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 𝑋=𝑈[𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌)], where 𝜓(𝑥) and 𝑈(𝑥) are twice differentiable monotone increasing functions, and 𝐴(𝛼),𝐵(𝛼) are deterministic functions of the affine parameter vector 𝛼 such that 𝐵(𝛼)>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 𝑋,𝑌 be random variables with distribution functions 𝐹𝑋,𝐹𝑌 and densities 𝑓𝑋,𝑓𝑌 (provided they exist). Suppose that the distributions and densities depend on a parameter vector 𝜃=(𝛼,𝛾) with values in the parameter space Θ𝑅𝑚, where 𝛼=(𝛼1,,𝛼𝑟) is a vector of affine parameters, 𝛾=(𝛾1,,𝛾𝑠) is a vector of shape parameters, and 𝑚=𝑟+𝑠. We assume that the functions 𝜓(𝑥) and 𝑈(𝑥) are continuous twice-differentiable monotone increasing with inverses 𝜑(𝑥)=𝜓1(𝑥) and 𝑇(𝑥)=𝑈1(𝑥). Moreover, these functions do not depend on 𝛼 but may depend on 𝛾.

Definition 2.1. The general affine transform X of Y (GATF) is the random variable defined by 𝑋=𝑈[𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌)] via a three-stage transformation. First, Y is nonlinearly transformed to 𝜓(𝑌), then positively linear transformed to 𝑇(𝑌)=𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌), with 𝐵(𝛼)>0, and again nonlinearly transformed to 𝑋=𝑈[𝑇(𝑌)]. The constants 𝐴(𝛼) and 𝐵(𝛼) are called location and scale parameters. A GATF family 𝐹{𝑌}={𝑋=𝑈[𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌)]𝐹𝑋(𝑥;𝜃),𝜃=(𝛼,𝛾)Θ} is a set of parameterised GATF X of Y whose distributions and densities satisfy the relationships 𝐹𝑋(𝑥)=𝐹𝑌𝜑𝑇(𝑥)𝐴(𝛼)𝑓𝐵(𝛼),(2.1)𝑋1(𝑥)=𝐵(𝛼)𝑇(𝑥)𝜑𝑇(𝑥)𝐴(𝛼)𝐵(𝛼)𝑓𝑌𝜑𝑇(𝑥)𝐴(𝛼)𝐵(𝛼).(2.2) 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 𝜉=(𝑋1,,𝑋𝑛) of size𝑛, where 𝑋𝑖 are independent and identically distributed random variables, and denote the common random variable by X. For a real function 𝐻(𝑥), we define and denote the mean value of 𝐻(𝜉) by

1𝐻(𝜉)=𝑛𝑛𝑖=1𝐻𝑋𝑖.(3.1) It is assumed that sample mean value equations like 𝐻(𝜉/𝛼)=1 have a unique solution 𝛼=𝛼(𝜉,𝐻). 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 𝑋=𝑈[𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌)] with support 𝐼𝑋=[𝑎𝑋,𝑏𝑋] and affine parameter vector 𝛼=(𝛼1,,𝛼𝑟)Θ𝑅𝑟. Suppose that the distribution function 𝐹𝑋(𝑥) of 𝑋 is twice differentiable, and that the MLE of the 𝑘th affine parameter 𝛼𝑘 is solution of one of the following mean value equations. Case 1 :. 𝐵𝑘=𝜕𝐵(𝛼)𝜕𝛼𝑘0,𝐴𝑘=𝜕𝐴(𝛼)𝜕𝛼𝑘arbitrary,𝑘1,,𝑟1,𝑆𝑘𝑇(𝜉)𝐴(𝛼)+𝐴𝐵(𝛼)𝑘𝐵𝑘=1,(3.2) with some real function 𝑆𝑘(𝑥).Case 2 :. 𝐵𝑘=𝜕𝐵(𝛼)𝜕𝛼𝑘0,𝐴𝑘=𝜕𝐴(𝛼)𝜕𝛼𝑘𝑟0,𝑘1,+1,,𝑟𝐿𝑘𝑇(𝜉)𝐴(𝛼)𝐵(𝛼)=0,(3.3) with some real function 𝐿𝑘(𝑥).
Then there exists a twice-differentiable and monotone increasing function 𝑄𝑘(𝑥) with derivative 𝑞𝑘(𝑥)=𝑄𝑘(𝑥), and constants 𝑐𝑘,𝑑𝑘0 such that
𝑐𝑘𝑆𝑘(𝑥)+1𝑐𝑘𝑑=𝑥𝑞𝑑𝑥ln𝑘𝑑(𝑥),inCase1,(3.4)𝑘𝐿𝑘𝑑(𝑥)=𝑞𝑑𝑥ln𝑘(𝑥),inCase2.(3.5) Furthermore, for simultaneous maximum likelihood estimation of the affine parameters, the following compatibility conditions must be satisfied: 𝐴𝑥+𝑗𝐵𝑗𝑐𝑖𝑆𝑖𝐴𝑥+𝑖𝐵𝑖+1𝑐𝑖=𝐴𝑥+𝑖𝐵𝑖𝑐𝑗𝑆𝑗𝐴𝑥+𝑗𝐵𝑗+1𝑐𝑗,𝑖,𝑗1,,𝑟1,𝑐(3.6)𝑖𝑆𝑖𝐴𝑥+𝑖𝐵𝑖+1𝑐𝑖=𝐴𝑥+𝑖𝐵𝑖𝑑𝑗𝐿𝑗(𝑥),𝑖1,,𝑟1𝑟,𝑗1𝑑+1,,𝑟,(3.7)𝑖𝐿𝑖(𝑥)=𝑑𝑗𝐿𝑗𝑟(𝑥),𝑖,𝑗1+1,,𝑟.(3.8) Under these conditions, the distribution function has the unique representation 𝐹𝑋𝑄(𝑥)=𝑖𝐴(𝑇(𝑥)𝐴)/𝐵+𝑖/𝐵𝑖𝑄𝑖𝑇𝑎𝑋𝐴𝐴/𝐵+𝑖/𝐵𝑖𝑄𝑖𝑇𝑏𝑋𝐴𝐴/𝐵+𝑖/𝐵𝑖𝑄𝑖𝑇𝑎𝑋𝐴𝐴/𝐵+𝑖/𝐵𝑖=𝑄𝑗((𝑇(𝑥)𝐴)/𝐵)𝑄𝑗𝑇𝑎𝑋𝐴/𝐵𝑄𝑗𝑇𝑏𝑋𝐴/𝐵𝑄𝑗𝑇𝑎𝑋,𝐴/𝐵(3.9) for all 𝑥𝐼𝑋=[𝑎𝑋,𝑏𝑋],𝑖{1,,𝑟1},𝑗{𝑟1+1,,𝑟}.

Proof. We proceed as in [5, proof of Theorem  2.1].Case 1 (𝑘{1,,𝑟1}). Using (2.2) and the relations 𝑌=𝜑((𝑇(𝑋)𝐴)/𝐵),𝜑[𝜓(𝑌)]=𝜓(𝑌)1, one obtains for the negative of the random log-likelihood of X the expression (𝑋)=ln𝐵(𝛼)ln𝑇(𝑋)+ln𝜓(𝑌)ln𝑓𝑌(𝑌).(3.10) Denoting partial derivatives with respect to 𝛼𝑘 with a lower index 𝑘 and making use of 𝑌𝑘=𝜑𝑇(𝑋)𝐴𝐵𝐴𝑘𝐵(𝑇(𝑋)𝐴)𝐵𝑘𝐵2𝐴=𝑘+𝐵𝑘𝜓(𝑌)𝐵𝜓(𝑌),(3.11) one obtains from (3.10) the expression for the partial derivative 𝐵𝐵𝑘𝑘𝐴(𝑋)=1𝜓(𝑌)+𝑘/𝐵𝑘𝜓𝜓(𝑌)(𝑌)𝜓𝑑(𝑌)𝑓𝑑𝑌ln𝑌(𝑌).(3.12) By assumption (3.2), one has using [12, Theorem  2.2] that 𝐵𝐵𝑘𝑘(𝑋)=𝑐𝑘1𝑆𝑘𝐴𝜓(𝑌)+𝑘𝐵𝑘(3.13) for some constant 𝑐𝑘0. By comparison 𝑦(𝑥)=𝜓(𝑥)+(𝐴𝑘/𝐵𝑘) solves the second-order differential equation 𝑦𝑦𝑐𝑘𝑆𝑘(𝑦)+1𝑐𝑘𝑦𝑦=𝑑𝑓𝑑𝑥ln𝑌(𝑥).(3.14) Setting 𝑔𝑘(𝑥)=(𝑐𝑘𝑆𝑘(𝑥)+(1𝑐𝑘))/𝑥 and multiplying with 𝑦, this simplifies to 𝑦𝑑𝑓𝑑𝑥ln𝑌(𝑥)𝑦𝑔𝑘(𝑦)𝑦2=0.(3.15) Transform it to the equivalent system of first-order equations in (𝑦1=𝑦,𝑦2) [13, Chapter 19]: 𝑦1=𝑦2,𝑦2=𝑑𝑓𝑑𝑥ln𝑌(𝑥)𝑦2+𝑔𝑘𝑦1𝑦22.(3.16) The second differential equation is of Bernoulli type [13, Chapter 2]. Setting 𝑦2=𝑧21, this is equivalent to the simpler system in (𝑦1,𝑧2): 𝑦1=𝑧21,𝑧2𝑑=𝑓𝑑𝑥ln𝑌(𝑥)𝑧2+𝑔𝑘𝑦1.(3.17) The second equation is linear inhomogeneous of first order and has the homogeneous solution 𝑧2=𝐶𝑘𝑓𝑌(𝑥)1. By variation of the constant, one sees that 𝐶𝑘(𝑥)=𝑔𝑘(𝑦1)𝑓𝑌(𝑥). On the other side, from the first equation in (3.17), one has 𝑦=𝑦1=𝑧21=𝐶𝑘(𝑥)1𝑓𝑌(𝑥), hence 𝑓𝑌(𝑥)=𝑦1𝐶𝑘(𝑥). Together, this shows the following separated differential equation: 𝑑𝐶𝑑𝑥ln𝑘(𝑥)=𝑔𝑘(𝑦)𝑦.(3.18) Assume momentary that 𝑔𝑘(𝑥) has an integral such that 𝐺𝑘(𝑥)=𝑔𝑘(𝑥) for some 𝐺𝑘(𝑥). Then, (𝑑/𝑑𝑥)ln{𝐶𝑘(𝑥)}=(𝑑/𝑑𝑥)𝐺𝑘(𝑦) has the solution 𝐶𝑘(𝑥)=𝐶𝑘1exp{𝐺(𝑦)},𝐶𝑘>0. It follows that the general solution of the second differential equation in (3.17) is given by 𝑧2=exp𝐺𝑘(𝑦)𝐶𝑘𝑓𝑌(𝑥).(3.19) The first differential equation in (3.17) implies the separated differential equation 𝑦exp𝐺𝑘(𝑦)=𝐶𝑘𝑓𝑌(𝑥).(3.20) Assume momentary that there exists a twice-differentiable function 𝑄𝑘(𝑥) such that 𝐺𝑘(𝑥)=ln{𝑄𝑘(𝑥)}(𝑔𝑘(𝑥)=𝐺𝑘(𝑥)=(𝑄𝑘(𝑥)/𝑄𝑘(𝑥))). The general solution to (3.20) yields the relationship 𝐹𝑌1(𝑥)=𝐶𝑘𝑄𝑘(𝑦)+𝐷𝑘,𝐶𝑘>0,𝐷𝑘𝑅.(3.21) Setting 𝑥=𝑌 and using that 𝑦(𝑥)=𝜓(𝑌)+(𝐴𝑘/𝐵𝑘)=(𝑇(𝑋)𝐴)/𝐵+(𝐴𝑘/𝐵𝑘), one gets the random relation 𝐹𝑌(𝑌)=(1/𝐶𝑘){𝑄𝑘((𝑇(𝑋)𝐴)/𝐵+(𝐴𝑘/𝐵𝑘))+𝐷𝑘}, which implies by (2.1) that 𝐹𝑋1(𝑥)=𝐶𝑘𝑄𝑘𝑇(𝑥)𝐴𝐵+𝐴𝑘𝐵𝑘+𝐷𝑘,𝑥𝐼𝑋.(3.22) Setting 𝑞𝑘(𝑥)=𝑄𝑘(𝑥), one obtains the density function 𝑓𝑋𝑇(𝑥)=(𝑥)𝐵𝐶𝑘𝑞𝑘𝑇(𝑥)𝐴𝐵+𝐴𝑘𝐵𝑘,𝑥𝐼𝑋.(3.23) The side conditions 𝑏𝑋𝑎𝑋𝑓𝑋(𝑥)𝑑𝑥=1, 𝐹𝑋(𝑏𝑋)=1, imply that the constants are determined by 𝐶𝑘=𝑄𝑘𝑇𝑏𝑋𝐴𝐵+𝐴𝑘𝐵𝑘𝑄𝑘𝑇𝑎𝑋𝐴𝐵+𝐴𝑘𝐵𝑘,𝐷𝑘=𝑄𝑘𝑇𝑎𝑋𝐴𝐵+𝐴𝑘𝐵𝑘.(3.24) The validity of the representation (3.9) for 𝑖{1,,𝑟1} is shown. Since 𝐹𝑌(𝑥) has been assumed twice differentiable, so is 𝑄𝑘(𝑥), and 𝑐𝑘𝑆𝑘(𝑥)+1𝑐𝑘=𝑥𝑔𝑘(𝑥)=𝑥𝐺𝑘𝑑(𝑥)=𝑥𝑞𝑑𝑥ln𝑘(𝑥),(3.25) as claimed in (3.4). In particular, the two momentary assumptions made above, that is, 𝑔𝑘(𝑥)=𝐺𝑘(𝑥) and 𝐺𝑘(𝑥)=ln{𝑄𝑘(𝑥)}, are fulfilled.Case 2 (𝑘{𝑟1+1,,𝑟}). Since 𝐵𝑘0, one has similarly to (3.11) the relationship 𝑌𝑘𝐴=𝑘𝐵𝜓(𝑌).(3.26) From (3.10), one obtains for the partial derivative of the random log-likelihood the relation 𝐵𝐴𝑘𝑘1(𝑋)=𝜓(𝑌)𝜓(𝑌)𝑑𝜓(𝑌)𝑓𝑑𝑌ln𝑌(𝑌).(3.27) By assumption (3.2) and again in [12, Theorem  2.2], one has 𝐵𝐴𝑘𝑘(𝑋)=𝑑𝑘𝐿𝑘(𝜓(𝑌))(3.28) for some constant 𝑑𝑘0. Through comparison, it follows that 𝑦(𝑥)=𝜓(𝑥) must solve 𝑦𝑑𝑓𝑑𝑥ln𝑌(𝑥)𝑦𝑑𝑘𝐿𝑘(𝑦)𝑦2=0.(3.29) Proceeding as in Case 1, one obtains a twice-differentiable function 𝑄𝑘(𝑥), with derivative 𝑞𝑘(𝑥)=𝑄𝑘(𝑥), such that 𝑑𝑘𝐿𝑘(𝑥)=(𝑑/𝑑𝑥)ln{𝑞𝑘(𝑥)} and 𝐹𝑌(𝑥)=(1/𝐶𝑘){𝑄𝑘(𝑦)+𝐷𝑘},𝐶𝑘>0,𝐷𝑘𝑅. As in Case 1, one concludes that (3.9) for 𝑗{𝑟1+1,,𝑟} must hold.
It remains to show the compatibility conditions (3.6)–(3.8). Through differentiation of (3.9), one obtains the probability density functions
𝑓𝑋𝑇(𝑥)=(𝑥)𝐵𝐶𝑖𝑞𝑖𝑇(𝑥)𝐴𝐵+𝐴𝑖𝐵𝑖=𝑇(𝑥)𝐵𝐶𝑗𝑞𝑗𝑇(𝑥)𝐴𝐵,(3.30) for all 𝑥𝐼𝑋,𝑖{1,,𝑟1},𝑗{𝑟1+1,,𝑟}. Three subcases are possible.Subcase 1 (𝑖,𝑗{1,,𝑟1}). From (3.30), one gets that 𝑞𝑗(𝑥+(𝐴𝑖/𝐵𝑖))=𝐶𝑞𝑖(𝑥+(𝐴𝑗/𝐵𝑗)) with 𝐶=𝐶𝑗/𝐶𝑖. Using (3.4), one obtains without difficulty the compatibility condition (3.6).Subcase 2 (𝑖{1,,𝑟1},𝑗{𝑟1+1,,𝑟}). From (3.30), one sees that 𝑞𝑗(𝑥)=𝐶𝑞𝑖(𝑥+(𝐴𝑗/𝐵𝑗)) with 𝐶=(𝐶𝑗/𝐶𝑖). Using (3.4) and (3.5), one shows without difficulty condition (3.7).Subcase 3 (𝑖,𝑗{𝑟1+1,,𝑟}). From (3.30), one obtains that 𝑞𝑗(𝑥)=𝐶𝑞𝑖(𝑥) with 𝐶=𝐶𝑗/𝐶𝑖. 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 𝑋=𝐴(𝛼)+𝐵(𝛼)𝜓(𝑌) with 𝜓(𝑥)=exp(𝛾1𝑥),𝛾1>0, Y exponential with mean one, 𝐴(𝛼)=𝛼2𝛼1, 𝐵(𝛼)=𝛼1, 𝛼=(𝛼1,𝛼2)𝑅2+, 𝜃=(𝛼1,𝛼2,𝛾1)Θ=𝑅3+. Its probability density function is

𝑓𝑋1(𝑥)=𝛼1𝛾11+𝑥𝛼2𝛼1(1+(1/𝛾1)),𝑥𝛼2.(4.1) Applying Theorem 3.1, one sees that the MLE of 𝛼1,𝛼2 are determined by the real functions

𝑆1(𝑥)=1+𝛾11+𝑥,𝐿2(𝑥)=1+𝛾1𝛾1𝑥.(4.2) According to Theorem 3.1, there are functions

𝑞1(𝑥)=(1+𝑥)(1+(𝛾1/𝛾1)),𝑞2(𝑥)=𝑥𝛾1+1/𝛾1,(4.3) and constants 𝑐1=𝛾11,𝑑2=1 such that

𝑐1𝑆1(𝑥)+1𝑐1𝑑=𝑥𝑞𝑑𝑥ln1(𝑥),𝑑2𝐿2𝑑(𝑥)=𝑞𝑑𝑥ln2(𝑥),(4.4) and the compatibility condition (3.7) is fulfilled. For any random sample 𝜉=(𝑋1,,𝑋𝑛) from this family, one observes that the simultaneous maximum likelihood equations

1+𝛾11+𝜉𝛼2/𝛼1=1,11+𝜉𝛼2/𝛼1=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 𝛼1 that is determined by the mean value equation 𝑆1((𝜉𝛼2)/𝛼1)=1?. By Theorem 3.1, Case 1, there must exist a constant 𝛾2 and a function 𝑞1(𝑥) such that

𝛾2𝑆1(𝑥)+1𝛾2𝑑=𝑥𝑞𝑑𝑥ln1(𝑥).(4.6) Using [5], formula (3.1) one obtains

𝑞1(𝑥)=𝑥𝛾21exp𝛾2𝑆1(𝑥)𝑥𝑑𝑥=𝑥1+𝛾1𝛾2(1+𝑥)1+𝛾1𝛾2.(4.7) A corresponding probability density function is

𝑓𝑋1(𝑥)=𝐶𝛼1𝑥𝛼2𝛼11+𝛾1𝛾21+𝑥𝛼2𝛼11+𝛾1𝛾2,𝑥𝛼2.(4.8) One notes that two well-known subfamilies are included, namely, the generalised Pareto (4.1) obtained by setting 𝛾1𝛾2=1, and the Beta of type II obtained by setting 𝑝=𝛾1𝛾2>0,𝑞=𝛾2>0. 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 𝛼1 and 𝛼2 are determined by

𝑆1(𝑥)=1+𝛾11+𝑥,𝐿2(𝑥)=1+𝛾1𝛾2𝑥1+𝛾1𝛾2𝑥1.(4.9) There are functions

𝑞1(𝑥)=𝑥1+𝛾1𝛾2(1+𝑥)1+𝛾1𝛾2,𝑞2(𝑥)=(𝑥1)1+𝛾1𝛾2𝑥1+𝛾1𝛾2,(4.10) and constants 𝑐1=𝛾2,𝑑2=1 such that

𝑐1𝑆1(𝑥)+1𝑐1𝑑=𝑥𝑞𝑑𝑥ln1(𝑥),𝑑2𝐿2𝑑(𝑥)=𝑞𝑑𝑥ln2(𝑥),(4.11) and the compatibility condition (3.7), that is,

𝛾2𝑆1(𝑥1)+1𝛾2=(𝑥1)𝐿2(𝑥),(4.12) is fulfilled. For a random sample 𝜉=(𝑋1,,𝑋𝑛), the MLE of 𝛼1 and 𝛼2 solves the simultaneous equations

1+𝛾11+𝜉𝛼2/𝛼1=1,1+𝛾1𝛾2𝜉𝛼2/𝛼1=𝛾2.(4.13) The value of the normalising constant in (4.8) depends only on the shape vector 𝛾=(𝛾1,𝛾2).

Proposition 4.1. Assume that 𝛾2,𝛾1𝛾2 are not integers. Then the normalising constant of the Pareto type IV model (4.8) is determined by the infinite series expansion 𝛾𝐶=𝐶1,𝛾2=𝑘=01+𝛾1𝛾2𝑘2𝑘1+𝛾1𝛾2𝑘𝛾2𝑘𝛾1𝛾2,(4.14) where (𝛼𝑘)=(𝛼(𝛼1)(𝛼𝑘+1))/𝑘!,𝑘1,(𝛼0)=1, is a generalised binomial coefficient.

Proof. From the observation made above, one notes that 𝐶=0𝑞1(𝑥)𝑑𝑥=0𝑥1+𝛾1𝛾2(1+𝑥)1+𝛾1𝛾2𝑑𝑥=0𝑥𝛾211+𝑥11+𝛾1𝛾2𝑑𝑥.(4.15) To obtain convergent integrals, separate calculation in two parts and make a substitution to get 𝐶=10𝑥1+𝛾1𝛾2(1+𝑥)1+𝛾1𝛾2𝑑𝑥+10𝑥1+𝛾2(1+𝑥)1+𝛾1𝛾2𝑑𝑥.(4.16) The binomial expansion (1+𝑥)𝛼=𝑘=0(𝛼𝑘)𝑥𝑘, valid for 𝑥(0,1) [14, (18.7), page 134], yields the series 𝐶=𝑘=01+𝛾1𝛾2𝑘10𝑥𝑘1𝛾1𝛾2𝑑𝑥+10𝑥𝑘1𝛾2𝑑𝑥.(4.17) Under the assumption 𝛾2,𝛾1𝛾2𝑘, 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

𝑓𝑋(𝑐𝑥)=𝛼1𝛾1𝑥𝛼2𝛼1𝑐11+𝑥𝛼2𝛼1𝑐1+1/𝛾1,𝑥𝛼2,(5.1) 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

𝑆𝑋(𝑥)=1+𝑥𝛼2𝛼1𝑐1+1/𝛾1,𝑥𝛼2.(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 “closest” to the full model will be retained. A detailed comparison of these two four parameter Pareto families is left to further research.


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