MATH4427 L4 ver4 Part1.pdf - MATH4427 Loss Models and their...

1 MATH4427 Loss Model and their applications
Lecture Note 4: Parameter estimation in parametric model
MATH4427 Loss Models and their applications
Lecture Note 4
Parameter Estimation in parametric model

2 MATH4427 Loss Model and their applications
Lecture Note 4: Parameter estimation in parametric model
Introduction
Recall that the construction of a loss model consists of three main stages:
Stage 1:
Choose probability
distributions for claim
severity ? and claim
frequency ?
Stage 2:
Estimate the parameter
values in the probability
distributions of ? and ?.
( Parameter estimation )
Stage 3:
Study the aggregate
loss 𝑆 or related
quantities using suitable
model.
Pareto
Gamma
Lognormal
Claim
severity
?
Tail distribution
+ nature of risk
Poisson
Binomial
Negative binomial
Claim
frequency
?
??
?
?
?−1
?~????? ሺ?, 𝜃ሻ
?~𝑃??????ሺ?ሻ
Estimate the values of
?, 𝜃 and ? using the
sample data
𝑆 = ?
1
+ ?
2
+ ⋯ + ?
?
(where ?
?
? are
independent)
𝑆 = ?
1
+ ?
2
+ ⋯ + ?
𝑁
(where ?
?
s are iid)
Individual risk model
Collective risk model

3 MATH4427 Loss Model and their applications
Lecture Note 4: Parameter estimation in parametric model
So far, we have explored how to execute the Stage 1 (model selection) and
Stage 3 (Aggregate loss model) in Chapter 2 and Chapter 3 respectively. In
order to complete the construction of the loss model, we will study how to
estimate the parameter values in the probability distributions (tuning) using
the sample data and the following methods:
Case 1: The probability distribution is fixed (parameter is fixed) but unknown
 Method of moments
 Maximum likelihood method
Case 2: The probability distribution is not fixed (parameters are random
variables)
 Bayesian estimation: How to update the probability distribution of the
parameter 𝜃 when there is an updated sample data ( prior distribution and
posterior distribution )? How to estimate the value of the parameter from
the distribution (called Bayes estimate )?

4 MATH4427 Loss Model and their applications
Lecture Note 4: Parameter estimation in parametric model
How to do the estimation? Estimation through random sampling
In most real-life statistical problems, the size of population is large for us to
investigate them all. Instead, we try to draw conclusions about the population
by examining a portion of the population (called random sample ).
Estimated Mean = ? Ƹ
Estimated Standard Deviation = 𝜎 ො
???? = ? (Unknown)
Standard Deviation = 𝜎
(Unknown)
Population
Sample
Random
Sampling
Investigation
Prediction

5 MATH4427 Loss Model and their applications
Lecture Note 4: Parameter estimation in parametric model
This process is called Statistical Interference . It is the process by which
information from sample data is used to draw conclusions about the
population from which the sample was selected.
Since the sample is randomly drawn from the whole population, the
estimations obtained from the sample data are also random quantities in
general. Thus the accuracy (or reliability) of the estimation method becomes an
important issue.