Non-negative Support Research Articles

Generalised local polynomial estimators of smooth functionals of a distribution function with nonnegative support

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for x ∈ [ 0 , ∞ ) , and for a functional, Φ ( x , F ) of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space L 2 ( q ) , where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.

Limit Distributions for the Estimates of the Digamma Distribution Parameters Constructed from a Random Size Sample

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

Probability Distribution Modeling of Scanner Prices and Relative Prices

The article deals with the problem of the proper selection of the theoretical distribution to describe the empirical distribution of scanner prices. In the empirical study we use scanner data from one retail chain in Poland, i.e. monthly data on natural yoghurt, drinking yoghurt, long grain rice and coffee powder sold in 212 outlets in January and February 2022. Prices and price relatives were modeled using selected ten probability distributions with non-negative support, including two, three and four-parameter family of distributions In addition to the visual assessment in the form of empirical PDF and CDF figures, numerical criteria were used. These include information criteria values such as AIC, BIC, HQIC and p values calculated for the K-S, AD and CVM goodness-of-fit tests. Our research showed that at least two models could be distinguished as very accurate, which provides a good background for simulation research on price indices or for the construction of so-called population price indices.

The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

Distributionally Robust Chance Constrained Geometric Optimization

This paper discusses distributionally robust geometric programs with individual or joint chance constraints. Several groups of uncertainty sets are considered: uncertainty sets with first two order moments information; uncertainty sets with known first order or first two order moments information under nonnegative support; uncertainty sets constrained by the Kullback–Leibler divergence with a normal or discrete reference distribution; uncertainty sets constrained by the Wasserstein distance under discrete, full, or nonnegative real-space support; and joint uncertainty sets for the product of random variables. Under each group of uncertainty sets, we find deterministic reformulations of the distributionally robust geometric programs with individual or joint chance constraints. Convexity, solution methods, and relationships of the reformulation programs are discussed. Finally, numerical tests are carried out on a shape optimization problem.

Efficient Sampling of Bernoulli-Gaussian-Mixtures for Sparse Signal Restoration

This paper introduces a new family of prior models called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bernoulli-Gaussian-Mixtures</i> (BGM), with a view to efficiently address sparse linear inverse problems or sparse linear regression, in the Bayesian framework. The BGM family is based on continuous Location and Scale Mixtures of Gaussians (LSMG), which includes a wide range of symmetric and asymmetric heavy-tailed probability distributions. Particular attention is paid to the decomposition of probability laws as Gaussian mixtures, from which we derive a Partially Collapsed Gibbs Sampler (PCGS) for the BGM, in a systematic way. PCGS is shown to be more efficient than the standard Gibbs sampler, both in terms of number of iterations and CPU time. Moreover, special attention is paid to BGM involving a density defined over a real half-line. An asymptotically exact LSMG approximation is introduced, which allows us to expand the applicability of PCGS to cases such as BGM models with a non-negative support.

On Generating Distributions with the Memoryless Property

The exponential and geometric distribution are well-known continuous and discrete family of distributions with the memoryless property, respectively. The memoryless property is emphasized in introductory probability and statistics textbooks even though no distribution beyond these two families of distributions has been explored in detail. By examining the relationship between these two families of distributions, we propose a general algorithm for generating distributions with the memoryless property. Then, we show that the general algorithm uniquely determines the distribution with the memoryless property given the parameter value, and non-negative support which contains zero and is closed under addition. Furthermore, we present a few nontrivial examples and their applications to demonstrate the richness of such distributions.

Bayesian Estimation for the Coefficients of Variation of Birnbaum–Saunders Distributions

The Birnbaum–Saunders (BS) distribution, which is asymmetric with non-negative support, can be transformed to a normal distribution, which is symmetric. Therefore, the BS distribution is useful for describing data comprising values greater than zero. The coefficient of variation (CV), which is an important descriptive statistic for explaining variation within a dataset, has not previously been used for statistical inference on a BS distribution. The aim of this study is to present four methods for constructing confidence intervals for the CV, and the difference between the CVs of BS distributions. The proposed methods are based on the generalized confidence interval (GCI), a bootstrapped confidence interval (BCI), a Bayesian credible interval (BayCI), and the highest posterior density (HPD) interval. A Monte Carlo simulation study was conducted to evaluate their performances in terms of coverage probability and average length. The results indicate that the HPD interval was the best-performing method overall. PM 2.5 concentration data for Chiang Mai, Thailand, collected in March and April 2019, were used to illustrate the efficacies of the proposed methods, the results of which were in good agreement with the simulation study findings.

A Quasi-Normal Distribution and its Application in Parameter Estimation on Heavily Censored Data

Parameter estimation on heavily censored data is a challenging issue. This paper addresses this issue using a two-step single-parameter maximum likelihood method to estimate mean time to failure (MTTF) and Weibull shape parameter (WSP). The first step fits the data to three one-parameter auxiliary models, which are special cases of a two-parameter quasi-normal distribution with nonnegative support, to obtain three estimates of the MTTF. The second step estimates the WSP through fixing each of the MTTF estimates. The best estimates are selected from three pairs of estimates based on appropriate rules. A simple numerical method is also proposed to construct the confidence intervals of the estimated MTTF and WSP. The auxiliary models are derived, the model selection rules are specified, and a numerical experiment is carried out to illustrate the accuracy of the proposed method.

Recursive asymmetric kernel density estimation for nonnegative data

Recursive nonparametric density estimation for nonnegative data is considered, using an asymmetric kernel with nonnegative support. It has a computational advantage in a situation where a huge number of data are sequentially collected. The recursive asymmetric kernel estimator keeps desirable asymptotic properties similar to the ordinary non-recursive asymmetric kernel estimator. Also, simulation studies and a real data analysis are performed for illustration.

Asymptotically Normal Estimators for the Parameters of the Gamma-Exponential Distribution

Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

Games with distributionally robust joint chance constraints

This paper studies an n-player non-cooperative game where each player has expected-value payoff function and chance-constrained strategy set. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong to a certain distributional uncertainty set. The chance-constrained strategy sets are defined using a distributionally robust framework. We consider one density based uncertainty set and four two-moments based uncertainty sets. One of the considered uncertainty sets is based on a nonnegative support. Under the standard assumptions on the players’ payoff functions, we show that there exists a Nash equilibrium of a distributionally robust chance-constrained game for each uncertainty set. As an application, we study Cournot competition in electricity market and perform the numerical experiments for the case of two electricity firms.

A New Kernel Estimator Based on Scaled Inverse Chi-Squared Density Function

In this work, a new kernel estimator based on scaled inverse chi-squared distribution is proposed to estimate densities having nonnegative support. The optimal rates of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE) are obtained. Adaptive Bayesian bandwidth selection method with Lindley approximation is used for heavy tailed distributions. Simulation studies are performed to compare the performance of the average integrated square error (ISE) by using the bandwidths obtained from the global least squares cross-validation bandwidth selection method and the bandwidths obtained from adaptive Bayesian method with Lindley approximation. Finally, real data sets are presented to illustrate the findings.

Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Distribution estimation for noisy data via density deconvolution is a notoriously difficult problem, especially for typical noise distributions like Gaussian. We develop a density deconvolution estimator based on quadratic programming (QP) that can achieve better estimation than kernel density deconvolution methods. The QP approach appears to have a more favorable regularization tradeoff between oversmoothing versus oscillation, especially at the tails of the distribution. An additional advantage is that it is straightforward to incorporate a number of common density constraints such as nonnegativity, integration-to-one, unimodality, tail convexity, tail monotonicity, and support constraints. We demonstrate that the QP approach has favorable estimation performance relative to existing methods. Its performance is superior when only the universally applicable nonnegativity and integration-to-one constraints are incorporated, and incorporating additional common constraints when applicable (e.g., nonnegative support, unimodality, tail monotonicity or convexity, etc.) can further substantially improve the estimation. Supplementary materials for this article are available online and include R code, the R package QPdecon, a vignette for the QPdecon package, the sodium dataset that is used as an example, and appendices with a proof and additional figures.

New type of gamma kernel density estimator

We discuss a new kernel type estimator for density function $$f_X(x)$$ with nonnegative support. Here, we use a type of gamma density as a kernel function and modify it with expansions of exponential and logarithmic functions. Our modified gamma kernel density estimator is not only free of the boundary bias, but the variance is also in smaller orders, which are $$O(n^{-1}h^{-1/4})$$ in the interior and $$O(n^{-1}h^{-3/4})$$ in the boundary region. Furthermore, the optimal orders of its mean squared error are $$O(n^{-8/9})$$ in the interior and $$O(n^{-8/11})$$ in the boundary region. Simulation results that demonstrate the proposed method’s performances are also presented.

Loewner’s Differential Equation and Spidernets

We regard Loewner’s differential equation with non-negative continuous driving functions (and more generally driving measures with non-negative support) from a quantum probability point of view and approximate the underlying quantum process by the adjacency matrices of growing graphs which arise from the comb product of certain spidernets. In the general case of an arbitrary continuous driving function we find a weaker approximation by comb products of graphs with weighted loops.

When Not to Classify: Anomaly Detection of Attacks (ADA) on DNN Classifiers at Test Time.

A significant threat to the recent, wide deployment of machine learning-based systems, including deep neural networks (DNNs), is adversarial learning attacks. The main focus here is on evasion attacks against DNN-based classifiers at test time. While much work has focused on devising attacks that make small perturbations to a test pattern (e.g., an image) that induce a change in the classifier's decision, until recently there has been a relative paucity of work defending against such attacks. Some works robustify the classifier to make correct decisions on perturbed patterns. This is an important objective for some applications and for natural adversary scenarios. However, we analyze the possible digital evasion attack mechanisms and show that in some important cases, when the pattern (image) has been attacked, correctly classifying it has no utility---when the image to be attacked is (even arbitrarily) selected from the attacker's cache and when the sole recipient of the classifier's decision is the attacker. Moreover, in some application domains and scenarios, it is highly actionable to detect the attack irrespective of correctly classifying in the face of it (with classification still performed if no attack is detected). We hypothesize that adversarial perturbations are machine detectable even if they are small. We propose a purely unsupervised anomaly detector (AD) that, unlike previous works, (1) models the joint density of a deep layer using highly suitable null hypothesis density models (matched in particular to the nonnegative support for rectified linear unit (ReLU) layers); (2) exploits multiple DNN layers; and (3) leverages a source and destination class concept, source class uncertainty, the class confusion matrix, and DNN weight information in constructing a novel decision statistic grounded in the Kullback-Leibler divergence. Tested on MNIST and CIFAR image databases under three prominent attack strategies, our approach outperforms previous detection methods, achieving strong receiver operating characteristic area under the curve detection accuracy on two attacks and better accuracy than recently reported for a variety of methods on the strongest (CW) attack. We also evaluate a fully white box attack on our system and demonstrate that our method can be leveraged to strong effect in detecting reverse engineering attacks. Finally, we evaluate other important performance measures such as classification accuracy versus true detection rate and multiple measures versus attack strength.

Exploring a copula-based alternative to additive error models—for non-negative and autocorrelated time series in hydrology

Inaccurate description of uncertainty in the error models can cause biases in parameter estimation. When the parameters of the deterministic model and the error model are inferred jointly from the observations, the posterior converges to regions that reflect the processes in both high and low flows. If the nature of errors in low and high flows is different to the extent that the same error description cannot be used for both, biases in inference are introduced. In such cases, the parameter posterior will adjust to the region of the hydrograph with longer proportionate presence in the calibration time series. In this paper we demonstrate that the autoregressive order 1 (AR1) description of errors can lead to sub-optimally performing predictive models if the calibration period has substantial sections of inadequately modelled flows. Inference is performed within the Bayesian framework. We show this for a synthetic example as well as a case study. We also see that the predictive uncertainty bands that we get using the AR1 description can be overconfident and also admit negative values. To mitigate this, we analyze an alternative to additive error models. We use a distribution with a non-negative support, gamma in this study, reflecting the uncertainty in the system response at every time step. The gamma distribution is conditioned on the deterministic model output, which determines its mode and standard deviation. We capture autocorrelation in time using copulas. Given that copulas can capture dependence between different marginals, we use different specifications of the marginal distribution for high and low flows. The results show that 1) biases in parameter estimation can be reduced if a representative error description is attained using the flexibility of a copula-based likelihood. 2) The non-negative support allows to make more realistic uncertainty intervals for low flows. 3) However, the autocorrelation parameter in copulas severely interacts with the model and heteroscedasticity parameters. 4) While the formulation, in principle, should be of added value for parameter inference, in case of less informative priors, the flexibility of this description can produce non-robust inference.

Nonparametric density estimation based on beta prime kernel

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.

Randomization Inference for Outcomes with Clumping at Zero

ABSTRACTWhile randomization inference is well developed for continuous and binary outcomes, there has been comparatively little work for outcomes with nonnegative support and clumping at zero. Typically, outcomes of this type have been modeled using parametric models that impose strong distributional assumptions. This article proposes new randomization inference procedures for nonnegative outcomes with clumping at zero. Instead of making distributional assumptions, we propose various assumptions about the nature of the response to treatment and use permutation inference for both testing and estimation. This approach allows for some natural goodness-of-fit tests for model assessment, as well as flexibility in selecting test statistics sensitive to different potential alternatives. We illustrate our approach using two randomized trials, where job training interventions were designed to increase earnings of participants.