Nonparametric Statistics

A new approach to estimation of the length-weight relationship of Pollicipes pollicipes (Gmelin, 1789) on the Atlantic coast of Galicia (Northwest Spain): Some aspects of its biology and management

by Marta Sestelo

This study was undertaken using data drawn from 5 sites along the Atlantic shoreline of Galicia (Northwest Spain) for... more This study was undertaken using data drawn from 5 sites along the Atlantic shoreline of Galicia (Northwest Spain) for a period of 2 y. The length–weight relationship of Pollicipes pollicipes (Gmelin, 1789) was estimated to observe the way in which individuals of this species gain weight as they increase in size. A classic allometric model was used for the purpose. As an alternative, a more general nonparametric model was also estimated, using local linear kernel smoothers. Comparison of these two models showed that use of the nonparametric model resulted in a better fit of the data. In addition, derivatives were used for estimating a size of capture for this species. For the same purpose, we also estimated this crustacean’s mean size at sexual maturation (L50) and the number of broods that it spawns per annum. Individuals! weight gain, a female maturity size of 15.7 mm, and P. pollicipes! estimated 1.73 broods per annum tend to suggest a size of capture based on a rostrocarinal length of 21.50 mm.

On the Grenander estimator at zero

by Marios Pavlides

Statistica Sinica 21 (2011), 873-899

We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the... more We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density $f_0$ is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations by use of tools from convex analysis. The theory is applied to a problem involving mixtures.

Nonparametric estimation of multivariate scale mixtures of uniform densities

by Marios Pavlides

Journal of Multivariate Analysis, Volume 107, May 2012, Pages 71–89

Suppose that $\boldsymbol{U} = (U_1, \ldots , U_d) $ has a Uniform$([0,1]^d)$ distribution, that $\boldsymbol{Y} =... more Suppose that $\boldsymbol{U} = (U_1, \ldots , U_d) $ has a Uniform$([0,1]^d)$ distribution, that $\boldsymbol{Y} = (Y_1 , \ldots , Y_d) $ has the distribution $G$ on $\mathds{R}_+^d$, and let $\m{X} = (X_1 , \ldots , X_d) = (U_1 Y_1 , \ldots , U_d Y_d )$. The resulting class of distributions of $\boldsymbol{X}$ (as $G$ varies over all distributions on $\mathds{R}_+^d$) is called the {\sl Scale Mixture of Uniforms} class of distributions, and the corresponding class of densities on $\mathds{R}_+^d$ is denoted by $\mathcal {F}_{\text{SMU}}(d)$. We study maximum likelihood estimation in the family $\mathcal {F}_{\text{SMU}}(d)$. We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in $\mathcal {F}_{\text{SMU}}(d)$. We also provide an asymptotic minimax lower bound for estimating the functional $f \mapsto f(\boldsybol{x})$ under reasonable differentiability assumptions on $f\in\mathcal {F}_{\text{SMU}}(d)$ in a neighborhood of $\boldsymbol{x}$. We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE.

Local Asymptotic Minimax Theory for Block-Decreasing Densities

by Marios Pavlides

Journal of Statistical Planning and Inference, Volume 142, Issue 8, August 2012, Pages 2322–2329

In this paper, we study Lebesgue densities on $\s$ that are
non-increasing in each coordinate, while keeping all... more In this paper, we study Lebesgue densities on $\s$ that are
non-increasing in each coordinate, while keeping all other coordinates fixed, from the perspective of local asymptotic minimax lower bound theory. In particular, we establish a local optimal rate of convergence of the order $n^{-1/(d+2)}$.

A Nonparametric Bayesian Approach Towards Robot Learning by Demonstration.

by Dimitrios Korkinof

In the past years, many authors have considered application of machine learning methodologies to effect robot learning... more In the past years, many authors have considered application of machine learning methodologies to effect robot learning by demonstration. Gaussian mixture regression (GMR) is one of the most successful methodologies used for this purpose. A major limitation of GMR models concerns automatic selection of the proper number of model states, i.e., the number of model component densities. Existing methods, including likelihood- or entropy-based criteria, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori. Under this motivation, to resolve the aforementioned issues of GMR-based methods for robot learning by demonstration, in this paper we introduce a nonparametric Bayesian formulation for the GMR model, the Dirichlet process GMR model. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally investigate its efficacy as a robot learning by demonstration methodology, considering a number of demanding robot learning by demonstration scenarios.

Automatic repeated-loess decomposition of data consisting of sums of oscillatory curves

by David Foster

Repeated loess is a nonparametric procedure that uses progressive smoothing and differencing to decompose data... more Repeated loess is a nonparametric procedure that uses progressive smoothing and differencing to decompose data consisting of sums of curves. Smoothing is by locally weighted polynomial regression. Here the procedure was developed so that the decomposition into components was controlled automatically by the number of maxima in each component. The level of smoothing of each component was chosen to maximize the estimated probability of the observed number of maxima. No assumptions were made about the periodicity of components and only very weak assumptions about their shapes. The automatic procedure was applied to simulated data and to experimental data on human visual sensitivity to line orientation.

Comparison of generalized estimating equations and quadratic inference functions using data from the National Longitudinal Survey of Children and Youth (NLSCY) …

by Dillon T. Browne

Statistical Methods Used in Gifted Education Journals 2006-2010

by Nicola L. Ritter, M.Ed.

Warne, R., Lazo, M., Ramos, T., & Ritter, N. (in press). Statistical Methods Used in Gifted Education Journals 2006-2010. Gifted Child Quarterly.

Hypothesis testing

by Christos Dimitrakakis

Multiple groups of orientation-selective visual mechanisms underlying rapid orientated-line detection

by David Foster

Visual search for an edge or line element differing in orientation from a background of other edge or line elements... more Visual search for an edge or line element differing in orientation from a background of other edge or line elements can be performed rapidly and effortlessly. In this study, based on psychophysical measurements with ten human observers, threshold values of the angle between a target and background line elements were obtained as functions of background-element orientation, in brief masked displays. A repeated-loess analysis of the threshold functions suggested the existence of several groups of orientation-selective mechanisms contributing to rapid orientated-line detection; specifically, coarse, intermediate and fine mechanisms with preferred orientations spaced at angles of approximately 90 deg, 35-50 deg and 10-25 deg, respectively. The preferred orientations of coarse and some intermediate mechanisms coincided with the vertical or horizontal of the frontoparallel plane, but the preferred orientations of fine mechanisms varied randomly from observer to observer, possibly reflecting individual variations in neuronal sampling characteristics.

Bootstrap variance estimators for the parameters of small-sample sensory-performance functions

by David Foster

The bootstrap method, due to Bradley Efron, is a powerful, general method for estimating a variance or standard... more The bootstrap method, due to Bradley Efron, is a powerful, general method for estimating a variance or standard deviation by repeatedly resampling the given set of experimental data. The method is applied here to the problem of estimating the standard deviation of the estimated midpoint and spread of a sensory-performance function based on data sets comprising 15-25 trials. The performance of the bootstrap estimator was assessed in Monte Carlo studies against another general estimator obtained by the classical "combination-of-observations" or incremental method. The bootstrap method proved clearly superior to the incremental method, yielding much smaller percentage biases and much greater efficiencies. Its use in the analysis of sensory-performance data may be particularly appropriate when traditional asymptotic procedures, including the probit transformation approach, become unreliable.

Comparison of two-dimensional binned data distributions using the energy test

by Peter Hobson

Computationally efficient algorithms for the two-dimensional Kolmogorov–Smirnov test

by Peter Hobson

The two-dimensional Kolmogorov-Smirnov test

by Peter Hobson

Nonparametric Estimation of a Trend Based Upon Sampled Continuous Processes

by David Degras

Comptes Rendus de l'Academie des Sciences de Paris, Ser. I 347 (2009) 191–194

Asymptotics for the Nonparametric Estimation of the Mean Function of a Random Process

by David Degras

Local Polynomial Regression Based on Functional Data

by David Degras

Co-authored with Karim Benhenni (Université de Grenoble). Submitted.

Suppose that $n$ statistical units are observed, each following the model $Y(x_j)=m(x_j)+ \epsilon(x_j),\, j=1,...,N,$... more Suppose that $n$ statistical units are observed, each following the model $Y(x_j)=m(x_j)+ \epsilon(x_j),\, j=1,...,N,$ where $m$ is a regression function, $0 \leq x_1 <...<x_N \leq 1$ are observation times spaced according to a sampling density $f$, and $\epsilon$ is a continuous-time error process having mean zero and regular covariance function. Considering the local polynomial estimation of $m$ and its derivatives, we derive asymptotic expressions for the bias and variance as $n,N\to\infty$. Such results are particularly relevant in the context of functional data where essential information is contained in the derivatives. Based on these results, we deduce optimal sampling densities, optimal bandwidths and asymptotic normality of the estimator. Simulations are conducted in order to compare the performances of local polynomial estimators based on exact optimal bandwidths, asymptotic optimal bandwidths, and cross-validated bandwidths.

Model-free estimation of the psychometric function

by David Foster

A subject’s response to the strength of a stimulus is described by the psychometric function, from which summary... more A subject’s response to the strength of a stimulus is described by the psychometric function, from which summary measures, such as a threshold or a slope, may be derived. Traditionally, this function is estimated by fitting a parametric model to the experimental data, usually the proportion of successful trials at each stimulus level. Common models include the Gaussian and Weibull cumulative distribution functions. This approach works well if the model is correct, but it can mislead if not. In practice, the correct model is rarely known. Here, a nonparametric approach based on local linear fitting is advocated. No assumption is made about the true model underlying the data, except that the function is smooth. The critical role of the bandwidth is identified, and its optimum value is estimated by a cross-validation procedure. As a demonstration, seven vision and hearing data sets were fitted by the local linear method and by several parametric models. The local linear method frequently performed better and never worse than the parametric ones. Supplemental materials for this article can be downloaded from app.psychonomic-journals.org/content/supplemental.

Variance Ranklets: orientation–-selective rank features for contrast modulations

by George Azzopardi

We introduce a novel type of orientation–selective rank features that are sensitive to contrast modulations... more We introduce a novel type of orientation–selective rank features that are sensitive to contrast modulations (second–order stimuli). Variance Ranklets are designed in close analogy with the standard Ranklets, but use the Siegel–Tukey statistics for dispersion instead of the Wilcoxon statistics. Their response shows the same orientation selectivity pattern of Haar wavelets on second–order signals that are not detectable by linear filters. To the best of our knowledge, this is the first family of rank filters designed to detect orientation in variance modulations. We validate our descriptors with an application to texture classification over a subset of the VisTex and Brodatz databases. The combination of standard (intensity) Ranklets with Variance Ranklets greatly improves on the performance of Ranklets alone. Comparison with other published results shows that state–of–the–art recognition rates can be achieved with a simple Nearest Neighbour classifier

Rank-Based Filter with Frequency-and Orientation-Selective Response

by George Azzopardi

This was my Master's project at Queen Mary University of London.

You can have a look at the below paper which is an extension of the project.

http://www.bmva.org/bmvc/2009/Papers/Paper456/Paper456.pdf