Nonparametric Families Research Articles

Point spread function modelling for astronomical telescopes: a review focused on weak gravitational lensing studies

The accurate modelling of the point spread function (PSF) is of paramount importance in astronomical observations, as it allows for the correction of distortions and blurring caused by the telescope and atmosphere. PSF modelling is crucial for accurately measuring celestial objects’ properties. The last decades have brought us a steady increase in the power and complexity of astronomical telescopes and instruments. Upcoming galaxy surveys like Euclid and Legacy Survey of Space and Time (LSST) will observe an unprecedented amount and quality of data. Modelling the PSF for these new facilities and surveys requires novel modelling techniques that can cope with the ever-tightening error requirements. The purpose of this review is threefold. Firstly, we introduce the optical background required for a more physically motivated PSF modelling and propose an observational model that can be reused for future developments. Secondly, we provide an overview of the different physical contributors of the PSF, which includes the optic- and detector-level contributors and atmosphere. We expect that the overview will help better understand the modelled effects. Thirdly, we discuss the different methods for PSF modelling from the parametric and non-parametric families for ground- and space-based telescopes, with their advantages and limitations. Validation methods for PSF models are then addressed, with several metrics related to weak-lensing studies discussed in detail. Finally, we explore current challenges and future directions in PSF modelling for astronomical telescopes.

Identifying Selection Bias from Observational Data

Access to a representative sample from the population is an assumption that underpins all of machine learning. Selection effects can cause observations to instead come from a subpopulation, by which our inferences may be subject to bias. It is therefore important to know whether or not a sample is affected by selection effects. We study under which conditions we can identify selection bias and give results for both parametric and non-parametric families of distributions. Based on these results we develop two practical methods to determine whether or not an observed sample comes from a distribution subject to selection bias. Through extensive evaluation on synthetic and real world data we verify that our methods beat the state of the art both in detecting as well as characterizing selection bias.

Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit a non-trivial covariance structure. The pushforward measure gives rise to new non-parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including SPD(3)=GL+(3)/SO(3) and S2=SO(3)/SO(2).

Kernel Flows: From learning kernels from data into the abyss

Learning can be seen as approximating an unknown function by interpolating the training data. Although Kriging offers a solution to this problem, it requires the prior specification of a kernel and it is not scalable to large datasets. We explore a numerical approximation approach to kernel selection/construction based on the simple premise that a kernel must be good if the number of interpolation points can be halved without significant loss in accuracy (measured using the intrinsic RKHS norm ‖⋅‖ associated with the kernel). We first test and motivate this idea on a simple problem of recovering the Green's function of an elliptic PDE (with inhomogeneous coefficients) from the sparse observation of one of its solutions. Next we consider the problem of learning non-parametric families of deep kernels of the form K1(Fn(x),Fn(x′)) with Fn+1=(Id+ϵGn+1)∘Fn and Gn+1∈span{K1(Fn(xi),⋅)}. With the proposed approach constructing the kernel becomes equivalent to integrating a stochastic data driven dynamical system, which allows for the training of very deep (bottomless) networks and the exploration of their properties. These networks learn by constructing flow maps in the kernel and input spaces via incremental data-dependent deformations/perturbations (appearing as the cooperative counterpart of adversarial examples) and, at profound depths, they (1) can achieve accurate classification from only one data point per class (2) appear to learn archetypes of each class (3) expand distances between points that are in different classes and contract distances between points in the same class. For kernels parameterized by the weights of Convolutional Neural Networks, minimizing approximation errors incurred by halving random subsets of interpolation points, appears to outperform training (the same CNN architecture) with relative entropy and dropout.

Regionalized Design Rainfall Estimation: an Appraisal of Inundation Mapping for Flood Management Under Data-Scarce Situations

Availability of regionalized design rainfall is crucial for flood modeling, particularly over the regions with sparse raingauge networks. This study proposes a new comprehensive framework for generating regionalized design rainfall time series for data-poor catchments involving non-linear and non-parametric optimization approaches. A large set of parametric and non-parametric families of distribution were considered for multivariate rainfall frequency analysis using at-site station data, while a unique design temporal pattern over the region was derived by quantifying the flood causing potential of design hyetographs. The regionalized design rainfall time series was used as one of the inputs to a two-dimensional (2D) flood model. The accuracy and performance of the derived regionalized design rainfall for flood inundation modeling were evaluated by comparing with those derived from different spatial interpolation methods. There was a high consensus of the former with those of widely used kriging and spline interpolation methods. A severely flood-prone and data-poor (no raingauge available within study area) large coastal catchment lying along the coast of the Bay of Bengal, India, was chosen for a demonstration of the proposed framework. The study showed that the framework can be used for extreme events arising due to floods, even under changing climatic scenarios. It further invokes the necessity for incorporating the proposed framework into various commercially and freely available flood models along with other existing interpolation techniques to support improved flood management.

Life Distributions: A Brief Discussion

This article presents a brief description of some of the characteristics of life distributions that arise in survival analysis and reliability theory. Alternative definitions of a distribution are discussed and related to a variety of stochastic orders: hazard rate order, likelihood ratio order, convex order, Lorenz order. Nonparametric families, particularly log-concave densities, completely monotone distributions, increasing hazard rate families, new-better-than-used families, and bathtub hazard rate families are analyzed. A taxonomy for semiparametric families is presented and the effect of introducing parameters on various stochastic orders is shown. Finally, the introduction of covariate models in these families is developed.

Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order

Assume that \(X_1,\ldots , X_n\) are i.i.d. random variables with a common distribution function \(F\) which precedes a fixed distribution function \(W\) in the convex transform order. In particular, if \(W\) is either uniform or exponential distribution function, then \(F\) has increasing density and failure rate, respectively. We present sharp upper bounds on the expectations of single order statistics and spacings based on \(X_1,\ldots , X_n\), expressed in terms of the population mean and standard deviation, for the family of all parent distributions preceding \(W\) in the convex transform order. We also characterize the distributions which attain the bounds, and specify the general results for the distributions with increasing density function.

Estimation of finite mixtures with symmetric components

It may sometimes be clear from background knowledge that a population under investigation proportionally consists of a known number of subpopulations, whose distributions belong to the same, yet unknown, family. While a parametric family is commonly used in practice, one can also consider some nonparametric families to avoid distributional misspecification. In this article, we propose a solution using a mixture-based nonparametric family for the component distribution in a finite mixture model as opposed to some recent research that utilizes a kernel-based approach. In particular, we present a semiparametric maximum likelihood estimation procedure for the model parameters and tackle the bandwidth parameter selection problem via some popular means for model selection. Empirical comparisons through simulation studies and three real data sets suggest that estimators based on our mixture-based approach are more efficient than those based on the kernel-based approach, in terms of both parameter estimation and overall density estimation.

Unbiased risk estimation and scoring rules

Stein unbiased risk estimation is generalized twice, from the Gaussian shift model to nonparametric families of smooth densities, and from the quadratic risk to more general divergence type distances. The development relies on a connection with local proper scoring rules.

Sharp Negative Bounds On Differences of Small Quantiles and Means of Distributions from Nonparametric Families

For the distributions with decreasing density and decreasing failure rate, and decreasing density and decreasing failure rate on the average, the quantiles are less than the mean if their orders do not exceed fixed levels. We determine the sharp negative upper bounds on the differences of the small quantiles and means for the distributions from the respective families in terms of scale units generated by pth absolute central moments. We show that the bounds amount trivially to zero when p > 1, and are strictly negative for p = 1.

Spatio-Temporal Covariance Functions Generated by Mixtures

Spatio-temporal covariance functions are introduced in this paper by using two approaches: (1) positive power mixture of purely spatial and purely temporal covariances, and (2) scale mixture of purely spatial and purely temporal covariances. Various parametric and nonparametric families of nonseparable spatio-temporal covariance functions are obtained with appropriate selections of the mixing function and covariances being mixed.

Moments inequalities of aging families of distributions with hypotheses testing applications

Abstract Nonparametric families of aging distributions have been the subject of investigation for more than three decades. Both probabilistic and statistical properties of these distributions were studied for such families as “increasing failure rate”, “new better than used”, “new better than used in expectation”, and “harmonic new better than used in expectation”. In the present work, moments inequalities are derived for the above-mentioned four families that demonstrate that if the mean life is finite for any of them then all higher-order moments exist. Next, based on these inequalities, new testing procedures for exponentiality against any one of the above classes are introduced and studied showing that they are simpler than most earlier ones and hold high relative efficiency for some commonly used alternatives.

Lower bounds on expected redundancy for nonparametric classes

The article focuses on lower bound results on expected redundancy for universal coding of independent and identically distributed data on [0, 1] from parametric and nonparametric families. After reviewing existing lower bounds, we provide a new proof for minimax lower bounds on expected redundancy over nonparametric density classes. This new proof is based on the calculation of a mutual information quantity, or it utilizes the relationship between redundancy and Shannon capacity. It therefore unifies the minimax redundancy lower bound proofs in the parametric and nonparametric cases.

Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures

The paper gives sufficient conditions for strong consistency of maximum likelihood estimators in certain nonparametric families. The main application is to families of mixtures, with emphasis on mixtures over exponential families.

Classes of Discrete Decreasing and Increasing Mean-Residual-Life Distributions

Discrete failure-time distributions can be appropriate to model lifetimes. This paper represents two well-known non-parametric families of discrete distributions: decreasing and increasing mean-residual-life. It provides two parametric families of discrete distributions which are suitable for fitting decreasing and increasing mean-residual-life models to discrete life-test data.

Strong Consistency of Approximate Maximum Likelihood Estimators with Applications in Nonparametrics

Wald's general analytic conditions that imply strong consistency of the approximate maximum likelihood estimators (AMLEs) have been extended by Le Cam, Kiefer and Wolfowitz, Huber, Bahadur, and Perlman. All these conditions use the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_0)\rbrack$, where $\theta_0$ is the true value of the parameter. However these methods usually fail in the nonparametric case. Thus, in this paper, for each $\theta \neq \theta_0$, we look at the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_r(\theta))\rbrack$, where $\theta_r(\theta)$ is a certain parameter selected in a neighborhood $V_r$ of $\theta_0$. Some general analytic conditions that imply strong consistency of the AMLE are given. The results are shown to be applicable to several nonparametric families having densities, e.g., concave distributions functions, and increasing failure rate distributions. In particular, they can be applied to several censored data cases.

Unbiased Estimation for Some Non-Parametric Families of Distributions

This paper is concerned with the theory of unbiased estimation for nonparametric families of distributions subject to "generalised moment" restrictions. Necessary and sufficient conditions under which symmetric statistics are unique minimum variance unbiased estimators of their expectations are obtained, and some new boundedly complete families of distributions are exhibited.

On the Completeness of Minimal Sufficient Statistics with Censored Observations

The property of completeness, for the minimal sufficient statistics, is investigated in the context of life testing when the set of observations is censored at a fixed time or at a fixed order statistic. Nonparametric families are shown to retain completeness for the observed order statistics and some implications regarding unbiased estimators and similar tests are presented. Most of the common parametric models fail to possess completeness of minimal sufficient statistics under censored sampling in the one-sample, two-sample and regression situations.