Zhanna N. Zenkova - Associate Professor, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University; SRO “Association of professional actuaries”
Address: 36, Lenin Street, Tomsk, 63403047, Russian Federation  
E-mail: zhanna.zenkova@mail.tsu.ru

Elizaveta A. Krainova - Doctoral Student, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University;
Address: 36, Lenin Street, Tomsk, 63403047, Russian Federation  
E-mail: lanshakoval@gmail.com

      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 q-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 q-quantile, and then the modified empirical cumulative distribution function is substituted into the integral that yields the mean value. This allows us to obtain a modified estimation of mean value using additional information about the q-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 modification is also smaller. Therefore, the modified estimation is more accurate than the classical one for a large sample size.
      The influence 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 significantly than for symmetrical ones.
      The modified 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 modified technique, the net premium becomes higher and the bankruptcy risk is reduced as well.
      This paper contains practically significant results which make it possible to give important recommendations to an insurance company.