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Том 225, № 5 (2017)
- Год: 2017
- Статей: 11
- URL: https://bakhtiniada.ru/1072-3374/issue/view/14853
Article
On Outliers Detection for Location-Scale and Shape-Scale Families
Аннотация
The problem of multiple upper outliers detection in samples from location-scale and shape-scale families is considered. A new test statistic is proposed. The critical values of the new test statistic are tabulated by simulation. The power of the new test and other available tests are compared by simulation.
723-732
Probabilistic Representations and Numerical Algorithms for Classical and Viscosity Solutions of the Cauchy Problem for Quasilinear Parabolic Systems
Аннотация
We propose two approaches which allow us to construct probabilistic representations of classical and viscosity solutions of the Cauchy problem for a system of quasilinear parabolic equations. Based on these representations, we develop two numerical algorithms to construct the required solution. The system under consideration arises as a mathematical model of parabolic conservation laws. Bibliography: 14 titles.
733-750
On Consistent Hypothesis Testing
Аннотация
We study natural links between various types of consistency: usual consistency, strong consistency, uniform consistency, and pointwise consistency. On the base of these results, we provide both sufficient conditions and necessary conditions for the existence of various types of consistent tests for a wide spectrum of problems of hypothesis testing which arise in statistics: on a probability measure of an independent sample, on a mean measure of a Poisson process, on a solution of an ill-posed linear problem in a Gaussian noise, on a solution of the deconvolution problem, for signal detection in a Gaussian white noise. In the last three cases, the necessary and sufficient conditions coincide.
751-769
Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables
Аннотация
An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. We also show that the mean width of the regular simplex with 2n vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. We establish several formulas conjectured by S. Finch on the projection length W of the regular cube, simplex, and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for W as the dimension of the regular polytope goes to ∞. Bibliography: 25 titles.
770-787
788-790
791-801
802-804
Tightness of Sums of Independent Identically Distributed Pseudo-Poisson Processes in the Skorokhod Space
Аннотация
We consider a pseudo-Poisson process of the following simple type. This process is a Poissonian subordinator for a sequence of i.i.d. random variables with finite variance. Further we consider sums of i.i.d. copies of a pseudo-Poisson process. For a family of distributions of these random sums, we prove the tightness (relative compactness) in the Skorokhod space. Under the conditions of the Central Limit Theorem for vectors, we establish the weak convergence in the functional Skorokhod space of the examined sums to the Ornstein–Uhlenbeck process.
805-811
On Stochastic Algorithms for Solving Boundary-Value Problems for the Laplace Operator
Аннотация
The paper deals with the mixed boundary-value problem for the Poisson equation. Random walks inside the domain are constructed and unbiased estimators for the solution of the boundary-value problem are defined on their trajectories. The finiteness of the estimator variance is proved. The possibility of practical realization of the estimators by the Monte Carlo method is studied.
812-817
On an Interval of Faultless Work for a System of Two Independent Alternating Renewal Processes
Аннотация
We consider a system of two independent alternating renewal processes with states 0 and 1 and an initial shift t0 of one process relative to the other one. An integral equation with respect to the expectation of time T (the first time when both processes have state 0) is derived. To derive this equation, we use the method of so-called minimal chains of overlapping 1-intervals. Such a chain generates some breaking semi-Markov process of intervals composing the interval (0, T ). A solution of the integral equation is obtained for the case where the lengths of 1-intervals have exponential distributions and lengths of 0-intervals have arbitrary distributions. For more general distributions of 1-intervals, the Monte Carlo method is applied when both processes are simulated numerically by a computer. A histogram for estimates of the expectation of T as a function of t0 is demonstrated. Bibliography: 4 titles.
818-832
Large Deviations for Sums of Bounded Functions of a Normalized Sample Under Gamma Distribution
Аннотация
We study large deviations of a widely used class of scale-free statistics under gamma distribution. We show that the constraints on the functions defining these statistics can be relaxed with respect to the previously obtained result. The result is applied to a recent exponentiality test. Bibliography: 7 titles.
833-840