Asymptotic behavior of the reserve of an organization exposed to risk is considered for when the number of factors leading to a loss is random. A specific distribution of losses (the Burr distribution) is considered. The activities of these organizations are compared asymptotically in terms of the required additional number of such factors. Two examples are given to illustrate the obtained results. The first example is related to the maximum loss; the second considers truncated binomial and Poisson distributions that describe the number of random factors leading to losses.

The representability of the Burr distribution as a mixture of Weibull distribution is studied in order to justify its utility for modelling the statistical regularities in extreme values registered in non-stationary flows of informative events. A result of [24] is improved by extending the domain of admissible values of the parameters which provide the representability of the (generalized) Burr distribution as a scale mixture of the Weibull distribution. This result gives an argument in favour of application of the Burr distribution as a model of statistical regularities of extreme values registered within moderate regular time intervals, say, daily (short-term) extremes. In turn, if we are interested in the statistical regularities of the behaviour of the absolute extreme observation over a long period, say, a decade (the long-term extreme), then it can be noted that the daily extreme values form a sample of the Burr-distributed random variables. As is known, the Burr distribution belongs to the domain of max-attraction of the Frechet distribution. The problem of improving the accuracy of the approximation of the distribution of the absolute extreme by the Frechet distribution by the construction of an asymptotic expansion for the distribution of the extreme order statistics in the sample of independent identically Burr-distributed random variables is also considered. These results are illustrated by an example of fitting the Burr distribution to the data representing the extreme values of characteristics of the interplanetary magnetic field.

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized Poisson sum for a given α∈(0,1). This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1−α)-quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution.

In the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized sum for a given α∈(0,1), and second, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the probability for the normalized sum to fall into a given interval. Both problems are solved under the condition that possible distributions of random summands possess coinciding three first moments. In both settings the best distribution delivers the smallest number of summands. Along with distributions of a non-random number of summands, we consider the case of random summation and introduce an analog of deficiency which can be used to compare the distributions of sums with random and non-random number of summands. The main mathematical tools used in the paper are asymptotic expansions for the distributions of R-valued functions of random vectors, in particular, normalized sums of independent identically distributed r.v.s and their quantiles. Along with the general case, main attention is paid to the situation where the summarized random variables are independent and identically distributed. The approach under consideration is applied to determination of the distribution of insurance payments providing the least insurance portfolio size under prescribed Value-at-Risk or non-ruin probability.

An asymptotical approach to the statistical estimation problem is considered under the assumption that the number of observations is a random variable. This leads to distributions with heavy tails and changes in the efficiency of the normally used statistical procedures. Statistical estimates based on random-size and nonrandom-size samples are asymptotically compared against one another. The concept of asymptotic deficiency (the number of additional observations needed for a given estimate to achieve the same quality as the optimum estimate) is used to do so. Asymptotic expansions are also obtained for the risk functions of estimates based on random-size samples.

The purpose of this paper is to use deficiency to compare estimators constructed from samples with random size with those constructed from samples with non-random size. The deficiency can be a characteristic of a possible loss of accuracy of statistical inference if a random-size sample is erroneously regarded as a sample with non-random size. It is heuristically shown that if the asymptotic distribution of the sample size normalized by its expectation is not degenerate, then the deficiency of a statistic constructed from a sample with random size of expectation n with respect to the same statistic constructed as if the sample size were non-random and equal to n, grows almost linearly as n grows. A non-trivial behavior of the deficiency is only possible if the random sample size is asymptotically degenerate. This is the case considered in this paper, where we study the deficiencies of statistics constructed from samples whose sizes have the Poisson, binomial and special three-point distributions. We also give some basic properties of estimators based on samples with random sizes.

In the paper general theorems concerning the asymptotic deficiencies of sample median based on the sample of random size are presented. These results make it possible to compare the quality of the sample median constructed from samples with both random and non-random sizes in terms of additional observations. The cases of the binomial distribution and the distribution concentrated at three points are considered.

The paper contains an introduction to the asymptotic theory of hypothesis testing and a review of recent results of the author and his students. We consider only the asymptotic approach, for which with increasing sample size n the test size is separated from zero, and the sequences of local alternatives, for which the power is separated from one. This paper focuses on asymptotically efficient tests when testing a simple hypothesis in the case of a one-parameter family. We study the difference between the powers of the best and asymptotically efficient tests. This difference is closely related to the notion of asymptotic test deficiency. We consider the formula for the limiting deviation of the power of asymptotically optimal test from the power of the best test in the case of Laplace distribution. Due to the irregularity of the Laplace distribution, this deviation is of order n −1/2, in contrast to the usual regular families for which this order is n −1. We also study the Bayesian settings and the case of increasing parameter dimension.

A class of inhomogeneous Markovian queuing systems with possible catastrophic failures and group arrival of customers in the case of empty queue is considered; basic estimates of the rate of convergence and stability for this class are obtained.

In the paper, we discuss the transformation of the asymptotic distribution of asymptotically normal statistics if the sample size is replaced by a random variable. We also discuss the asymptotic expansions of the distribution function and concentration functions of statistics constructed from samples with random sizes.

Statistical regularities of the information flows in contemporary communication, computational and other information systems are characterized be the presence of the so-called "heavy tails". The outlying observations make the traditional moment-type location estimators inaccurate. In this case the robust median-type location estimators are preferable. On the other hand, the random character of the intensity of the flow of informative events results in that the available sample size (traditionally this is the number of observations registered within a certain time interval) is random. The randomness of the sample size crucially changes the asymptotic properties of the estimators. In the paper, asymptotic expansions are obtained for the distribution function of the sample median constructed from a sample with random size. A general theorem on the asymptotic expansion is proved for this case. The cases of the Laplace, Student and Cauchy distributions are considered. Special attention is paid to the situations in which the heavy-tailed distributions (Cauchy, Laplace) are inherent in both the original sample and the asymptotic regularities of the sample median (Student, Laplace) due to the randomness of the sample size. This approach can be successfully used for big data mining and analysis of information flows in high-performance computing.

Some estimates are presented for the accuracy of approximation of the negative binomial distribution by the gamma distribution. These results are applied to estimate the convergence rate to the Student distribution for the distributions of some statistics constructed from samples with random sizes.

We obtain perturbation bounds for an inhomogeneous Markovian queueing model with group services.

In the paper, the asymptotic behavior of the power function of the most powerful test in the problem of testing a simple hypothesis against a simple alternative from a homogeneous sample of independent observations is studied under the assumption that the likelihood ratio has a heavy-tailed distribution belonging to the domain of attraction of a stable law.

We present a general transfer theorem for random sequences with independent random indexes in the double array limit setting. We also prove its partial inverse providing necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the cases of random sums of independent not necessarily identically distributed random variables and statistics constructed from samples with random sizes. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance-mean mixtures.

We consider time-inhomogeneous Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process. Specific queueing models are shown as examples.

We consider a class of inhomogeneous Markovian queueing models with batch arrivals and group services. Bounds on the truncation errors in weak ergodic case are obtained. Two concrete queueing models are studied as examples.