Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

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

Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

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

Higher Order Asymptotic Optimality in Testing Problems With Nuisance Parameters

We consider testing hypotheses about the location parameter of a symmetric distribution when a finite-dimensional nuisance parameter is present. For local alternatives, we study the power loss of asymptotically efficient tests in this problem, which is the difference between the power of the most powerful test for a given value of the nuisance parameter (as if it were known) and the power of the test at hand. The power loss is typically of order n \Gamma1 and is closely related to the deficiency of the test. In particular, we obtain the lower bound for the power loss in a locally asymptotically minimax sense similar to that used in the estimation theory and indicate a test on which this bound is attained. This bound corresponds to the envelope power function obtained by Pfanzagl and Wefelmeyer (1978) for test statistics of a specific structure

Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case

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

Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case

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

Stability bounds for M t /M t /N/N + R queue

ABSTRACT We study Mt/Mt/N/N +R queue and obtain stability bounds for main characteristics of the respective queue-length process