Robert M. Mnatsakanov

Nonparametric estimation of ruin probabilities given a random sample of claims

In this paper the well-known insurance ruin problem is reconsidered. The ruin probability is estimated in the case of an unknown claims density, assuming a sample of claims is given. An important step in the construction of the estimator is the application of a regularized version of the inverse of the Laplace transform. A rate of convergence in probability for the integrated squared error (ISE) is derived and a simulation study is included.

Kn-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their kn-nearest neighbors, where {kn: n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.

Consistent Estimation of the Structural Distribution Function

Motivated by problems in linguistics we consider a multinomial random vector for which the number of cells N is not much smaller than the sum of the cell frequencies, i.e. the sample size n. The distribution function of the uniform distribution on the set of all cell probabilities multiplied by N is called the structural distribution function of the cell probabilities. Conditions are given that guarantee that the structural distribution function can be estimated consistently as n increases indefinitely although n/N does not. The natural estimator is inconsistent and we prove consistency of essentially two alternative estimators.

A note on recovering the distributions from exponential moments

The problem of recovering a cumulative distribution function of a positive random variable via the scaled Laplace transform inversion is studied. The uniform upper bound of proposed approximation is derived. The approximation of a compound Poisson distribution as well as the estimation of a distribution function of the summands given the sample from a compound Poisson distribution are investigated. Applying the simulation study, the question of selecting the optimal scaling parameter of the proposed Laplace transform inversion is considered. The behavior of the approximants are demonstrated via plots and table.

Approximation of the ruin probability using the scaled Laplace transform inversion

The problem of recovering the ruin probability in the classical risk model based on the scaled Laplace transform inversion is studied. It is shown how to overcome the problem of evaluating the ruin probability at large values of an initial surplus process. Comparisons of proposed approximations with the ones based on the Laplace transform inversions using a fixed Talbot algorithm as well as on the ones using the Trefethen-Weideman-Schmelzer and maximum entropy methods are presented via a simulation study.

Some results for moment-empirical cumulative distribution functions

An unknown probability cumulative distribution function (CDF) can be recovered from its moments and estimated from its empirical moments. In this paper, some further results for such moment-empirical CDFs’ are considered, in particular for certain models where the sample is not directly drawn from the distribution of actual interest, as in biased sampling.

Recovery of functions from transformed moments: A unified approach

ABSTRACT Two approximations recovering the functions from their transformed moments are proposed. The upper bounds for the uniform rate of convergence are derived. In addition, the comparisons of the estimates of the cumulative distribution function and its density function with the empirical distribution and the kernel density estimates are conducted via a simulation study. The plots of recovered functions are presented for several examples as well.

Recovery of a quantile function from moments

The problem of recovering a quantile function of a positive random variable via the values of moments or given the values of its Laplace transform is studied. Two new approximations as well as two new estimates of a quantile function given the sample from underlying distribution are proposed. The uniform and L 1 upper bounds of proposed estimates are derived. The plots illustrate the behavior of the recovered approximants for the moderate and large sample sizes.

On the moment-recovered approximations of regression and derivative functions with applications

In this paper three formulas for recovering the conditional mean and conditional variance based on product moments are proposed. The upper bounds for the uniform rate of approximations of regression and derivatives of some moment-determinate function are derived. Two cases where the support of underlying functions is bounded and unbounded from above are studied. Based on the proposed approximations, novel nonparametric estimates of the distribution function and its density in multiplicative-censoring model are constructed. Simulation study justifies the consistency of the estimates.

On moment-density estimation in some biased models

It is known from the famous “moment problem” that under suitable conditions a probability distribution can be recovered from its moments. In Mnatsakanov and Ruymgaart [5, 6] an attempt has been made to exploit this idea and estimate a cdf or pdf, concentrated on the positive half-line, from its empirical moments. The ensuing density estimators turned out to be of kernel type with a convolution kernel, provided that convolution is considered on the positive half-line with multiplication as a group operation (rather than addition on the entire real line). This does not seem to be unnatural when densities on the positive half-line are to be estimated; the present estimators have been shown to behave better in the right hand tail (at the level of constants) than the traditional estimator (Mnatsakanov and Ruymgaart [6]). Apart from being an alternative to the usual density estimation techniques, the approach is particularly interesting in certain inverse problems, where the moments of the density of interest are related to those of the actually sampled density in a simple explicit manner. This occurs, for instance, in biased sampling models. In such models the pdf f (or cdf F ) of a positive random variable X is of actual interest, but one observes a random sample Y1, . . . , Yn of n copies of a random variable Y with density