.20) Assume momentary that there exists a twice-differentiable function ๐‘„๐‘˜(๐‘ฅ) such that ๐บ๐‘˜(๐‘ฅ)=โˆ’ln{๐‘„๎…ž๐‘˜(๐‘ฅ)}(๐‘”๐‘˜(๐‘ฅ)=๐บ๎…ž๐‘˜(๐‘ฅ)=โˆ’(๐‘„๐‘˜๎…ž๎…ž(๐‘ฅ)/๐‘„โ€ฒ๐‘˜(๐‘ฅ))). The general solution to (3.20) yields the relationship ๐น๐‘Œ1(๐‘ฅ)=๐ถ๐‘˜๎€ฝ๐‘„๐‘˜(๐‘ฆ)+๐ท๐‘˜๎€พ,๐ถ๐‘˜>0,๐ท๐‘˜โˆˆ๐‘….(3.21) Setting ๐‘ฅ=๐‘Œ and using that ๐‘ฆ(๐‘ฅ)=๐œ“(๐‘Œ)+(๐ด๐‘˜/๐ต๐‘˜)=(๐‘‡(๐‘‹)โˆ’๐ด)/๐ต+(๐ด๐‘˜/๐ต๐‘˜), one gets the random relation ๐น๐‘Œ(๐‘Œ)=(1/๐ถ๐‘˜){๐‘„๐‘˜((๐‘‡(๐‘‹)โˆ’๐ด)/๐ต+(๐ด๐‘˜/๐ต๐‘˜))+๐ท๐‘˜}, which implies by (2.1) that ๐น๐‘‹1(๐‘ฅ)=๐ถ๐‘˜๎‚ป๐‘„๐‘˜๎‚ต๐‘‡(๐‘ฅ)โˆ’๐ด๐ต+๐ด๐‘˜๐ต๐‘˜๎‚ถ+๐ท๐‘˜๎‚ผ,๐‘ฅโˆˆ๐ผ๐‘‹.(3.22) Setting ๐‘ž๐‘˜(๐‘ฅ)=๐‘„๎…ž๐‘˜(๐‘ฅ), one obtains the density function ๐‘“๐‘‹๐‘‡(๐‘ฅ)=๎…ž(๐‘ฅ)๐ต๐ถ๐‘˜๐‘ž๐‘˜๎‚ต๐‘‡(๐‘ฅ)โˆ’๐ด๐ต+๐ด๐‘˜๐ต๐‘˜๎‚ถ,๐‘ฅโˆˆ๐ผ๐‘‹.(3.23) The side conditions โˆซ๐‘๐‘‹๐‘Ž๐‘‹๐‘“๐‘‹(๐‘ฅ)๐‘‘๐‘ฅ=1, ๐น๐‘‹(๐‘๐‘‹)=1, imply that the constants are determined by ๐ถ๐‘˜=๐‘„๐‘˜๎ƒฉ๐‘‡๎€ท๐‘๐‘‹๎€ธโˆ’๐ด๐ต+๐ด๐‘˜๐ต๐‘˜๎ƒชโˆ’๐‘„๐‘˜๎ƒฉ๐‘‡๎€ท๐‘Ž๐‘‹๎€ธโˆ’๐ด๐ต+๐ด๐‘˜๐ต๐‘˜๎ƒช,๐ท๐‘˜=โˆ’๐‘„๐‘˜๎ƒฉ๐‘‡๎€ท๐‘Ž๐‘‹๎€ธโˆ’๐ด๐ต+๐ด๐‘˜๐ต๐‘˜๎ƒช.(3.24) The validity of the representation (3.9) for ๐‘–โˆˆ{1,โ€ฆ,๐‘Ÿ1} is shown. Since ๐น๐‘Œ(๐‘ฅ) has been assumed twice differentiable, so is ๐‘„๐‘˜(๐‘ฅ), and ๐‘๐‘˜๐‘†๐‘˜(๐‘ฅ)+1โˆ’๐‘๐‘˜=๐‘ฅ๐‘”๐‘˜(๐‘ฅ)=๐‘ฅ๐บ๎…ž๐‘˜๐‘‘(๐‘ฅ)=โˆ’๐‘ฅโ‹…๎€ฝ๐‘ž๐‘‘๐‘ฅln๐‘˜๎€พ(๐‘ฅ),(3.25) as claimed in (3.4). In particular, the two momentary assumptions made above, that is, ๐‘”๐‘˜(๐‘ฅ)=๐บ๎…ž๐‘˜(๐‘ฅ) and ๐บ๐‘˜(๐‘ฅ)=โˆ’ln{๐‘„๎…ž๐‘˜(๐‘ฅ)}, are fulfilled.Case 2 (๐‘˜โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ}). Since ๐ต๐‘˜โ‰ก0, one has similarly to (3.11) the relationship ๐‘Œ๐‘˜๐ด=โˆ’๐‘˜๐ต๐œ“๎…ž(๐‘Œ).(3.26) From (3.10), one obtains for the partial derivative of the random log-likelihood the relation โˆ’๐ต๐ด๐‘˜โ‹…โ„“๐‘˜1(๐‘‹)=โ‹…๎‚ต๐œ“โ€ฒ(๐‘Œ)๐œ“โ€ฒโ€ฒ(๐‘Œ)โˆ’๐‘‘๐œ“โ€ฒ(๐‘Œ)๎€ฝ๐‘“๐‘‘๐‘Œln๐‘Œ๎€พ๎‚ถ(๐‘Œ).(3.27) By assumption (3.2) and again in [12, Theorem โ€‰2.2], one has โˆ’๐ต๐ด๐‘˜โ‹…โ„“๐‘˜(๐‘‹)=๐‘‘๐‘˜โ‹…๐ฟ๐‘˜(๐œ“(๐‘Œ))(3.28) for some constant ๐‘‘๐‘˜โ‰ 0. Through comparison, it follows that ๐‘ฆ(๐‘ฅ)=๐œ“(๐‘ฅ) must solve ๐‘ฆ๎…ž๎…žโˆ’๐‘‘๎€ฝ๐‘“๐‘‘๐‘ฅln๐‘Œ๎€พ(๐‘ฅ)โ‹…๐‘ฆ๎…žโˆ’๐‘‘๐‘˜โ‹…๐ฟ๐‘˜(๐‘ฆ)โ‹…๐‘ฆ๎…ž2=0.(3.29) Proceeding as in Case 1, one obtains a twice-differentiable function ๐‘„๐‘˜(๐‘ฅ), with derivative ๐‘ž๐‘˜(๐‘ฅ)=๐‘„๎…ž๐‘˜(๐‘ฅ), such that ๐‘‘๐‘˜๐ฟ๐‘˜(๐‘ฅ)=โˆ’(๐‘‘/๐‘‘๐‘ฅ)ln{๐‘ž๐‘˜(๐‘ฅ)} and ๐น๐‘Œ(๐‘ฅ)=(1/๐ถ๐‘˜){๐‘„๐‘˜(๐‘ฆ)+๐ท๐‘˜},๐ถ๐‘˜>0,๐ท๐‘˜โˆˆ๐‘…. As in Case 1, one concludes that (3.9) for ๐‘—โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ} must hold.
It remains to show the compatibility conditions (3.6)โ€“(3.8). Through differentiation of (3.9), one obtains the probability density functions
๐‘“๐‘‹๐‘‡(๐‘ฅ)=๎…ž(๐‘ฅ)๐ต๐ถ๐‘–๐‘ž๐‘–๎‚ต๐‘‡(๐‘ฅ)โˆ’๐ด๐ต+๐ด๐‘–๐ต๐‘–๎‚ถ=๐‘‡๎…ž(๐‘ฅ)๐ต๐ถ๐‘—๐‘ž๐‘—๎‚ต๐‘‡(๐‘ฅ)โˆ’๐ด๐ต๎‚ถ,(3.30) for all ๐‘ฅโˆˆ๐ผ๐‘‹,๐‘–โˆˆ{1,โ€ฆ,๐‘Ÿ1},๐‘—โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ}. Three subcases are possible.Subcase 1 (๐‘–,๐‘—โˆˆ{1,โ€ฆ,๐‘Ÿ1}). From (3.30), one gets that ๐‘ž๐‘—(๐‘ฅ+(๐ด๐‘–/๐ต๐‘–))=๐ถโ‹…๐‘ž๐‘–(๐‘ฅ+(๐ด๐‘—/๐ต๐‘—)) with ๐ถ=๐ถ๐‘—/๐ถ๐‘–. Using (3.4), one obtains without difficulty the compatibility condition (3.6).Subcase 2 (๐‘–โˆˆ{1,โ€ฆ,๐‘Ÿ1},๐‘—โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ}). From (3.30), one sees that ๐‘ž๐‘—(๐‘ฅ)=๐ถโ‹…๐‘ž๐‘–(๐‘ฅ+(๐ด๐‘—/๐ต๐‘—)) with ๐ถ=(๐ถ๐‘—/๐ถ๐‘–). Using (3.4) and (3.5), one shows without difficulty condition (3.7).Subcase 3 (๐‘–,๐‘—โˆˆ{๐‘Ÿ1+1,โ€ฆ,๐‘Ÿ}). From (3.30), one obtains that ๐‘ž๐‘—(๐‘ฅ)=๐ถโ‹…๐‘ž๐‘–(๐‘ฅ) with ๐ถ=๐ถ๐‘—/๐ถ๐‘–. 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 ๐‘‹=๐ด(๐›ผ)+๐ต(๐›ผ)โ‹…๐œ“(๐‘Œ) with ๐œ“(๐‘ฅ)=exp(๐›พ1๐‘ฅ),๐›พ1>0, Y exponential with mean one, ๐ด(๐›ผ)=๐›ผ2โˆ’๐›ผ1, ๐ต(๐›ผ)=๐›ผ1, ๐›ผ=(๐›ผ1,๐›ผ2)โˆˆ๐‘…2+, ๐œƒ=(๐›ผ1,๐›ผ2,๐›พ1)โˆˆฮ˜=๐‘…3+. Its probability density function is

๐‘“๐‘‹1(๐‘ฅ)=๐›ผ1๐›พ1๎‚ต1+๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถโˆ’(1+(1/๐›พ1)),๐‘ฅโ‰ฅ๐›ผ2.(4.1) Applying Theorem 3.1, one sees that the MLE of ๐›ผ1,๐›ผ2 are determined by the real functions

๐‘†1(๐‘ฅ)=1+๐›พ11+๐‘ฅ,๐ฟ2(๐‘ฅ)=โˆ’1+๐›พ1๐›พ1๐‘ฅ.(4.2) According to Theorem 3.1, there are functions

๐‘ž1(๐‘ฅ)=(1+๐‘ฅ)โˆ’(1+(๐›พ1/๐›พ1)),๐‘ž2(๐‘ฅ)=๐‘ฅ๎€ท๐›พโˆ’1+1/๐›พ1๎€ธ,(4.3) and constants ๐‘1=โˆ’๐›พ1โˆ’1,๐‘‘2=โˆ’1 such that

๐‘1๐‘†1(๐‘ฅ)+1โˆ’๐‘1๐‘‘=โˆ’๐‘ฅโ‹…๎€ฝ๐‘ž๐‘‘๐‘ฅln1๎€พ(๐‘ฅ),๐‘‘2๐ฟ2๐‘‘(๐‘ฅ)=โˆ’๎€ฝ๐‘ž๐‘‘๐‘ฅln2๎€พ(๐‘ฅ),(4.4) and the compatibility condition (3.7) is fulfilled. For any random sample ๐œ‰=(๐‘‹1,โ€ฆ,๐‘‹๐‘›) from this family, one observes that the simultaneous maximum likelihood equations

1+๐›พ11+๎€ท๎€ท๐œ‰โˆ’๐›ผ2๎€ธ/๐›ผ1๎€ธ=1,11+๎€ท๎€ท๐œ‰โˆ’๐›ผ2๎€ธ/๐›ผ1๎€ธ=0,(4.5) 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 ๐›ผ1 that is determined by the mean value equation ๐‘†1((๐œ‰โˆ’๐›ผ2)/๐›ผ1)=1?. By Theorem 3.1, Case 1, there must exist a constant ๐›พ2 and a function ๐‘ž1(๐‘ฅ) such that

๐›พ2๐‘†1(๐‘ฅ)+1โˆ’๐›พ2๐‘‘=โˆ’๐‘ฅโ‹…๎€ฝ๐‘ž๐‘‘๐‘ฅln1๎€พ(๐‘ฅ).(4.6) Using [5], formula (3.1) one obtains

๐‘ž1(๐‘ฅ)=๐‘ฅ๐›พ2โˆ’1๎‚ปโ‹…expโˆ’๐›พ2๎€œ๐‘†1(๐‘ฅ)๐‘ฅ๎‚ผ๐‘‘๐‘ฅ=๐‘ฅโˆ’๎€ท1+๐›พ1๐›พ2๎€ธโ‹…(1+๐‘ฅ)๎€ท1+๐›พ1๎€ธ๐›พ2.(4.7) A corresponding probability density function is

๐‘“๐‘‹1(๐‘ฅ)=๐ถ๐›ผ1โ‹…๎‚ต๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถโˆ’๎€ท1+๐›พ1๐›พ2๎€ธโ‹…๎‚ต1+๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถ๎€ท1+๐›พ1๎€ธ๐›พ2,๐‘ฅโ‰ฅ๐›ผ2.(4.8) One notes that two well-known subfamilies are included, namely, the generalised Pareto (4.1) obtained by setting ๐›พ1๐›พ2=โˆ’1, and the Beta of type II obtained by setting ๐‘=โˆ’๐›พ1๐›พ2>0,๐‘ž=โˆ’๐›พ2>0. This suggests the name โ€œgeneralised Pareto-Betaโ€ but we prefer the simpler nomenclature โ€œPareto type IV modelโ€ for the new four-parameter family (4.8). Applying Theorem 3.1, one sees that the MLE of ๐›ผ1 and ๐›ผ2 are determined by

๐‘†1(๐‘ฅ)=1+๐›พ11+๐‘ฅ,๐ฟ2๎€ท(๐‘ฅ)=1+๐›พ1๎€ธ๐›พ2๐‘ฅโˆ’1+๐›พ1๐›พ2๐‘ฅโˆ’1.(4.9) There are functions

๐‘ž1(๐‘ฅ)=๐‘ฅโˆ’๎€ท1+๐›พ1๐›พ2๎€ธโ‹…(1+๐‘ฅ)๎€ท1+๐›พ1๎€ธ๐›พ2,๐‘ž2(๐‘ฅ)=(๐‘ฅโˆ’1)โˆ’๎€ท1+๐›พ1๐›พ2๎€ธโ‹…๐‘ฅ๎€ท1+๐›พ1๎€ธ๐›พ2,(4.10) and constants ๐‘1=๐›พ2,๐‘‘2=โˆ’1 such that

๐‘1๐‘†1(๐‘ฅ)+1โˆ’๐‘1๐‘‘=โˆ’๐‘ฅโ‹…๎€ฝ๐‘ž๐‘‘๐‘ฅln1๎€พ(๐‘ฅ),๐‘‘2๐ฟ2๐‘‘(๐‘ฅ)=โˆ’๎€ฝ๐‘ž๐‘‘๐‘ฅln2๎€พ(๐‘ฅ),(4.11) and the compatibility condition (3.7), that is,

๐›พ2๐‘†1(๐‘ฅโˆ’1)+1โˆ’๐›พ2=โˆ’(๐‘ฅโˆ’1)๐ฟ2(๐‘ฅ),(4.12) is fulfilled. For a random sample ๐œ‰=(๐‘‹1,โ€ฆ,๐‘‹๐‘›), the MLE of ๐›ผ1 and ๐›ผ2 solves the simultaneous equations

1+๐›พ11+๎€ท๎€ท๐œ‰โˆ’๐›ผ2๎€ธ/๐›ผ1๎€ธ=1,1+๐›พ1๐›พ2๎€ท๐œ‰โˆ’๐›ผ2๎€ธ/๐›ผ1=๐›พ2.(4.13) The value of the normalising constant in (4.8) depends only on the shape vector ๐›พ=(๐›พ1,๐›พ2).

Proposition 4.1. Assume that ๐›พ2,๐›พ1๐›พ2 are not integers. Then the normalising constant of the Pareto type IV model (4.8) is determined by the infinite series expansion ๎€ท๐›พ๐ถ=๐ถ1,๐›พ2๎€ธ=โˆž๎“๐‘˜=0๎‚ต๎€ท1+๐›พ1๎€ธ๐›พ2๐‘˜๎‚ถ๎€ท2๐‘˜โˆ’1+๐›พ1๎€ธ๐›พ2๎€ท๐‘˜โˆ’๐›พ2๎€ธ๎€ท๐‘˜โˆ’๐›พ1๐›พ2๎€ธ,(4.14) where (๐›ผ๐‘˜)=(๐›ผ(๐›ผโˆ’1)โ€ฆ(๐›ผโˆ’๐‘˜+1))/๐‘˜!,๐‘˜โ‰ฅ1,(๐›ผ0)=1, is a generalised binomial coefficient.

Proof. From the observation made above, one notes that ๎€œ๐ถ=โˆž0๐‘ž1๎€œ(๐‘ฅ)๐‘‘๐‘ฅ=โˆž0๐‘ฅโˆ’๎€ท1+๐›พ1๐›พ2๎€ธ(1+๐‘ฅ)๎€ท1+๐›พ1๎€ธ๐›พ2๎€œ๐‘‘๐‘ฅ=โˆž0๐‘ฅ๐›พ2โˆ’1๎€ท1+๐‘ฅโˆ’1๎€ธ๎€ท1+๐›พ1๎€ธ๐›พ2๐‘‘๐‘ฅ.(4.15) To obtain convergent integrals, separate calculation in two parts and make a substitution to get ๎€œ๐ถ=10๐‘ฅโˆ’๎€ท1+๐›พ1๐›พ2๎€ธ(1+๐‘ฅ)๎€ท1+๐›พ1๎€ธ๐›พ2๎€œ๐‘‘๐‘ฅ+10๐‘ฅโˆ’๎€ท1+๐›พ2๎€ธ(1+๐‘ฅ)๎€ท1+๐›พ1๎€ธ๐›พ2๐‘‘๐‘ฅ.(4.16) The binomial expansion (1+๐‘ฅ)๐›ผ=โˆ‘โˆž๐‘˜=0(๐›ผ๐‘˜)๐‘ฅ๐‘˜, valid for ๐‘ฅโˆˆ(0,1) [14, (18.7), page 134], yields the series ๐ถ=โˆž๎“๐‘˜=0๎‚ต๎€ท1+๐›พ1๎€ธ๐›พ2๐‘˜๎‚ถโ‹…๎‚ป๎€œ10๐‘ฅ๐‘˜โˆ’1โˆ’๐›พ1๐›พ2๎€œ๐‘‘๐‘ฅ+10๐‘ฅ๐‘˜โˆ’1โˆ’๐›พ2๎‚ผ๐‘‘๐‘ฅ.(4.17) Under the assumption ๐›พ2,๐›พ1๐›พ2โ‰ ๐‘˜, 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

๐‘“๐‘‹(๐‘๐‘ฅ)=๐›ผ1๐›พ1โ‹…๎‚ต๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถ๐‘โˆ’1โ‹…๎‚ต๎‚ต1+๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถ๐‘๎‚ถโˆ’๎€ท๎€ท1+1/๐›พ1๎€ธ๎€ธ,๐‘ฅโ‰ฅ๐›ผ2,(5.1) which looks similar to (4.8), except for the โ€œpower lawโ€ 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

๐‘†๐‘‹(๎‚ต๎‚ต๐‘ฅ)=1+๐‘ฅโˆ’๐›ผ2๐›ผ1๎‚ถ๐‘๎‚ถโˆ’๎€ท1+1/๐›พ1๎€ธ,๐‘ฅโ‰ฅ๐›ผ2.(5.2) 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 โ€œclosestโ€ 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.

References

  1. B. Efron, โ€œTransformation theory: how normal is a family of distributions?โ€ The Annals of Statistics, vol. 10, no. 2, pp. 323โ€“339, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. W. Hürlimann, โ€œGeneral location transform of the order statistics from the exponential, Pareto and Weibull, with application to maximum likelihood estimation,โ€ Communications in Statistics: Theory and Methods, vol. 29, no. 11, pp. 2535โ€“2545, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. W. Hürlimann, โ€œGeneral affine transform families: why is the Pareto an exponential transform?โ€ Statistical Papers, vol. 44, no. 4, pp. 499โ€“519, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. E. J. Gumbel, Statistics of Extremes, Columbia University Press, New York, NY, USA, 1958. View at: MathSciNet
  5. W. Hürlimann, โ€œOn the characterization of maximum likelihood estimators for location-scale families,โ€ Communications in Statistics: Theory and Methods, vol. 27, no. 2, pp. 495โ€“508, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. P. Embrechts, C. Klüppelberg, and Th. Mikosch, Modelling Extremal Events for Insurance and Finance, vol. 33 of Applications of Mathematics, Springer, Berlin, Germany, 1997. View at: MathSciNet
  7. S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications, Imperial College Press, London, UK, 2000. View at: MathSciNet
  8. W. Hürlimann, โ€œHigher-degree stop-loss transforms and stochastic orders (II) applications,โ€ Blätter der Deutschen Gesellschaft für Versicherungsmathematik, vol. 24, no. 3, pp. 465โ€“476, 2000. View at: Google Scholar
  9. A. M. Abd Elfattah, E. A. Elsherpieny, and E. A. Hussein, โ€œA new generalized Pareto distribution,โ€ 2007, http://interstat.statjournals.net/YEAR/2007/abstracts/0712001.php. View at: Google Scholar
  10. G. Bottazzi, โ€œOn the Pareto type III distribution,โ€ Sant'Anna School of Advanced Studies, Pisa, Italy, 2007, http://www.lem.sssup.it/WPLem/files/2007-07.pdf. View at: Google Scholar
  11. C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2003. View at: MathSciNet
  12. A. K. Gupta and T. Varga, โ€œAn empirical estimation procedure,โ€ Metron, vol. 52, no. 1-2, pp. 67โ€“70, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. W. Walter, Gewöhnliche Differentialgleichungen, Eine Einführung. Heidelberger Taschenbücher, Band 110, Springer, Berlin, Germany, 1972. View at: MathSciNet
  14. Ch. Blatter, Analysis II, Heidelberger Taschenbücher, Band 152, Springer, Berlin, Germany, 1974. View at: MathSciNet

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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views960
Downloads748
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.