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Unformatted text preview: ta With incomplete data The available data may present some issues, e.g.
the exact dates of entry and exit from observation have not
been recorded
For example, only the number of policies in force on annual
valuation data (usually 1 Jan in the UK) is available.
the deﬁnition of age does not correspond exactly to the age
interval x to x + 1 (for integer x )
We will discuss how to deal with those issues. 5/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
Introduction 1 Introduction
Central vs Initial Exposed to Risk
Complete data
Incomplete data
2 Census approximations
Introduction
“Calendar Year” rate interval
“Policy Year” rate interval
3 Examples
Deﬁnition of x
Trapezium approximation
Estimation
Principle of correspondence
Calendar year rate intervals
Policy year rate intervals
6/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
Introduction Sampling in time
Denote the number of people at risk of age x at time t as Px ,t so
that
c
Ex =
Px ,t dt
time of study Often, the data that is available is not continuous in time (we
don’t have this for t ≥ 0), but corresponds to a (yearly) ‘picture’
or ‘count’ of the population under risk—what we call a census (we
have it for t = 0, 1, 2, . . .). So we need to approximate this
integral. Assuming Px ,t is linear between census dates we have
c
Ex ≈
time of study 1
[Px ,t + Px ,t +1 ]
2 This is called the trapezium approximation.
6/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
Introduction Deﬁnitions of t and x
The times at which the ‘picture’ is taken (the deﬁnition of t ) can
be either:
calendar years;
policy years.
Similarly, the deﬁnition of age x can diﬀer. For example, it can be
Age last birthday
Age nearest birthday
Age next birthday
This is because it is very unlikely that everyone will be exactly x at
time t . . .
7/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
Introduction Recall what we are doing. . .
We aim to estimate
d
q = Ex estimates q at the actual age at the start of the rate
x
interval
d
µ = Exc estimates µ at the actual age in the middle of the rate
x
interval Individuals will be considered “age x” from age [start] to age [end]:
Actual age at the start Actual age at the end
Deﬁnition of x
of the rate interval
of the rate interval
Age last birthday
x
x +1
1
x−1
Age nearest birthday
x+2
2
Age next birthday
x −1
x
8/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
Introduction Principle of Correspondence “A life alive at time t should be included in the exposure Px ,t (at
age x at time t ) if and only if, were that life to die immediately,
they would be counted in the deaths data dx (at age x )”
dx is the number of deaths over the time of study according
to some (not necessarily matching) deﬁnition of x .
It is always the death data (deﬁnition of x in dx ) that
determines what rate interval to use: if the death data and
the census data use diﬀerent deﬁnitions of age, we must
adjust the census data. 9/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Calendar Year” rate interval 1 Introduction
Central vs Initial Exposed to Risk
Complete data
Incomplete data
2 Census approximations
Introduction
“Calendar Year” rate interval
“Policy Year” rate interval
3 Examples
Deﬁnition of x
Trapezium approximation
Estimation
Principle of correspondence
Calen...
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