Page 130 - Contributed Paper Session (CPS) - Volume 2

CPS1461 Michal P. et. al
companies in various countries collect claim amounts paid by every insurance
company each year. The data are represented by cumulative claim payments,
which can be seen in terms of the panel data structure, where the given
insurance company i{1,…,N} provides the overall claim amount Yi,t paid at the
given time t{1,…,T} (i.e., annual payments). The follow-up period may be
relatively very short (only 10–15 years) and it is not reasonable to assume that
T tends to infinity as it can be assumed for the number of available companies
N.
The model which we assume for the scenario described above can be
expressed as
= + { > } + + , = 1, … . , , = 1, … , ; (1)
,

,
where µi R are the panel specific mean parameters, τ {1, … , } is some
common changepoint time (same for all considered panels) with the
corresponding jump magnitudes δi R. Thus, if there is some common
changepoint in model (1) present at time τ < T, then the corresponding panel
specific means change from µi before the change to µi +δi after the change.

This formulation also allows for a specific case where δi =0 meaning no jump
is present for some given panel i. The panel specific variance scaling
parameters σi > 0 mimic heteroscedasticity of the panels. The random factors
ξt’s are used to introduce a mutual dependence between individual panels
where the level of dependence is modeled by the magnitude of unknown
loadings ζi R.

Assumption A 1. The vectors [ , … , ] and [ , … , ] exist on a
⊺
⊺
,1
,1
,
,
probability space (Ω, ℱ,P) and are independent for = 1, . . . , . Moreover,
[ , … , ] are iid for = 1, . . . , with Eεi,t = 0 and Varεi,t = 1, having the
⊺
,
,1
autocorrelation function
ρt =Corr ( , ,+ ) = Cov ( , ,+ ),∀s {1, . . . , − },
,
,
which is independent of the time s, the cumulative autocorrelation function
() = Var ∑ =1 , = ∑ ||< ( − ||) ,

and the shifted cumulative correlation function

(, ) = Cov (∑ , ∑ ) = ∑ ∑ − , < ;
,
,
=1 =+1 =1 =+1
for all = 1, . . . , and , = 1, . . . , .

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