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

Authors: Werner Hürlimann
Journal: Journal of Probability and Statistics
 (2009)

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## 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.


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## Body

## 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.Section2 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

LetX,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 Θ⊂Rm, 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)=ψ-1(x)  and T(x)=U-1(x). Moreover, these functions do not depend on  α  but may depend on γ.Definition 2.1.
Thegeneral 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 familyF{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
(2.1)FX(x)=FY{φ[T(x)-A(α)B(α)]},(2.2)fX(x)=1B(α)·T′(x)·φ′[T(x)-A(α)B(α)]·fY{φ[T(x)-A(α)B(α)]}.
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 sampleξ=(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(3.1)H(ξ)¯=1n∑i=1nH(Xi).
It is assumed that sample mean value equations like  H(ξ/α̂)¯=1  have a unique solution α̂=α̂(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 GATFX=U[A(α)+B(α)·ψ(Y)]  with support  IX=[aX,bX]  and affine parameter vector  α=(α1,…,αr)∈Θ⊂Rr. 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 :.
(3.2)Bk=∂B(α)∂αk≠0,Ak=∂A(α)∂αkarbitrary,k∈{1,…,r1},Sk(T(ξ)-A(α̂)B(α̂)+AkBk)¯=1,
with some real function  Sk(x).Case 2 :.
(3.3)Bk=∂B(α)∂αk≡0,Ak=∂A(α)∂αk≠0,k∈{r1+1,…,r},Lk(T(ξ)-A(α̂)B(α̂))¯=0,
with some real function Lk(x).
Then there exists a twice-differentiable and monotone increasing functionQk(x)  with derivative  qk(x)=Qk′(x), and constants  ck,dk≠0  such that
(3.4)ckSk(x)+1-ck=-x·ddxln{qk(x)},inCase1,(3.5)dkLk(x)=-ddxln{qk(x)},inCase2.
Furthermore, for simultaneous maximum likelihood estimation of the affine parameters, the following compatibility conditions must be satisfied:
(3.6)(x+AjBj)·(ciSi(x+AiBi)+1-ci)=(x+AiBi)·(cjSj(x+AjBj)+1-cj),i,j∈{1,…,r1},(3.7)ciSi(x+AiBi)+1-ci=(x+AiBi)·djLj(x),i∈{1,…,r1},j∈{r1+1,…,r},(3.8)diLi(x)=djLj(x),i,j∈{r1+1,…,r}.
Under these conditions, the distribution function has the unique representation
(3.9)FX(x)=Qi((T(x)-A)/B+(Ai/Bi))-Qi((T(aX)-A)/B+(Ai/Bi))Qi((T(bX)-A)/B+(Ai/Bi))-Qi((T(aX)-A)/B+(Ai/Bi))=Qj((T(x)-A)/B)-Qj(T(aX)-A/B)Qj(T(bX)-A/B)-Qj(T(aX)-A/B),
for all  x∈IX=[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
(3.10)-ℓ(X)=lnB(α)-lnT′(X)+lnψ′(Y)-lnfY(Y).
Denoting partial derivatives with respect to  αk  with a lower index k  and making use of
(3.11)Yk=φ′(T(X)-AB)·-AkB-(T(X)-A)BkB2=-Ak+Bkψ(Y)Bψ′(Y),
one obtains from (3.10) the expression for the partial derivative
(3.12)-BBk·ℓk(X)=1-(ψ(Y)+(Ak/Bk)ψ’(Y))·(ψ′′(Y)ψ′(Y)-ddYln{fY(Y)}).
By assumption (3.2), one has using [12, Theorem  2.2] that
(3.13)-BBk·ℓk(X)=ck·{1-Sk(ψ(Y)+AkBk)}
for some constant  ck≠0. By comparison  y(x)=ψ(x)+(Ak/Bk)  solves the second-order differential equation
(3.14)y′′y′-{ckSk(y)+1-ck}·y′y=ddxln{fY(x)}.
Setting  gk(x)=(ckSk(x)+(1-ck))/x  and multiplying with  y′,  this simplifies to
(3.15)y′′-ddxln{fY(x)}·y′-gk(y)·y′2=0.
Transform it to the equivalent system of first-order equations in  (y1=y,y2) [13, Chapter 19]:
(3.16)y1′=y2,y2′=ddxln{fY(x)}·y2+gk(y1)·y22.
The second differential equation is of Bernoulli type [13, Chapter 2]. Setting  y2=z2-1, this is equivalent to the simpler system in  (y1,z2):
(3.17)y1′=z2-1,z2′=-ddxln{fY(x)}·z2+gk(y1).
The second equation is linear inhomogeneous of first order and has the homogeneous solution z2=Ck·fY(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′=z2-1=Ck(x)-1·fY(x), hence  fY(x)=y1′·Ck(x). Together, this shows the following separated differential equation:
(3.18)ddxln{Ck(x)}=-gk(y)·y′.
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)=Ck-1·exp{-G(y)},Ck>0. It follows that the general solution of the second differential equation in (3.17) is given by
(3.19)z2=exp{-Gk(y)}CkfY(x).
The first differential equation in (3.17) implies the separated differential equation
(3.20)y′·exp{-Gk(y)}=Ck·fY(x).
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
(3.21)FY(x)=1Ck{Qk(y)+Dk},Ck>0,Dk∈R.
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
(3.22)FX(x)=1Ck{Qk(T(x)-AB+AkBk)+Dk},x∈IX.
Setting  qk(x)=Qk′(x),  one obtains the density function
(3.23)fX(x)=T′(x)BCkqk(T(x)-AB+AkBk),x∈IX.
The side conditions  ∫aXbXfX(x)dx=1, FX(bX)=1,  imply that the constants are determined by
(3.24)Ck=Qk(T(bX)-AB+AkBk)-Qk(T(aX)-AB+AkBk),Dk=-Qk(T(aX)-AB+AkBk).
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
(3.25)ckSk(x)+1-ck=xgk(x)=xGk′(x)=-x·ddxln{qk(x)},
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}).
SinceBk≡0, one has similarly to (3.11) the relationship
(3.26)Yk=-AkBψ′(Y).
From (3.10), one obtains for the partial derivative of the random log-likelihood the relation
(3.27)-BAk·ℓk(X)=1ψ’(Y)·(ψ’’(Y)ψ’(Y)-ddYln{fY(Y)}).
By assumption (3.2) and again in [12, Theorem  2.2], one has
(3.28)-BAk·ℓk(X)=dk·Lk(ψ(Y))
for some constant  dk≠0. Through comparison, it follows that  y(x)=ψ(x)  must solve
(3.29)y′′-ddxln{fY(x)}·y′-dk·Lk(y)·y′2=0.
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,Dk∈R. 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
(3.30)fX(x)=T′(x)BCiqi(T(x)-AB+AiBi)=T′(x)BCjqj(T(x)-AB),
for all  x∈IX,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))=C·qi(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)=C·qi(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)=C·qi(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 byX=A(α)+B(α)·ψ(Y)  with  ψ(x)=exp(γ1x),γ1>0, Y  exponential with mean one,  A(α)=α2-α1,  B(α)=α1,  α=(α1,α2)∈R+2,  θ=(α1,α2,γ1)∈Θ=R+3. Its probability density function is(4.1)fX(x)=1α1γ1(1+x-α2α1)-(1+(1/γ1)),x≥α2.
Applying Theorem 3.1, one sees that the MLE of  α1,α2  are determined by the real functions(4.2)S1(x)=1+γ11+x,L2(x)=-1+γ1γ1x.
According to Theorem 3.1, there are functions(4.3)q1(x)=(1+x)-(1+(γ1/γ1)),q2(x)=x-1+(γ1/γ1),
and constants  c1=-γ1-1,d2=-1  such that(4.4)c1S1(x)+1-c1=-x·ddxln{q1(x)},d2L2(x)=-ddxln{q2(x)},
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(4.5)1+γ11+((ξ-α2)/α1)¯=1,11+((ξ-α2)/α1)¯=0,
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 Theorem3.1. 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((ξ-α2)/α1)¯=1?. By Theorem 3.1, Case 1, there must exist a constant  γ2  and a function  q1(x)  such that(4.6)γ2S1(x)+1-γ2=-x·ddxln{q1(x)}.
Using [5], formula (3.1) one obtains(4.7)q1(x)=xγ2-1·exp{-γ2∫S1(x)xdx}=x-(1+γ1γ2)·(1+x)(1+γ1)γ2.
A corresponding probability density function is(4.8)fX(x)=1Cα1·(x-α2α1)-(1+γ1γ2)·(1+x-α2α1)(1+γ1)γ2,x≥α2.
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(4.9)S1(x)=1+γ11+x,L2(x)=(1+γ1)γ2x-1+γ1γ2x-1.
There are functions(4.10)q1(x)=x-(1+γ1γ2)·(1+x)(1+γ1)γ2,q2(x)=(x-1)-(1+γ1γ2)·x(1+γ1)γ2,
and constants  c1=γ2,d2=-1  such that(4.11)c1S1(x)+1-c1=-x·ddxln{q1(x)},d2L2(x)=-ddxln{q2(x)},
and the compatibility condition (3.7), that is,(4.12)γ2S1(x-1)+1-γ2=-(x-1)L2(x),
is fulfilled. For a random sample  ξ=(X1,…,Xn),  the MLE of  α1  and  α2  solves the simultaneous equations(4.13)1+γ11+((ξ-α2)/α1)¯=1,1+γ1γ2(ξ-α2)/α1¯=γ2.
The value of the normalising constant in (4.8) depends only on the shape vector  γ=(γ1,γ2).Proposition 4.1.
Assume thatγ2,γ1γ2  are not integers. Then the normalising constant of the Pareto type IV model (4.8) is determined by the infinite series expansion
(4.14)C=C(γ1,γ2)=∑k=0∞((1+γ1)γ2k)2k-(1+γ1)γ2(k-γ2)(k-γ1γ2),
where  (αk)=(α(α-1)…(α-k+1))/k!,k≥1,(α0)=1, is a generalised binomial coefficient.Proof.
From the observation made above, one notes that(4.15)C=∫0∞q1(x)dx=∫0∞x-(1+γ1γ2)(1+x)(1+γ1)γ2dx=∫0∞xγ2-1(1+x-1)(1+γ1)γ2dx.
To obtain convergent integrals, separate calculation in two parts and make a substitution to get
(4.16)C=∫01x-(1+γ1γ2)(1+x)(1+γ1)γ2dx+∫01x-(1+γ2)(1+x)(1+γ1)γ2dx.
The binomial expansion  (1+x)α=∑k=0∞(αk)xk, valid for  x∈(0,1) [14, (18.7), page 134], yields the series
(4.17)C=∑k=0∞((1+γ1)γ2k)·{∫01xk-1-γ1γ2dx+∫01xk-1-γ2dx}.
Under the assumption  γ2,γ1γ2≠k,  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(5.1)fX(x)=cα1γ1·(x-α2α1)c-1·(1+(x-α2α1)c)-(1+(1/γ1)),x≥α2,
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(5.2)SX(x)=(1+(x-α2α1)c)-(1+1/γ1),x≥α2.
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.



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