Modern research directions and some new results related to statistical estimates of the principle information characteristics are indicated. Independent, identically distributed observations are used to construct such estimates.
We consider the problem of approximating functions of one variable within the class of nonlinear approximations with a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a given function that depends nonlinearly on the second parameter. The computational algorithm is based on residual minimization in a Hilbert space. The first key point of the developed approach is related to the actual transition to a standard linear problem of best approximation of functions when the second parameter is given on an extended set of points within the interval of permissible values. The second key point consists in determining the set of linear approximation parameters at each separate iteration of the classical nonnegative least squares (NNLS) method. Numerical results are presented that illustrate the capabilities of this computational algorithm for nonlinear approximation of functions.
Asymptotic behavior of test powers for random-size samples is considered in the problem of testing a simple hypothesis concerning a univariate parameter against a sequence of closely related alternatives. The concept of test power is introduced in this case. Specific tests are asymptotically compared (for normal samples) using the concept of deficiency, which is the additional number of observations required for a competing test to asymptotically achieve the power of the best test. Two examples are considered to illustrate the results obtained. The first example concerns a truncated binomial distribution, and the second example considers a truncated Poisson distribution. These distributions describe random sample sizes.
An initial-boundary value problem for an evolution equation with involution in the spatial variable and a nonlocal boundary condition is studied. The uniqueness of the solution is established, and classes of initial-value functions that ensure its existence and stability are described.
A nonlinear, two-phase mathematical model of production at a mining facility is proposed. In this model, material flows are represented as two phases (ore and the useful substance contained within it), linked through the concentration of the useful substance. The process of material balance calculation is studied, which represents the solution to an optimization problem with nonlinear constraints at nodes with product accumulation. The solution to this problem is proposed to be sought using the interior point method. Known measured values of useful substance concentrations are considered, as well as the results of indirect measurements, ranges of flow values, stockpiles, and unknown concentration values. The results of test calculations for material balance calculations at a small section of a mining facility are presented.
