Delta Method,Moment Convergence,and Inference

Statistics, as functions of the observations, are usually given by well-behaved functions. This fact is used to obtain limit distributions for statistics whose components are given by asymptotically linear functions. These results are then extended to the moments of distributions, covariance matrices and confidence regions for parameters of interest. These regions may be used to test, through duality, hypothesis on these parameters. A theoretical application is presented.     

Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling

In this article, we revisit the importance of the generalized jackknife in the construction of reliable semi-parametric estimates of some parameters of extreme or even rare events. The generalized jackknife statistic is applied to a minimum-variance reduced-bias estimator of a positive extreme value index—a primary parameter in statistics of extremes. A couple of refinements are proposed and a simulation study shows that these are able to achieve a lower mean square error. A real data illustration is also provided.     

Quadratic perturbation expansions of certain functions of eigenvalues and eigenvectors and their application to sensitivity analysis in multivariate methods

Tanaka (1988) lias derived the influence functions, which are equivalent to the perturbation expansions up to linear terms, of two functions of eigenvalues and eigenvectors of a real symmetric matrix, and applied them to principal component analysis. The present paper deals with the perturbation expansions up to quadratic terms of the same functions and discusses their application to sensitivity analysis in multivariate methods, in particular, principal component analysis and principal factor analysis. Numerical examples are given to show how the approximation improves with the quadratic terms.     

Waiting time to exhaust lottery numbers

Questions related to lotteries are usually of interest to the public since people think there is a magic formula which will help them to win lottery draws. This note shows how to compute the expected waiting time to observe specific numbers in a sequence of lottery draws and show that surprising facts are expected to occur.     

A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.     

Nonparametric product partition models for multiple change-points analysis

ABSTRACT

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data.     

Strong consistency of presmoothed Kaplan–Meier integrals when covariables are present

It is known that the Kaplan–Meier estimation may be improved via presmoothing methods. In this article, we introduce an extended presmoothed Kaplan–Meier estimator in the presence of covariates. The main result is the strong consistency of general empirical integrals based on such an estimator. As applications, one can obtain a consis-tent multivariate empirical distribution under censoring, and also can obtain a consistent estimation of regression parameters. We illustrate the new estimation methods through simulations and real data analysis.     

On time-irreversibility and other non-linear features in time series

Time-irreversibility, asymmetry of the distribution, and the occurrence of sudden bursts are considered, amongst others, as non-linear features in time series modeling. The implication is often made that time series showing these features must be analyzed using non-linear models. In contrast, this paper shows that time-irreversible asymmetric time series showing certain types of sudden bursts may be generated by linear models with adequate input sequences. Thus some non-linear time series features may be caused by the pattern in the input sequence rather than by non-linearity in the model. Examples are considered to illustrate the situation.