1. Short introduction to the model
We will use the same notation as in Siegenthaler (2017) except for the position of some indices. In order to make the understanding easier, upper indices n and m will indicate the time of the estimation, whereas lower indices i and h refer to the origin period of the corresponding claims. The indices k refer to time (development years) between the origin year and the year of estimation, i.e., i + k = n.
The model introduced in Siegenthaler (2017) tries to analyze estimates ˆUni of ultimates U_{i}. The aim of the original paper is to present estimators for the oneyear uncertainty (solvency uncertainty) and the ultimate uncertainty of the estimated ultimates independent of the underlying claim data and the reserving methodology used to estimate ˆUni. Because of Solvency II, the Swiss Solvency Test and the upcoming IFRS 17, such uncertainty estimators are becoming more and more important. Up to now, reserving models based on claim data are used to estimate such uncertainties, for instance, Merz and Wüthrich (2008), Mack (1991), Dahms (2012), or bootstrapping approaches. Often the resulting estimates for uncertainties strongly depend on the reserving method used. Since real uncertainties cannot be observed, it is impossible to decide which model is the best. Therefore, an estimation method that is independent of the underlying reserving method would be very helpful. Unfortunately, the model cannot fulfill these promises. It only corrects for some bias in the estimated ultimates (see Section 3). Therefore, we will call it bias correction model.
The bias correction model assumes that there exist constants g_{k} and σ2k such that
ˆUni≔ˆE[Uiℱn]
ˆUi+Ji=Ui
E[ˆUi+k+1iDi+kk]=gkˆUi+ki
Var[ˆUi+k+1iDi+kk]=σ2kˆUi+ki
where Fn represents all at the time of estimation n available information and Dnk consists of all information contained in the estimated ultimates ˆUmh that have been or will be estimated before or at max(n,n−i+h), i.e.,
Dnk :=σ(ˆUmh :m≤max(n,n−i+h)).
2. Model assumption
2.1. Simple example
The definition 1.1 of the estimated ultimates ˆUni can be rephrased as:
ˆUni are random variables measurable with respect to Fn.
Therefore, it is possible that there is no relation between ˆUni and the real ultimate U_{i} at all, except for the tail condition 1.2.
Indeed, take independent origin years with stable business, i.e., E[Ui]=g and Var[Ui]=σ2, for all origin years i, then the following estimated ultimates fulfill assumptions (1.1) to (1.4):
ˆUni≔{1,for n<i+J,Ui,for n=i+J,
with
gk≔{ 1,for k<J−1,g,for k=J−1, and σ2k≔{ 0,for k<J−1,σ2,for k=J−1.
In this simple example the solvency uncertainty equals zero, except for ˆUi+Ji at estimation time i + J − 1. At this time it is equal to the ultimate uncertainty σ2, which can be estimated by the empirical variance
ˆσ2 :=1i−1i−1∑h=0(Uh−¯U)2 with ¯U≔1ii−1∑h=0Uh.
The bias correction model leads to other estimates for the uncertainties, because it considers the bias correction ˆˆUni:=ˆgn−i.....ˆgJ−1ˆUni instead of the estimation behind ˆUni (see Section 3).
2.2. Best estimates
In the last section we have seen that it may be helpful to put some restrictions on the allowed estimated ultimates. In practice, for instance under Solvency II, the Swiss Solvency Test or IFRS 17, one would take best estimates. Unfortunately, it is not exactly clear what that means mathematically. Let us look at two interpretations. Note, almost all standard stochastic reserving methods satisfy both interpretations.
Unbiased estimates: In most cases we would at least expect that our estimates ˆUni are unbiased. In this case we get
E[Ui]= E[ˆUi+k+1i]= E[E[ˆUi+k+1iDi+kk]]= gkE[ˆUi+ki]= gkE[Ui],
which leads to gk≡1.
Claims development result is predicted by zero: This is the oneyear version of unbiased estimates. We would like to have
E[ˆUi+k+1iFi+k]=ˆUi+ki.
Unfortunately, this is too strong. But we can weaken it by using the same estimation methodology for the left hand side like we have used to estimate ˆUi+ki, i.e., we assume
ˆE[ˆUi+k+1iFi+k]=ˆUi+ki.
In the case of the bias correction model the only way to predict the left hand side is
ˆE[ˆUi+k+1iFi+k] :=ˆE[ˆUi+k+1iDi+kk] :=ˆgkˆUi+ki,
which leads to ˆgk≡1. Since ˆgk are unbiased, it follows that gk≡1.
2.3. Parameter error
The solvency uncertainty can be split into
E[(ˆUn+1i−ˆUni)2Fn] = Var[ˆUn+1Fn]⏟random error+ (E[ˆUn+1iFn]−ˆUni)2⏟parameter error.
In the case of best estimates, assumption (1.3) of the bias correction model means that there is no parameter error, which is very unlikely in practice. Similar arguments lead to the same statement for the parameter error of the ultimate uncertainty. Note, our reasoning stays true if we replace Fn by an arbitrary σalgebra D with respect to which ˆUni is measurable.
3. Bias correction model
If we look closely at the derivation of the estimator for the solvency uncertainty in the bias correction model by Siegenthaler (2017), we note that

It is enough to assume
i) E[ˆUi+k+1iDk]=gkˆUi+ki,
ii) Var[ˆUi+k+1iDk]=σ2kˆUi+ki,
where Dk is an arbitrary σalgebra containing all ˆUh+kh, h≥0.

If gk≠1 then the term ˆUi+k+1i−ˆUi+ki is biased, which should be compensated for in the derivation of the uncertainty. Usually, this is done by subtracting
ˆE[ˆUi+k+1i−ˆUi+kiDk]=(ˆgk−1)ˆUi+ki.
This compensation is missing in the original paper.
Indeed, we get
E[(ˆUi+k+1i−ˆUi+ki−ˆE[ˆUi+k+1i−ˆUi+kiDk])2Dk]=E[(ˆUi+k+1i−ˆgkˆUi+ki)2Dk]=Var[ˆUi+k+1iDk]+(gk−ˆgk)2(ˆUi+ki)2.
The second term is approximated by the D_{k}conditional variance of ˆg_{k}. This leads to
E[(ˆUi+k+1i−ˆUi+ki−ˆE[ˆUi+k+1i−ˆUi+kiDk])2Dk]≈Var[ˆUi+k+1i˜Di,k]+Var[ˆgkDk](ˆUi+kk)2≈ˆσ2k(1ˆUi+ki+1∑i−1