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Том 25, № 2 (2016)
- Год: 2016
- Статей: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13888
Article
Operator-based intensity functions for the nonhomogeneous Poisson process
Аннотация
A nonhomogeneous Poisson process (NHPP) plays an important role in a variety of applications as reliability of repairable systems, software reliability and actuarial studies. An NHPP is characterized by its intensity function m(t), which provides information on the time-dependent nature of the reliability of the system. Various intensity functions, which describe different behavior (from reliability decay to reliability growth along with monotonicity, convexity or concavity), have been suggested for NHPP’s for modeling repairable systems. Perhaps one of the most frequently utilized NHPP is the power lawprocess (PLP) in which m(t) is a power function of t. Inthis studywe present a general method for constructing new intensity functions for NHPP’s yielding new classes of NHPP’s. This method utilizes certain operators Ln, n ∈ N0, acting on some suitable functions L0 = f (termed base functions). We call these classesOBIF’s (operator-based intensity functions). These classes are represented in terms of three parameters of which one is an indexing parameter n ∈ N0 and two others are scale and shape parameters. The fact that n ∈ N0 is also a parameter provides a flexibility in the choice of the appropriate statistical model for NHPP’s data. In particular, we consider the exponential operator acting on the PLP intensity function f and realize that Ln’s, n ≥ 2, inherit properties similar to those of L1 (convexity and concavity) and thus are suitable for modelling bathtub data. We also consider a more comprehensive treatment of OBIF classes where both, the operator and base functions, are general. All of the introduced operators are demonstrated with illustrative examples.
79-98
On score-functions and goodness-of-fit tests for stochastic processes
Аннотация
The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.
99-120
On testing sphericity and identity of a covariance matrix with large dimensions
Аннотация
Tests for certain covariance structures, including sphericity, are presented when the data may be high-dimensional but not necessarily normal. The tests are formulated as functions of location-invariant estimators defined as U-statistics of higher order kernels. Under a few mild assumptions, the limit distributions of the tests are shown to be normal. The accuracy of the tests is demonstrated by simulations.
121-132
Risk-optimal estimators for survey procedures with certain indirect questions
Аннотация
Surveys usually consist of a list of direct questions. However respondents reluctantly provide direct information on sensitive topics such as socially undesired behavior (e.g., social fraud, discrimination, tax evasion), income or political preferences. For this reason, the diagonal technique (DT), an indirect questioning procedure has been proposed in the literature. In this paper, we consider multiple categorical target variables where all or some of the variables are gathered by the DT. The maximum likelihood (ML) estimator for the joint distribution depends on the setup of the survey procedure, i.e., on certain parameters to adjust. We conduct a decision-theoretic analysis and derive risk-optimal ML estimators. The special point of our investigation is the incorporation of the degree of privacy protection (DPP). In particular, in the class of ML estimators corresponding to a given DPP, we detect an estimator with the lowest risk, i.e., with the highest quality.
133-144
On likelihood ratio ordering of parallel systems with exponential components
Аннотация
Let T(λ1,...,λn) be the lifetime of a parallel system consisting of exponential components with hazard rates λ1,...,λn, respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ1, λ2) majorizes (γ1, γ2), then, T(λ1, λ2) is larger than T(γ1, γ2) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ1,...,λn) is d-larger than (γ1,...,γn), then T(λ1,...,λn) is larger than T(γ1,...,γn) in likelihood ratio order.
145-150