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In this paper, the stochastic finite‐time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode‐dependent limiting average dwell time (MD‐LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite‐time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results. [ABSTRACT FROM AUTHOR]

Bias approximation has played an important rôle in statistical inference, and numerous bias calculation techniques have been proposed under different contexts. We provide a unified approach to approximating the bias of the maximum likelihood estimator and the l2 penalized likelihood estimator for both linear and nonlinear models, where the design variables are allowed to be random and the sample size can be a stopping time. The proposed method is based on the Woodroofe–Stein identity and is justified by very weak approximations. The accuracy of the derived bias formulas is assessed by simulation for several examples. The bias of the ridge estimator in high‐dimensional settings is also discussed. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic properties of the stopping times based on the monitoring tests for a bubble, that is, ADF‐type and CUSUM‐type tests. We derive the stochastic orders of these stopping times and find that they depend on the relative position of the emerging date of a bubble. We show that when a bubble emerges early in the monitoring period, the diverging rate of the stopping time based on the CUSUM‐type detector is slower than that based on the ADF‐type detector, which implies that the former tends to find a bubble earlier than the latter, whereas this advantage disappears for the late emerging bubble. We also derive the limiting distribution of the stopping time based on the CUSUM detector. The finite sample simulations support the theoretical result. [ABSTRACT FROM AUTHOR]

It is well known that adaptive sequential nonparametric estimation of differentiable functions with assigned mean integrated squared error and minimax expected stopping time is impossible. In other words, no sequential estimator can compete with an oracle estimator that knows how many derivatives an estimated curve has. Differentiable functions are typical in probability density and regression models but not in spectral density models, where considered functions are typically smoother. This paper shows that for a large class of spectral densities, which includes spectral densities of classical autoregressive moving average processes, an adaptive minimax sequential estimation with assigned mean integrated squared error is possible. Furthermore, a two-stage sequential procedure is proposed, which is minimax and adaptive to smoothness of an underlying spectral density. [ABSTRACT FROM AUTHOR]

This paper introduces a mathematical model for a currently popular financial product called a stock loan. Quantitative analysis is carried out to establish explicitly the value of such a loan, as well as the ranges of fair values of the loan size and interest, and the fee for providing such a service. [ABSTRACT FROM AUTHOR]

Abstract A sequential procedure is constructed to provide a fixed-accuracy estimator for the number of faults in a system. This paper focuses on the case when faults are homogeneous. However, the method can be adapted to other models by choosing a more robust estimator. The accuracy of the estimator depends on the failure intensity, the length of testing period and the total number of faults in the system. Simulations illustrate the performance of the proposed procedure. The method is applied to an information system failure dataset. [ABSTRACT FROM AUTHOR]

The typical approach in change–point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. Based on the, perhaps, more realistic situation that the process can only be partially observed, we consider the counting process related to the original process observed at equidistant time points, after which action is taken or not depending on the number of observations between those time points. In order for the procedure to stop also when everything is in order, we introduce a fixed time horizon n at which we stop declaring “no change” if the observed data did not suggest any action until then. We propose some stopping rules and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes and extreme value asymptotics for Gaussian processes. [ABSTRACT FROM AUTHOR]