G the exact dates of entry and exit from observation

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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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