Testing Independence of Infinite Dimensional Random Elements: A Sup-norm Approach

In this article, we study the test for independence of two random elements $X$ and $Y$ lying in an infinite dimensional space ${\cal{H}}$ (specifically, a real separable Hilbert space equipped with the inner product $\langle ., .\rangle_{\cal{H}}$). In the course of this study, a measure of association is proposed based on the sup-norm difference between the joint probability density function of the bivariate random vector $(\langle l_{1}, X \rangle_{\cal{H}}, \langle l_{2}, Y \rangle_{\cal{H}})$ and the product of marginal probability density functions of the random variables $\langle l_{1}, X \rangle_{\cal{H}}$ and $\langle l_{2}, Y \rangle_{\cal{H}}$, where $l_{1}\in{\cal{H}}$ and $l_{2}\in{\cal{H}}$ are two arbitrary elements. It is established that the proposed measure of association equals zero if and only if the random elements are independent. In order to carry out the test whether $X$ and $Y$ are independent or not, the sample version of the proposed measure of association is considered as the test statistic after appropriate normalization, and the asymptotic distributions of the test statistic under the null and the local alternatives are derived. The performance of the new test is investigated for simulated data sets and the practicability of the test is shown for three real data sets related to climatology, biological science and chemical science.Comment: Remark 2.4 has been adde

Co-variance Operator of Banach Valued Random Elements: U-Statistic Approach

This article proposes a co-variance operator for Banach valued random elements using the concept of $U$-statistic. We then study the asymptotic distribution of the proposed co-variance operator along with related large sample properties. Moreover, specifically for Hilbert space valued random elements, the asymptotic distribution of the proposed estimator is derived even for dependent data under some mixing conditions.Comment: Preliminary version of an ongoing work. Comments are welcom

On Testing Homological Equivalence

In this article, we develop a test to check whether the support of the unknown distribution generating the data is homologically equivalent to the support of some specified distribution. Similarly, it is also checked whether the supports of two unknown distributions are homologically equivalent or not. In the course of this study, test statistics based on the Betti numbers are formulated, and the consistency of the tests are established. Moreover, some simulation studies are conducted when the specified population distributions are uniform distribution over circle and 3-D torus, which indicate that the proposed tests are performing well. Furthermore, the practicability of the tests are shown on two well-known real data sets also

Inspecting discrepancy between multivariate distributions using half-space depth based information criteria

This article inspects whether a multivariate distribution is different from a specified distribution or not, and it also tests the equality of two multivariate distributions. In the course of this study, a graphical tool-kit using well-known half-spaced depth based information criteria is proposed, which is a two-dimensional plot, regardless of the dimension of the data, and it is even useful in comparing high-dimensional distributions. The simple interpretability of the proposed graphical tool-kit motivates us to formulate test statistics to carry out the corresponding testing of hypothesis problems. It is established that the proposed tests based on the same information criteria are consistent, and moreover, the asymptotic distributions of the test statistics under contiguous/local alternatives are derived, which enable us to compute the asymptotic power of these tests. Furthermore, it is observed that the computations associated with the proposed tests are unburdensome. Besides, these tests perform better than many other tests available in the literature when data are generated from various distributions such as heavy tailed distributions, which indicates that the proposed methodology is robust as well. Finally, the usefulness of the proposed graphical tool-kit and tests is shown on two benchmark real data sets.Comment: Few results are rewritten for better understanding, and many remarks have been added to explain those results. The algorithms are also rewritten and few changes have been made in the numerical result

A study of the power and robustness of a new test for independence against contiguous alternatives

Various association measures have been proposed in the literature that equal zero when the associated random variables are independent. However many measures, (e.g., Kendall's tau), may equal zero even in the presence of an association between the random variables. In order to over- come this drawback, Bergsma and Dassios (2014) proposed a modification of Kendall's tau, (denoted as τ ∗), which is non-negative and zero if and only if independence holds. In this article, we investigate the robustness properties and the asymptotic distributions of τ ∗ and some other well-known measures of association under null and contiguous alternatives. Based on these asymptotic distributions under contiguous alternatives, we study the asymptotic power of the test based on τ ∗ under contiguous alternatives and compare its performance with the performance of other well-known tests available in the literature

Identifying shifts between two regression curves

This article studies the problem whether two convex (concave) regression functions modelling the relation between a response and covariate in two samples differ by a shift in the horizontal and/or vertical axis. We consider a nonparametric situation assuming only smoothness of the regression functions. A graphical tool based on the derivatives of the regression functions and their inverses is proposed to answer this question and studied in several examples. We also formalize this question in a corresponding hypothesis and develop a statistical test. The asymptotic properties of the corresponding test statistic are investigated under the null hypothesis and local alternatives. In contrast to most of the literature on comparing shape invariant models, which requires independent data the procedure is applicable for dependent and non-stationary data. We also illustrate the finite sample properties of the new test by means of a small simulation study and a real data example

Nanometer Probing of Operating Nano-Photonic Devices

The external performance of quantum optoelectronic devices is governed by the three-dimensional profiles of electric potentials determined by the distribution of charge carriers (electrons and holes) within the active regions of the devices. Charge carrier dynamics play a vital role in active photonic quantum/nano devices, such as electrically-pumped semiconductor lasers. As an example, in quantum cascade lasers (QCLs) the Electric Field Domain (EFD) hypothesis posits that the potential distribution might be simultaneously spatially non-uniform and temporally unstable. Until now, there are no experimental means of probing the inner potential profile directly and as a result the mechanisms responsible for sub-par device performance of QCLs remain the subject of speculation. Another example is interband cascade lasers (ICLs), in which the distribution of gain-providing charge carrier governs the operation and performance of the devices, but has not been experimentally measured prior to this study. This work presents a systematic experimental study of gain-providing charge carrier distribution in a lasing interband cascade laser and electric potential distribution in THz QCLs. The unique charge carrier distribution profile in the quantum-well active region is quantitatively measured at nanometer scales by using the non-invasive scanning voltage microscopy (SVM) technique. Experimental results clearly confirm the accumulation and spatial segregation of holes and electrons in the core of the ICL device. The measurement also shows that the charge carrier density is essentially clamped in the presence of stimulated emission in ICLs, thus conclusively differentiating the lasing from non-lasing devices. The SVM technique has been applied to lasing THz QCLs to verify the hypothesis of electric field domains in semiconductor quantum structures. The experimental results reveal that the multi-quantum-well active region is divided into multiple sections having distinctly different electric fields. The electric field across these serially-stacked quantum cascade modules are observed not to continuously increase in proportion to the gradual increase of the applied device bias, but rather jumps between discrete values related to tunneling resonances. Also in the THz QCLs the progression of the observed EFDs are carefully probed. Experimental evidences reveal that an incremental change in device bias leads to a hopping-style shift in the EFD boundary – the higher electric field domain expands at least one module each step at the expense of the lower field domain within the active region. The SVM findings in THz QCLs indicate the importance of quantum active region design for intrinsically more uniform and stable electric field profiles. The two showcase study examples demonstrate that the cryogenic-temperature SVM is an enabling technique, being able to measure and resolve nanometer scale features non-destructively on operating devices. This experimental approach allows directly mapping the electric field distribution as well as the charge carrier distribution inside operating semiconductor quantum devices at nanometer scales, thus connecting the inner workings with the external measures of the devices. The experimental approach is expected to facilitate a deeper understanding of fundamental processes that are governing the operation and performance of a wide range of nanoelectronic and nanophotonic devices.4 month