On the efficiency of the Asmussen–Kroese-estimator and its application to stop-loss transforms

Fixed Precision MCMC Estimation by Median of Products of Averages

“…We explain how it can be used to obtain exponential inequalities for unbounded variables and discuss applications to rare-event simulations [4,Chapter 6]. This is illustrated by one specific example, namely estimation of the tail probability of a random sum; see [5] and [16]. We suggest that an estimator which can be used here is a median of averages (MA) of independent and identically distributed (i.i.d.)…”

Improving the Asmussen–Kroese-Type Simulation Estimators

“…The paper [4] gives an Asmussen-Kroese type estimator of θ = E[(S n − u) + ]. Using that θ = nE[(S n − u) + 1 {Xn=Mn} ], [4] proposes to estimate θ by…”

“…Our approach could also be used to estimate E[((S N − u) + ) 2 ], the second stop-loss transform identity considered in [4].…”

“…heavy-tailed random variables, and N is a non-negative integer-valued random variable that is independent of X i 's. The estimation of these probabilities has applications in insurance risk, financial mathematics, and queueing theory, and their efficient Monte Carlo estimation has been subject of extensive research in the last decade, (see [1], [4], and the references there).…”

“…(The preceding also gives some intuition about why the AsmussenKroese estimator can be so good, namely the right hand side of the preceding equation is often small, and so its variance is too.) The paper [4] uses the ideas underlying Asmussen-Kroese estimators to estimate the stoploss transform identities, E[((S N − u) + ) k ], k = 1, 2, (stop-loss transforms have applications in the pricing of stop-loss reinsurance contracts and the valuation of catastrophe risk bonds). In section 2.3 we give an improved estimator of E[(S N − u) + ].…”