Abstract
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.
Highlights
We propose an approach to establishing stochastic order for the distributions of sums of independent random variables (r.v.s) based on the notion of deficiency that is well known in asymptotic statistics, see, e.g., [2] and later publications [3,4,5]
The paper deals with an approach to the comparison of distributions of sums of a finite number of independent random variables by deficiency
The formulas for the asymptotic deficiency were obtained as the solution of two problems, one of which deals with the description of the distribution of a separate summand minimizing the number of summands and providing a prescribed value of the (1 − α)quantile of the normalized sum for a given α ∈ (0, 1)
Summary
The problem considered in the paper is very close to the problem of stochastic ordering and even may be considered as a a version of this problem. The first problem to be considered below consists in determination of 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). The second problem considered in the paper consists in determination of 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. The distribution of a random vector (X1, . . . , Xn) will be denoted L(X1, . . . , Xn)
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