On the Construction of Minimax Optimal Nonparametric Tests with Kernel Embedding Methods

Kernel embedding methods have witnessed a great deal of practical success in the area of nonparametric hypothesis testing in recent years. But ever since its first proposal, there exists an inevitable problem that researchers in this area have been trying to answer--what kernel should be selected, because the performance of the associated nonparametric tests can vary dramatically with different kernels. While the way of kernel selection is usually ad hoc, we wonder if there exists a principled way of kernel selection so as to ensure that the associated nonparametric tests have good performance. As consistency results against fixed alternatives do not tell the full story about the power of the associated tests, we study their statistical performance within the minimax framework. First, focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel embedding based test could be suboptimal, and suggest a simple remedy by moderating the kernel. We prove that the moderated approach provides optimal tests for a wide range of deviations from the null and can also be made adaptive over a large collection of interpolation spaces. Then, we study the asymptotic properties of goodness-of-fit, homogeneity and independence tests using Gaussian kernels, arguably the most popular and successful among such tests. Our results provide theoretical justifications for this common practice by showing that tests using a Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings. In addition, our analysis also pinpoints the importance of choosing a diverging scaling parameter when using Gaussian kernels and suggests a data-driven choice of the scaling parameter that yields tests optimal, up to an iterated logarithmic factor, over a wide range of smooth alternatives. Numerical experiments are presented to further demonstrate the practical merits of our methodology

The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose single tests whose test statistics are U-statistics based on general kernel functions. The corresponding critical values are constructed from a non-asymptotic wild bootstrap approach, leading to level \alpha tests. Various choices for the kernel functions are possible, including projection, approximation or reproducing kernels. In this last case, we obtain a parametric rate of testing for a weak metric defined in the RKHS associated with the considered reproducing kernel. Then we introduce, in the other cases, an aggregation procedure, which allows us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. The resulting multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic oracle type conditions for the classical L2-norm. From these conditions, we deduce that they are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak Besov bodies in the univariate case, but also Sobolev and anisotropic Nikol'skii-Besov balls in the multivariate case

New Estimation Approaches for the Hierarchical Linear Ballistic Accumulator Model

The Linear Ballistic Accumulator (Brown & Heathcote, 2008) model is used as a measurement tool to answer questions about applied psychology. The analyses based on this model depend upon the model selected and its estimated parameters. Modern approaches use hierarchical Bayesian models and Markov chain Monte-Carlo (MCMC) methods to estimate the posterior distribution of the parameters. Although there are several approaches available for model selection, they are all based on the posterior samples produced via MCMC, which means that the model selection inference inherits the properties of the MCMC sampler. To improve on current approaches to LBA inference we propose two methods that are based on recent advances in particle MCMC methodology; they are qualitatively different from existing approaches as well as from each other. The first approach is particle Metropolis-within-Gibbs; the second approach is density tempered sequential Monte Carlo. Both new approaches provide very efficient sampling and can be applied to estimate the marginal likelihood, which provides Bayes factors for model selection. The first approach is usually faster. The second approach provides a direct estimate of the marginal likelihood, uses the first approach in its Markov move step and is very efficient to parallelize on high performance computers. The new methods are illustrated by applying them to simulated and real data, and through pseudo code. The code implementing the methods is freely available.Comment: 35 pages, 6 figures, 7 table

Data-driven rate-optimal specification testing in regression models

We propose new data-driven smooth tests for a parametric regression function. The smoothing parameter is selected through a new criterion that favors a large smoothing parameter under the null hypothesis. The resulting test is adaptive rate-optimal and consistent against Pitman local alternatives approaching the parametric model at a rate arbitrarily close to 1/\sqrtn. Asymptotic critical values come from the standard normal distribution and the bootstrap can be used in small samples. A general formalization allows one to consider a large class of linear smoothing methods, which can be tailored for detection of additive alternatives.Comment: Published at http://dx.doi.org/10.1214/009053604000001200 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Kernel alternatives to aproximate operational severity distribution: an empirical application

The estimation of severity loss distribution is one the main topic in operational risk estimation. Numerous parametric estimations have been suggested although very few work for both high frequency small losses and low frequency big losses. In this paper several estimation are explored. The good performance of the double transformation kernel estimation in the context of operational risk severity is worthy of a special mention. This method is based on the work of Bolancé and Guillén (2009), it was initially proposed in the context of the cost of claims insurance, and it means an advance in operational risk research

Nonparametric Kernel Testing in Semiparametric Autoregressive Conditional Duration Model

A crucially important advantage of the semiparametric regression approach to the nonlinear autoregressive conditional duration (ACD) model developed in Wongsaart et al. (2011), i.e. the so-called Semiparametric ACD (SEMI-ACD) model, is the fact that its estimation method does not require a parametric assumption on the conditional distribution of the standardized duration process and, therefore, the shape of the baseline hazard function. The research in this paper complements that of Wongsaart et al. (2011) by introducing a nonparametric procedure to test the parametric density function of ACD error through the use of the SEMI-ACD based residual. The hypothetical structure of the test is useful, not only to the establishment of a better parametric ACD model, but also to the specification testing of a number of financial market microstructure hypotheses, especially those related to the information asymmetry in finance. The testing procedure introduced in this paper differs in many ways from those discussed in existing literatures, for example Aït-Sahalia (1996), Gao and King (2004) and Fernandes and Grammig (2005). We show theoretically and experimentally the statistical validity of our testing procedure, while demonstrating its usefulness and practicality using datasets from New York and Australia Stock Exchange.Duration model, hazard rates and random measures, nonparametric kernel testing.