Introducing the L2,w space for building the projective estimation of probability density function

The task of recovering probability density function of continuous random variable from finite independent sample is considered in the paper. The author investigates the building of projective estimation of probability density function in the case when probability density f ( x ) is not square integrable, i. e. function f ( x ) is outside of the functional Hilbert space L 2. In this case a convergence condition of density estimation to true density doesn’t hold even with using optimal coefficients. Probability density functions, which is outside the L 2 space, occurs even in model distributions, for example, in chi-square distribution with number of freedom k = 1. For solving this task one introduces an L 2, w functional space, which is expansion of the L 2 space. Properties of the introduced space are investigated in the paper. One shows that for any positive Lebesgue measurable function w ( x ) it is also Hilbert. Moreover, discernibility of elements remains true in expansion from L 2 to L 2, w . A statement that probability density function of any continuous random variable belongs to some L 2, w space is proved. Besides, establishing separability of the introducing space is found important, because only in this case sequence of projective estimations converges to true density. The author proved that any space of L 2, w kind, which contains L 2, is separable, so it is possible to build projective estimation of probability density function in this space. Obtained theoretical results were tested on series of numerical experiments. Results are included in the paper. This paper contains the results which are about estimating of probability density function of chi-square distributed random variable and also variate which has probability density that is outside L 2 and contains two points where it converges to +¥. The results let us make a conclusion that suggested method can be used in probability density function estimating even in cases when that density is outside L 2.