The problem of optimal trajectory planning of the unmanned underwater vehicle (UUV) is considered and analytically solved. The task is to minimize the risk of detection of the moving object by a static sonar while moving between two given points on a plane. The detection is based on the primary acoustic field radiated by the object with a non-uniform radiation pattern. In the first part of the article, the probability of non-detection is derived. Further, it is used as an optimization criterion. The non-uniform radiation pattern of the object differentiates this work from previous research in the area. The optimal trajectory and velocity law of the moving object are found, as well as the criterion value on it.

We consider nonparametric estimation of the derivative of a probability density function with the bounded support on [0, ∞). Estimates are looked up in the class of estimates with asymmetric gamma kernel functions. The use of gamma kernels is due to the fact they are nonnegative, change their shape depending on the position on the semi-axis and possess other good properties. We found analytical expressions for bias, variance, mean integrated squared error (MISE) of the derivative estimate. An optimal bandwidth, the optimal MISE, and rate of mean square convergence of the estimates for density derivative have also been found.

In some applications it is necessary to estimate derivatives of probability densities defined on the positive semi-axis. The quality of nonparametric estimates of the probability densities and their derivatives are strongly influenced by smoothing parameters (bandwidths). In this paper an expression for the optimal smoothing parameter of the gamma kernel estimate of the density derivative is obtained. For this parameter data-driven estimates based on methods called "rule of thumb" are constructed. The quality of the estimates is verified and demonstrated on examples of density derivatives generated by Maxwell and Weibull distributions.

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