A uniform L1L^1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

We develop a new $L^1$ law of large numbers where the $i$-th summand is given by a function $h(\cdot)$ evaluated at $X_i - \theta_n$, and where $\theta_n \circeq \theta_n(X_1,X_2,\ldots,X_n)$ is an estimator converging in probability to some parameter $\theta\in \mathbb{R}$. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between $\theta$ and another consistent estimator $\theta_n^{\star}$. Our main contribution is the treatment of the case where $|h|$ blows up at $0$, which is not covered by standard uniform laws of large numbers.Comment: 10 pages, 1 figur

A study of seven asymmetric kernels for the estimation of cumulative distribution functions

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.Comment: 38 pages, 2 tables, 9 figure

PoweR: A Reproducible Research Tool to Ease Monte Carlo Power Simulation Studies for Goodness-of-fit Tests in R

The PoweR package aims to help obtain or verify empirical power studies for goodnessof-fit tests for independent and identically distributed data. The current version of our package is only valid for simple null hypotheses or for pivotal test statistics for which the set of critical values does not depend on a particular choice of a null distribution (and on nuisance parameters) under the non-simple null case. We also assume that the distribution of the test statistic is continuous. As a reproducible research computational tool it can be viewed as helping to simply reproducing (or detecting errors in) simulation results already published in the literature. Using our package helps also in designing new simulation studies. The empirical levels and powers for many statistical test statistics under a wide variety of alternative distributions can be obtained quickly and accurately using a C/C++ and R environment. The parallel package can be used to parallelize computations when a multicore processor is available. The results can be displayed using LATEX tables or specialized graphs, which can be directly incorporated into a report. This article gives an overview of the main design aims and principles of our package, as well as strategies for adaptation and extension. Hands-on illustrations are presented to help new users in getting started

A comprehensive empirical power comparison of univariate goodness-of-fit tests for the Laplace distribution

In this paper, we do a comprehensive survey of all univariate goodness-of-fit tests that we could find in the literature for the Laplace distribution, which amounts to a total of 45 different test statistics. After eliminating duplicates and considering parameters that yield the best power for each test, we obtain a total of 38 different test statistics. An empirical power comparison study of unmatched size is then conducted using Monte Carlo simulations, with 400 alternatives spanning over 20 families of distributions, for various sample sizes and confidence levels. A discussion of the results follows, where the best tests are selected for different classes of alternatives. A similar study was conducted for the normal distribution in Rom\~ao et al. (2010), although on a smaller scale. Our work improves significantly on Puig & Stephens (2000), which was previously the best-known reference of this kind for the Laplace distribution. All test statistics and alternatives considered here are integrated within the PoweR package for the R software.Comment: 37 pages, 1 figure, 20 table

Temporal and Spatial Independent Component Analysis for fMRI Data Sets Embedded in the AnalyzeFMRI R Package

For statistical analysis of functional magnetic resonance imaging (fMRI) data sets, we propose a data-driven approach based on independent component analysis (ICA) implemented in a new version of the AnalyzeFMRI R package. For fMRI data sets, spatial dimension being much greater than temporal dimension, spatial ICA is the computationally tractable approach generally proposed. However, for some neuroscientific applications, temporal independence of source signals can be assumed and temporal ICA becomes then an attractive exploratory technique. In this work, we use a classical linear algebra result ensuring the tractability of temporal ICA. We report several experiments on synthetic data and real MRI data sets that demonstrate the potential interest of our R package

Understanding convergence concepts: A visual-minded and graphical simulation-based approach

This paper describes the difficult concepts of convergence in probability, almost sure convergence, convergence in law and in r-th mean using a visual-minded and a graphical simulation-based approach. For this purpose, each probability of events is approximated by a frequency. An R package is available on CRAN which reproduces all the experiments done in this paper

Tests for circular symmetry of complex-valued random vectors

We propose tests for the null hypothesis that the law of a complex-valued random vector is circularly symmetric. The test criteria are formulated as $L^2$-type criteria based on empirical characteristic functions, and they are convenient from the computational point of view. Asymptotic as well as Monte-Carlo results are presented. Applications on real data are also reported. An R package called CircSymTest is available from the authors

Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution $F$ (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of $F$). This allows us to illustrate the extent of the 'failure' of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer $K$, new $K$-tuplewise independent sequences that are not mutually independent. For $K \geq 4$, it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.Comment: 15 pages, 5 figures, 1 tabl