Actuarial Statistics: Compound Poisson Distribution and Moment Generating Functions, Exams of Statistics

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M. PHIL. IN STATISTICAL SCIENCE

Friday, 4 June, 2010 9:00 am to 11:00 am

ACTUARIAL STATISTICS

Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.

STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS

Cover sheet None Treasury Tag Script paper

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

Let S = X 1 +... + XN (if N = 0 then S = 0) be a random sum with ‘steps’ X 1 , X 2 ,... given by independent and identically distributed (iid) positive random variables, and where the number N of summands is independent of the steps. Show that S has moment generating function MS (u) = GN [MX 1 (u)] where GN is the probability generating function of N and MX 1 is the moment generating function of X 1.

A portfolio consists of n independent policies where for policy i, i = 1,... , n , the number of claims during an accounting period is Ni independent of the sizes of claims which are iid with distribution function Fi.

(a) Suppose that Ni has a Poisson distribution with mean λi. Show that the distribution of the total amount T claimed on the portfolio during a typical accounting period has a compound Poisson distribution. In the case where F 1 = F 2 =... = Fn = F and F (x) = 1 − e−x^ , x > 0, show that the distribution function of T can be written in the form FT (x) = a + (1 − a) ˜FT (x) , x > 0 ,

where a is a constant in (0, 1) which you should specify, and where F˜T has density

f˜T (x) =

∑^ ∞

k = 1

λk (eλ^ − 1)k!

x k−^1 e−x k!

.

(b) Now suppose that, for i = 1,... , n , P(Ni = k) = qkp, k = 0, 1 ,... , where q = 1 − p and 0 < p < 1 , and that F 1 = F 2 =... = Fn = F where F is as in (a). Show that T is distributed as a random sum, and state the distribution of the steps and of the number of summands for this random sum. When n = 2 show that

FT (x) = b + (1 − b) FˇT (x) , x > 0 ,

where b is a constant between 0 and 1 and FˇT is a distribution function. State the value of b and write down a density for FˇT.

Actuarial Statistics

Consider the total amount claimed in a year on a particular risk where the number of claims N has P(N = n) = pn , n = 0, 1 , 2 ,... , and where claims are independent and identically distributed random variables X 1 , X 2 ,... independent of N. Suppose that

pn =

(

a +

b n

)

pn− 1 , n = 1, 2 ,...

for some constants a and b. Suppose also that the claim sizes are positive and discrete with P(X 1 = j) = fj , j = 1, 2 ,.... Let S be the total amount claimed in a year, and let gr = P(S = r), r = 0, 1 ,.... Show the following recursion for {gr }∞ r = 0 :

g 0 = p 0 , gr =

∑^ r

j = 1

(

a +

bj r

)

fj gr−j r > 1.

Now assume that N has a Poisson distribution with mean λ. Write down the recursion for {gr }∞ r = 0 for this case, and derive a recursion for {E(Sk)}∞ k = 1. Use this recursion to find E(S), var(S) and E

(

(S − E(S))^3

)

in terms of λ and the moments of X 1.

In the classical risk model, let MX (r) be the moment generating function of the claim sizes, let λ be the claim arrival rate, let the premium loading factor be θ > 0 (so that the premium rate is c = (1 + θ)λμ where μ is the expected claim size), and let ψ(u) be the probability of ruin with initial capital u > 0. You are given that there is a unique positive solution R of the equation MX (r) − 1 = (1 + θ)μr. Prove that ψ(u) 6 e−Ru^ for u > 0 (the Lundberg inequality).

(a) Suppose that the claim sizes are exponentially distributed with mean 1. Find R.

(b) Suppose that the claim sizes have density

fX (x) = e −^2 x^ +

e −^2 x/^3 , x > 0.

Write down the expected claim size. Show that R is the smaller root of

3(1 + θ) r^2 − (8 θ + 5) r + 4 θ = 0.

If mistakenly the claims are assumed to be exponentially distributed with the same mean, compare the resulting upper bounds on the probability of ruin given by the Lundberg inequality.

Actuarial Statistics [TURN OVER

Let Xi be the amount claimed on a risk in year i, i = 1, 2 ,... , and suppose that, given θ, the Xi’s are independent and identically distributed with density

f (x | θ) = θk^ e−θ/x x k+1(k − 1)!

, x > 0 ,

where k > 2 is a known positive integer. Suppose that the prior density of θ is

π(θ) = λα^ θα−^1 e−λθ (α − 1)!

, θ > 0 ,

where α is a known positive integer and λ > 0 is known. Suppose that X 1 ,... , Xn are observed and consider μ(θ) = E(Xn+1| θ). Find c 0 , c 1 ,... , cn in terms of known

quantities such that E

[(

μ(θ) − c 0 −

∑n i = 1 ciXi

) 2 ]

is minimised. Define what is meant

by a credibility estimate, and show that c 0 +

∑n i=1 ci^ Xi^ can be written in the form of a credibility estimate. Discuss the effect on the credibility factor as

(a) k increases while α and λ remain fixed;

(b) the prior variance of θ decreases while the prior mean of θ and k remain fixed.

Find the Bayesian estimate of E(Xn+1| θ) under quadratic loss. State whether or not this can be written in the form of a credibility estimate.

[ Hint:

(i) You may assume without proof that if Y has density

f (x) =

θke−θ/x xk+1(k − 1)! , x > 0 ,

where k ∈ { 3 , 4 ,.. .} and θ > 0 , then

E(Y ) =

θ (k − 1)

, E(Y 2 ) =

θ^2 (k − 1) (k − 2)

, and var(Y ) = θ^2 (k − 1)^2 (k − 2)

.

(ii) Let random variables U , V and W be such that U and V have finite second moments. Then cov (U, V ) = E

(

cov (U, V | W )

)

• cov

(

E(U | W ), E(V | W )

)

.

]

END OF PAPER

Actuarial Statistics