Approximations for the boundary crossing probabilities of moving sums of normal random variables

In this paper we study approximations for boundary crossing probabilities for the moving sums of i.i.d. normal random variables. We propose approximating a discrete time problem with a continuous time problem allowing us to apply developed theory for stationary Gaussian processes and to consider a number of approximations (some well known and some not). We bring particular attention to the strong performance of a newly developed approximation that corrects the use of continuous time results in a discrete time setting. Results of extensive numerical comparisons are reported. These results show that the developed approximation is very accurate even for small window length

Covering of high-dimensional cubes and quantization

As the main problem, we consider covering of a d-dimensional cube by n balls with reasonably large d (10 or more) and reasonably small n, like n = 100 or n = 1000. We do not require the full coverage but only 90% or 95% coverage. We establish that efficient covering schemes have several important properties which are not seen in small dimensions and in asymptotical considerations, for very large n. One of these properties can be termed ‘do not try to cover the vertices’ as the vertices of the cube and their close neighbourhoods are very hard to cover and for large d there are far too many of them. We clearly demonstrate that, contrary to a common belief, placing balls at points which form a low-discrepancy sequence in the cube, results in a very inefficient covering scheme. For a family of random coverings, we are able to provide very accurate approximations to the coverage probability. We then extend our results to the problems of coverage of a cube by smaller cubes and quantization, the latter being also referred to as facility location. Along with theoretical considerations and derivation of approximations, we provide results of a large-scale numerical investigation

Covering of high-dimensional sets

Let $(\mathcal{X},\rho)$ be a metric space and $\lambda$ be a Borel measure on this space defined on the $\sigma$-algebra generated by open subsets of $\mathcal{X}$; this measure $\lambda$ defines volumes of Borel subsets of $\mathcal{X}$. The principal case is where $\mathcal{X} = \mathbb{R}^d$, $\rho$ is the Euclidean metric, and $\lambda$ is the Lebesgue measure. In this article, we are not going to pay much attention to the case of small dimensions $d$ as the problem of construction of good covering schemes for small $d$ can be attacked by the brute-force optimization algorithms. On the contrary, for medium or large dimensions (say, $d\geq 10$), there is little chance of getting anything sensible without understanding the main issues related to construction of efficient covering designs

Online change-point detection for a transient change

We consider a popular online change-point problem of detecting a transient change in distributions of independent random variables. For this change-point problem, several change-point procedures are formulated and some advanced results for a particular procedure are surveyed. Some new approximations for the average run length to false alarm are offered and the power of these procedures for detecting a transient change in mean of a sequence of normal random variables is compared

Approximations of the boundary crossing probabilities for the maximum of moving weighted sums

We study approximations of boundary crossing probabilities for the maximum of moving weighted sums of i.i.d. random variables. We consider a particular case of weights obtained from a trapezoidal weight function which, under certain parameter choices, can also result in an unweighted sum. We demonstrate that the approximations based on classical results of extreme value theory provide some scope for improvement, particularly for a range of values required in practical applications