No abstract

In this paper, the task of increasing the accuracy of net premium estimations in non-life insurance is considered. Improvements are achieved by involving additional information about a known quantile of loss cumulative distribution function. The additional information is used by projection the empirical cumulative distribution function onto the class of cumulative distribution functions with a certain quantile, and then the modifi ed empirical cumulative distribution function is substituted into the integral that yields the mean value. This allows us to obtain a modifi ed estimation of mean value using additional information about the quantile which is unbiased and its variance is asymptotically less than the variance of the classical sample mean, so that the mean-square error of the modifi cation is also smaller. Therefore, the modifi ed estimation is more accurate than the classical one for a large sample size.The infl uence of a quantile value on the variance of the new estimation is studied for uniform, triangular and normal distributions. It is suggested that the minimum of the variance is reached when a known quantile is equal to the median (symmetry center) for symmetrical distribution. Based on Simpson triangular distribution, it was shown that for cases of skewed distributions involving the quantile allows one to decrease the variance more signifi cantly than for symmetrical ones.The modifi ed estimation of mean value is applied to a real data set for calculation of a net premium. The data contain information about payments for voluntary health insurance of some insurance company. It is demonstrated that the classical method underestimates the net premium, and so it could lead to the company's bankruptcy. After applying the new modifi ed technique, the net premium becomes higher and the bankruptcy risk is reduced as well.This paper contains practically signifi cant results which make it possible to give important recommendations to an insurance company.

The aim of this research is to determine optimal methods of using stimulants in a pre-sowing treatment of common pine seeds to increase their germination and obtain high-quality planting material. A study has been performed in forest nurseries of the State Forest Natural Reserve (SFNR) “Ertis Ormany” (Pavlodar region) and the Arykbalyk Branch of the State National Natural Park (SNNP) “Kokshetau” (Akmola region). The research is done with one-and two-year-old seedlings of ordinary pine sowing of 2017. A presowing seed preparation is carried out according to six options by using various types of stimulants. A method of mathematical processing of the thus obtained measurements of the seedlings with an insufficient number of measured plants is proposed by using a bootstrap analysis with and without quantile information. It allows one to estimate the influence of stimulants on the average growth of one-and two-year-old seedlings at a reliable level. As a result of these studies, it has been revealed that the pre-sowing treatment of pine seeds with ordinary stimulants increases the average height of annual seedlings. However, the results obtained in this experiment depend on the region of nursery location, but the use of soil irrigation with an “EridGrow” activator has increased the average height in both nurseries: by 13.9% in the “Kokshetau” SNNP for one-year-old seedlings, by 37.2% for two-year-old seedlings compared with the average height of the control seedlings. For pine seedlings in the “Ertis Ormany” SFNR nursery the influence of the stimulants is insignificant, but the positive effect on the average growth with the “EridGrow” soil irrigation is a 7.3% increase in the first year and a 24.7% one in the second year.

The newsvendor problem is a popular inventory management problem in supply chain management and logistics. Solutions to the newsvendor problem determine optimal inventory levels. This model is typically fully determined by a purchase and sale prices and a distribution of random market demand. From a statistical point of view, this problem is often considered as a quantile estimation of a critical fractile which maximizes anticipated profit. The distribution of demand is a random variable and is often estimated on historic data. In an ideal situation, when the probability distribution of the demand is known, one can determine the quantile of a critical fractile minimizing a particular loss function. Since maximum likelihood estimation is asymptotically efficient, under certain regularity assumptions, the maximum likelihood estimators are used for the quantile estimation problem. Then, the Cramer-Rao lower bound determines the lowest possible asymptotic variance. Can one find a quantile estimate with a smaller variance then the Cramer-Rao lower bound? If a relevant additional information is available then the answer is yes. Additional information may be available in different forms. This manuscript considers minimum variance and minimum mean squared error estimation for incorporating additional information for estimating optimal inventory levels. By a more precise assessment of optimal inventory levels, we maximize expected profit.

No abstract