Poisson-type random measure

Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution.[1] The PT family of distributions is also known as the Katz family of distributions,[2] the Panjer or (a,b,0) class of distributions[3] and may be retrieved through the Conway–Maxwell–Poisson distribution.[4]

A major contributor to this article appears to have a close connection with its subject.(May 2020)

. . . Poisson-type random measure . . .

Let

K{displaystyle K}

be a non-negative integer-valued random variable

KN0=N>0{0}{displaystyle Kin mathbb {N} _{geq 0}=mathbb {N} _{>0}cup {0}}

) with law

κ{displaystyle kappa }

, mean

c(0,){displaystyle cin (0,infty )}

and when it exists variance

δ2>0{displaystyle delta ^{2}>0}

. Let

ν{displaystyle nu }

be a probability measure on the measurable space

(E,E){displaystyle (E,{mathcal {E}})}

. Let

X={Xi}{displaystyle mathbf {X} ={X_{i}}}

be a collection of iid random variables (stones) taking values in

(E,E){displaystyle (E,{mathcal {E}})}

with law

ν{displaystyle nu }

.

The random counting measure

N{displaystyle N}

on

(E,E){displaystyle (E,{mathcal {E}})}

depends on the pair of deterministic probability measures

(κ,ν){displaystyle (kappa ,nu )}

through the stone throwing construction (STC) [5]

Nω(A)=N(ω,A)=i=1K(ω)IA(Xi(ω))forωΩ,AE{displaystyle quad N_{omega }(A)=N(omega ,A)=sum _{i=1}^{K(omega )}mathbb {I} _{A}(X_{i}(omega ))quad {text{for}}quad omega in Omega ,,,,Ain {mathcal {E}}}

where

K{displaystyle K}

has law

κ{displaystyle kappa }

and iid

X1,X2,{displaystyle X_{1},X_{2},dotsb }

have law

ν{displaystyle nu }

.

N{displaystyle N}

is a mixed binomial process[6]

Let

E+={f:ER+}{displaystyle {mathcal {E}}_{+}={f:Emapsto mathbb {R} _{+}}}

be the collection of positive

E{displaystyle {mathcal {E}}}

-measurable functions. The probability law of

N{displaystyle N}

is encoded in the Laplace functional

EeNf=E(Eef(X))K=E(νef)K=ψ(νef)forfE+{displaystyle quad mathbb {E} e^{-Nf}=mathbb {E} (mathbb {E} e^{-f(X)})^{K}=mathbb {E} (nu e^{-f})^{K}=psi (nu e^{-f})quad {text{for}}quad fin {mathcal {E}}_{+}}

where

ψ(){displaystyle psi (cdot )}

is the generating function of

K{displaystyle K}

. The mean and variance are given by

ENf=cνf{displaystyle quad mathbb {E} Nf=cnu f}

and

VarNf=cνf2+(δ2c)(νf)2{displaystyle quad mathbb {V} {text{ar}}Nf=cnu f^{2}+(delta ^{2}-c)(nu f)^{2}}

The covariance for arbitrary

f,gE+{displaystyle f,gin {mathcal {E}}_{+}}

is given by

Cov(Nf,Ng)=cν(fg)+(δ2c)νfνg{displaystyle quad mathbb {C} {text{ov}}(Nf,Ng)=cnu (fg)+(delta ^{2}-c)nu fnu g}

When

K{displaystyle K}

is Poisson, negative binomial, or binomial, it is said to be Poisson-type (PT). The joint distribution of the collection

N(A),,N(B){displaystyle N(A),ldots ,N(B)}

is for

i,,jN{displaystyle i,ldots ,jin mathbb {N} }

and

i++j=k{displaystyle i+cdots +j=k}

P(N(A)=i,,N(B)=j)=P(N(A)=i,,N(B)=j|K=k)P(K=k)=k!i!j!ν(A)iν(B)jP(K=k){displaystyle mathbb {P} (N(A)=i,ldots ,N(B)=j)=mathbb {P} (N(A)=i,ldots ,N(B)=j|K=k),mathbb {P} (K=k)={frac {k!}{i!cdots j!}},nu (A)^{i}cdots nu (B)^{j},mathbb {P} (K=k)}

The following result extends construction of a random measure

N=(κ,ν){displaystyle N=(kappa ,nu )}

to the case when the collection

X{displaystyle mathbf {X} }

is expanded to

(X,Y)={(Xi,Yi)}{displaystyle (mathbf {X} ,mathbf {Y} )={(X_{i},Y_{i})}}

where

Yi{displaystyle Y_{i}}

is a random transformation of

Xi{displaystyle X_{i}}

. Heuristically,

Yi{displaystyle Y_{i}}

represents some properties (marks) of

Xi{displaystyle X_{i}}

. We assume that the conditional law of

Y{displaystyle Y}

follows some transition kernel according to

P(YB|X=x)=Q(x,B){displaystyle mathbb {P} (Yin B|X=x)=Q(x,B)}

.

. . . Poisson-type random measure . . .

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. . . Poisson-type random measure . . .

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