Parameter Estimation - Brigham Young University

Parameter Estimation

Chapters 13-15

Stat 477 - Loss Models

Chapters 13-15 (Stat 477) Parameter Estimation Brian Hartman - BYU 1 / 23

Methods for parameter estimation

Methods for parameter estimation

Methods for estimating parameters in a parametric model:

method of moments

matching of quantiles (or percentiles)

maximum likelihood estimation

full/complete, individual datacomplete, grouped datatruncated or censored data

Bayes estimation


Method of moments

Method of momentsAssume we observe values of the random sample X1, . . . , Xn from apopulation with common distribution function F (x|θ) whereθ = (θ1, . . . , θp) is a vector of p parameters.

The observed sample values will be denoted by x1, . . . , xn.

Denote the k-th raw moment by µ′k(θ) = E(Xk|θ). Denote the empirical(sample) moments by

µ̂′k =1

n

n∑j=1

xkj .

The method of moments is fairly straightforward: equate the first p rawmoments to the corresponding p empirical moments and solve theresulting system of simultaneous equations:

µ′k(θ̂) =1

n

n∑j=1

xkj .


Method of moments Illustrations

Illustrations

Example 1: For a Normal distribution, derive expressions for themethod of moment estimators for the parameters µ and σ2.

Example 2: For a Gamma distribution with a shape parameter α anda scale parameter θ, derive expressions for their method of momentestimators.

Example 3: For a Pareto distribution, derive expressions for themethod of moment estimators for the parameters α and θ.


Method of moments Illustrations

SOA Example #6

For a sample of dental claims x1, x2, . . . , x10 you are given:∑xi = 3860 and

∑x2i = 4, 574, 802

Claims are assumed to follow a lognormal distribution withparameters µ and σ.

µ and σ are estimated using the method of moments.

Calculate E[X ∧ 500] for the fitted distribution. [∼259]


Matching of quantiles

Matching of quantiles

Denote the 100q-th quantile of the distribution by πq(θ) for which in thecase of continuous distribution is the solution to:

F (πq(θ)|θ) = q.

Denote the smoothed empirical estimate by π̂q. This is usually found bysolving for

π̂q = (1− h)x(j) + hx(j+1),

where j = b(n+ 1)qc and h = (n+ 1)q − j. Here x(1) ≤ · · · ≤ x(n)denote the order statistics of the sample.The quantile matching estimate of θ is any solution to the p equations:

πqk(θ̂) = π̂qk for k = 1, 2, . . . , p,

where the qk’s are p arbitrarily chosen quantiles.


Matching of quantiles Example

Example

Suppose the following observed sample

12, 15, 18, 21, 21, 23, 28, 32, 32, 32, 58

come from a Weibull with

F (x|λ, γ) = 1− e−λxγ .

Calculate the quantile matching estimates of λ and γ by matching themedian and the 75-th quantile.


Matching of quantiles Example

SOA Example #155

You are given the following data:

0.49 0.51 0.66 1.82 3.71 5.20 7.62 12.66 35.24

You use the method of percentile matching at the 40th and 80thpercentiles to fit an inverse Weibull distribution to these data. Calculatethe estimate of θ. [1.614]


Maximum likelihood estimation

The method of maximum likelihood

The maximum likelihood estimate of parameter vector θ is obtained bymaximizing the likelihood function. The likelihood contribution of anobservation is the probability of observing the data.

In many cases, it is more straightforward to maximize the logarithm of thelikelihood function.

The likelihood function will vary depending on whether it is completelyobserved, or if not, then possibly truncated and/or censored.


Likelihood function Complete, individual data

No truncation and no censoring

In the case where we have no truncation and no censoring and eachobservation Xi, for i = 1, . . . , n, is recorded, the likelihood function is

L(θ) =

n∏j=1

fXj (xj |θ).

The corresponding log-likelihood function is

`(θ) = log[L(θ)] =

n∑j=1

log fXj (xj |θ).


Likelihood function Illustrations

Illustrations

Assume we have n samples completely observed. Derive expressions forthe maximum likelihood estimates for the following distributions:

Exponential: fX(x) = λe−λx, for x > 0.

Uniform: fX(x) =1θ , for 0 < x < θ.


Likelihood function Illustrations

SOA Example #26

You are given:

Low-hazard risks have an exponential claim size distribution withmean θ.

Medium-hazard risks have an exponential claim size distribution withmean 2θ.

High-hazard risks have an exponential claim size distribution withmean 3θ.

No claims from low-hazard risks are observed.

Three claims from medium-hazard risks are observed, of sizes 1, 2 and3.

One claim from a high-hazard risk is observed, of size 15.

Calculate the maximum likelihood estimate of θ. [2]


Likelihood function Complete, grouped data

Complete, grouped data

Starting with a set of numbers c0 < c1 < · · · < ck where from the sample,we have nj observed values in the interval (cj−1, cj ].

For such data, the likelihood function is

L(θ) =

k∏j=1

[F (cj |θ)− F (cj−1|θ)]nj .


Likelihood function Example

Example

Suppose you are given the following observations for a loss randomvariable X:

Interval Number of Observations

(0, 2] 4(2, 4] 7(4, 6] 10(6, 8] 6

(8, ∞) 3

Total 30

Determine the log-likelihood function of the sample if X has a Pareto withparameters α and θ. If it is possible to maximize this log-likelihood andsolve explicitly, determine the MLE of the parameters.



SOA Example #44

You are given:

Losses follow an exponential distribution with mean θ.

A random sample of 20 losses is distributed as follows:

Loss Range Frequency

[0,1000] 7(1000, 2000] 6(2000, ∞) 7

Calculate the maximum likelihood estimate of θ. [1996.90]


Likelihood function Truncated and/or censored data

Truncated and/or censored dataConsider the case where we have a left truncated and right censoredobservation. It is usually best to represent this observation as (ti, xi, δi)where:

ti is the left truncation point;

xi is the value that produced the data point; and

δi is an indicator (1 or 0) whether data is right censored.

When observation is right censored, the contribution to the likelihood is[1− F (xi)1− F (ti)

]δi.

Otherwise if not right censored, the contribution to the likelihood is[f(xi)

1− F (ti)

]1−δi.

Note that the policy limit if reached would be equal to xi − ti.Chapters 13-15 (Stat 477) Parameter Estimation Brian Hartman - BYU 16 / 23

Likelihood function Truncated and/or censored data

SOA Example #4

You are given:

Losses follow a single-parameter Pareto distribution with densityfunction:

f(x) =α

xα+1x > 1 0 < α <∞

A random sample of size five produced three losses with values 3, 6and 14, and two losses exceeding 25.

Calculate the maximum likelihood estimate of α. [0.2507]



Illustrative example

An insurance company records the claim amounts from a portfolio ofpolicies with a current deductible of 100 and policy limit of 1100. Lossesless than 100 are not reported to the company and losses above the limitare recorded as 1000. The recorded claim amounts as

120, 180, 200, 270, 300, 1000, 1000.

Assume ground-up losses follow a Pareto with parameters α and θ = 400.

Use the maximum likelihood estimate of α to estimate the LossElimination Ratio for a policy with twice the current deductible.


Bayesian estimation

Definitions and notation

Assume that conditionally on θ, the observable random variablesX1, . . . , Xn are i.i.d. with conditional density fX|θ(x|θ). Denote therandom vector X = (X1, . . . , Xn)

′.

prior distribution: the probability distribution of θ with density π(θ).

joint distribution of (X, θ) is given by

fX,θ(x, θ) = fX|θ(x|θ) · π(θ)= fX|θ(x1|θ) · · · fX(xn|θ) · π(θ).


Bayesian estimation Posterior distribution

Posterior distribution

The conditional probability distribution of θ, given the observed dataX1, . . . , Xn, is called the posterior distribution.

This is denoted by π(θ|X) and is computed using

πθ|X(θ|x) =fX,θ(x, θ)∫fX,θ(x, θ)dθ

=fX|θ(x|θ)π(θ)∫fX|θ(x|θ)π(θ)dθ

.

We can conveniently write πθ|X(θ|x) = c · fX|θ(x1, . . . , xn|θ)π(θ),where c is the constant of integration.

More often, we write πθ|X(θ|x) ∝ fX|θ(x1, . . . , xn|θ)π(θ).

The mean of the posterior, E(θ|X), is called the Bayes estimator of θ.


Bayesian estimation Poisson with a Gamma prior

Poisson with a Gamma prior

Suppose that conditionally on θ, claims on an insurance policy X1, . . . , Xn

are distributed as Poisson with mean λ.

Let the prior distribution of λ be a Gamma with parameters α and θ (as intextbook).

Show that the posterior distribution is also Gamma distributed andfind expressions for its parameters.

Show that the resulting Bayes estimator can be expressed as theweighted combination of the sample mean and the prior mean.


Bayesian estimation SOA Exam Question

SOA Exam Question

You are given:

Conditionally, given β, an individual loss X has Exponentialdistribution with density:

f(x|β) = 1

βe−x/β, for x > 0.

The prior distribution of β is Inverse Gamma with density:

π(β) =c2

β3e−c/β, for β > 0.

Hint:

∫ ∞0

1

yne−a/ydy =

(n− 2)!

an−1, for n = 2, 3, . . .

Given that the observed loss is x, calculate the mean of the posteriordistribution of β.


Bayesian estimation SOA Exam Question

SOA Example #5

You are given

The annual number of claims for a policyholder has a binomialdistribution with probability function:

p(x|q) =(2

x

)qx(1− q)2−x, x = 0, 1, 2, . . .

The prior distribution is:

p(q) = 4q3, 0 < q < 1

This policyholder had one claim in each of Years 1 and 2. Calculate theBayesian estimate of the number of claims in Year 3. [4/3]


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Parameter Estimation - Brigham Young University