In this paper, we study the tail risk measures for several commonly used multivariate aggregate loss models where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies. We first develop formulas for the moment (or size biased) transforms of the multivariate aggregate losses, showing their relationship with the moment transforms of the claim frequencies and claim sizes. Then, we apply the formulas to compute some popular risk measures such as the tail conditional expectation and tail variance of the multivariate aggregated losses and to perform capital allocation analysis.

The quantification of operational risk has to deal with various concerns regarding data, much more than other types of risk which banks and insurers are obliged to manage. One of the main questions that worries both researchers and practitioners is the bias in the data on the operational losses amounts recorded. We support the assertions made by several authors and defend that this concern is serious when modeling operational losses data and, typically, is presented in all the databases. We show that it's possible, based on mild assumptions on the internal procedures put in place to manage operational losses, to make parametric inference using loss data statistics, that is, to estimate the parameters for the losses amounts, taking in consideration the bias that, not being considered, generates a two fold error in the estimators for the mean loss amount and the total loss amount, the former being overvalued and the last undervalued. In this paper, we do not consider the existence of a threshold for which, all losses above, are reported and available for analysis and estimation procedures. In this sense, we follow a different approach to the parametric inference. Here, we consider that the probability that a loss is reported and ends up recorded for analysis, increases with the size of the loss, what causes the bias in the database but, at the same time, we do not consider the existence of a threshold, above which, all losses are recorded. Hence, no loss has probability one of being recorded, in what we defend is a realist framework. We deduce the general formulae, present simulations for common theoretical distributions used to model (operational reported) losses amounts, estimate the impact for not considering the bias factor when estimating the value at risk and estimate the true total operational losses the bank incurred.

Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

We consider a system where units having magnitudes arrive according to a nonhomogeneous Poisson process, remain there for a random period and then depart. Eventually, at any point in time only a portion of those units which have entered the system remain. Of interest are the finite time properties and limiting behaviors of the distribution of magnitudes among the units present in the system and among those which have departed from the system. We will derive limiting results for the empirical distribution of magnitudes among the active (departed) units. These results are also shown to extend to systems having stages or steps through which units must proceed. Examples are given to illustrate these results.