Nonparametric measure-transportation-based methods for directional data

FORTRAN 77 program for constructing and plotting confidence bands for the distribution and quantile functions for truncated data

Let {Xn, n ≥ 1} be a sequence of stationary non-negative associated random variables with common marginal distribution function F(x) and quantile function Q(u), where Q(u) is defined as F(Q(u)) = u. Here we consider the smooth estimation of Q(u), adapted from generalized kernel smoothing (Cheng and Parzen J. Stat. Plann. Infer. 59, 291–307, 1997) of the empirical quantile function. Some asymptotic properties of the kernel quantile estimator, for associated sequences, are also established parallel to those in the i.i.d. case. Various estimators in this class of estimators are contrasted, through a simulation study, among themselves and with an indirect smooth quantile estimator obtained by inverting the Poisson weights based estimator of the distribution function studied in Chaubey et al. (Statist. Probab. Lett. 81, 267–276, 2011). The indirect smoothing estimator seems to be the best estimator on account of smaller MSE, however, a quantile estimator based on the Bernstein polynomials and that using the corrected Poisson weights turn out to be almost as good as the inverse distribution function estimator using Poisson weights.

Analysis of NMR Spin-Relaxation Data Using an Inverse Gaussian Distribution Function

This work is devoted to the study of rates of convergence of the empirical measures μ n = 1 n ∑ k = 1 n δ X k \mu _n = \frac {1}{n} \sum _{k=1}^n \delta _{X_k} , n ≥ 1 n \geq 1 , over a sample ( X k ) k ≥ 1 {(X_k)}_{k \geq 1} of independent identically distributed real-valued random variables towards the common distribution μ \mu in Kantorovich transport distances W p W_p . The focus is on finite range bounds on the expected Kantorovich distances E ( W p ( μ n , μ ) ) \mathbb {E}(W_p(\mu _n,\mu )) or [ E ( W p p ( μ n , μ ) ) ] 1 / p \big [ \mathbb {E}(W_p^p(\mu _n,\mu )) \big ]^{1/p} in terms of moments and analytic conditions on the measure μ \mu and its distribution function. The study describes a variety of rates, from the standard one 1 n \frac {1}{\sqrt n} to slower rates, and both lower and upper-bounds on E ( W p ( μ n , μ ) ) \mathbb {E}(W_p(\mu _n,\mu )) for fixed n n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

Quantile function or inverse cumulative distribution function (CDF) is heavily utilized in modelling, simulation, reliability analysis and random number generation. The use is often limited if the inversion method fails to estimate it from the cumulative distribution function. As a result, approximation becomes the other feasible option. The failure of the inversion method is often due to the intractable nature of the CDF of the distribution. Approximation may come in the form of series expansions, closed form or functional approximation, numerical algorithm and the closed form expression drafted in terms of the quantile function of another distribution. This work used the cubic spline interpolation to obtain the closed form of the inverse cumulative distribution function of the Nakagami-m distribution. Consequently, the closed form of the quantile function obtained for the selected parameters of the distribution serves as an approximation which compares favourably with the R software values. The result obtained was a significant improvement over some results surveyed from literature for four reasons. Firstly, the approximates produced better results in simulation as evidenced by the some values of the root mean square error of this work when compared with others. Secondly, the result obtained at the extreme tail of the distribution is better than others selected from the literature. Thirdly, the closed form estimates are easy to compute and save computation time. The closed form of the quantile function obtained in this work can be used in simulating Nakagami random variables which are used in modelling attenuation and fading channels in wireless communications and ultrasonic tissue characterization.

Norm-based inequality measures are developed within the unequally distributed/relative unequally distributed (UD/RUD) framework by applying the L1 norm and the squared L2 norm to the cumulative distribution and quantile function (CDQF), the quantile function (QF), and their combined representation. All six indices maintain key properties such as scale, replication, and translation invariance, as well as anonymity, and their behavior under Pigou-Dalton transfers and subgroup decomposition is examined analytically. The indices based on the cumulative distribution and quantile function combine the vertical and horizontal gaps, giving them representation decomposability and making them the most informative overall measures of inequality. A Monte Carlo study on six contrasting income distributions corroborates these theoretical advantages, confirming finite, stable values even for mixed-sign and heavy-tailed supports.

Methods for estimating extreme loads are used in design as well as risk assessment. Regression using maximum likelihood or least squares estimation is widely used in a univariate analysis but these methods favour solutions that fit observations in an average sense. Here we describe a new technique for estimating extremes using a quantile function model. A quantile of a distribution is most commonly termed a ‘return level’ in flood risk analysis. The quantile function of a random variable is the inverse function of its distribution function. Quantile function models are different from the conventional regression models, because a quantile function model estimates the quantiles of a variable conditional on some other variables, while a regression model studies the conditional mean of a variable. So quantile function models allow us to study the whole conditional distribution of a variable via its quantile function, whereas conventional regression models represent the average behaviour of a variable.Little work can be found in the literature about prediction from a quantile function model. This paper proposes a prediction method for quantile function models. We also compare different types of statistical models using sea level observations from Venice. Our study shows that quantile function models can be used to estimate directly the relationships between sea condition variables, and also to predict critical quantiles of a sea condition variable conditional on others. Our results show that the proposed quantile function model and the developed prediction method have the potential to be very useful in practice.

In this article, we introduce a new one-parameter model, which is named sine inverted exponential (SIE) distribution. The SIE distribution is a new extension of the inverse exponential (IE) distribution. The SIE distribution aims to provide the SIE model for data-fitting purposes. The SIE distribution is more flexible than the inverted exponential (IE) model, and it has many applications in physics, medicine, engineering, nanophysics, and nanoscience. The density function (PDFu) of SIE distribution can be unimodel shape and right skewed shape. The hazard rate function (HRFu) of SIE distribution can be J-shaped and increasing shaped. We investigated some fundamental statistical properties such as quantile function (QFu), moments (Mo), moment generating function (MGFu), incomplete moments (ICMo), conditional moments (CMo), and the SIE distribution parameters were estimated using the maximum likelihood (ML) method for estimation under censored samples (CS). Finally, the numerical results were investigated to evaluate the flexibility of the new model. The SIE distribution and the IE distribution are compared using two real datasets. The numerical results show the superiority of the SIE distribution.

International audience

In this article we study and obtain some general results on the quantile functions for the generalized beta family and the family of beta generated distributions. Having described the standardization rule, we have derived the quantile function of the 5 parameter generalized beta family of distributions [21, 22]. Further, quantile rules for distributional model building have been applied to generate quantile functions of several known and unknown distributions. Attempts have been made to obtain and study the quantile functions of size biased generalized beta distributions and generalized beta generated distributions. Finally, we have applied the proposed results to simulated as well as real datasets.

The C source code associated with the Simulation 101 preconference workshop (offered at the 2006 and 2007 Winter Simulation Conferences) is presented here. This paper begins with general instructions for downloading, compiling, and executing the software. This is followed by sections on four groups of the software, categorized by functionality: libraries, Monte Carlo simulations, discrete-event simulations, and utilities. The libraries contain code to generate random numbers, code to generate random variates, and code to evaluate probability density functions, cumulative distribution functions, and inverse distribution functions. The Monte Carlo simulations consist of six programs that estimate various probabilities associated with simple probability problems, some with known analytic solutions and others without analytic solutions. The discrete-event simulations consist of various applications from queueing and inventory systems. Finally, the utilities are used to calculate various point and interval estimators from data sets. 1

To generate random variates from an unknown continuous distribution via the inverse transform method, we present a flexible, computationally tractable procedure for estimating the associated inverse distribution function based on sample data. Previously proposed methods for estimating inverse distribution functions can fail in either the distribution-fitting or variate-generation stages of application. To avoid these difficulties, we have developed the procedure IDPF for estimating an Inverse Distribution with a Polynomial Filter. After a first-cut or reference distribution has been obtained by some standard technique, a front-end polynomial filter for the inverse of the reference distribution is estimated by constrained nonlinear regression so that the resulting inverse distribution has minimum distance from the empirical inverse distribution. The constraints on the regression ensure that the fitted inverse distribution function is nondefective and monotonically nondecreasing. A specific implementation of this procedure is based on well-known techniques for obtaining a reference fit from the Johnson translation system of distributions. We present the results of a Monte Carlo study to demonstrate the effectiveness of the method. Compared to the reference fit, procedure IDPF yields significantly better approximations not only to the empirical inverse distribution function but also to the underlying theoretical inverse distribution function.

The aim of this study was to address the issue of significant performance degradation in existing defogging algorithms under extreme fog conditions. Traditional Taylor series-based deformable convolutions are limited by local approximation errors, while the heavy-tailed characteristics of the Cauchy distribution can more successfully model outliers in fog images. The following improvements are made: (1) A displacement generator based on the inverse cumulative distribution function (ICDF) of the Cauchy distribution is designed to transform uniform noise into sampling points with a long-tailed distribution. A novel double-peak Cauchy ICDF is proposed to dynamically balance the heavy-tailed characteristics of the Cauchy ICDF, enhancing the modeling capability for sudden changes in fog concentration. (2) An innovative Cauchy–Gaussian fusion module is proposed to dynamically learn and generate hybrid coefficients, combining the complementary advantages of the two distributions to dynamically balance the representation of smooth regions and edge details. (3) Tree-based multi-path and cross-resolution feature aggregation is introduced, achieving local–global feature adaptive fusion through adjustable window sizes (3/5/7/11) for parallel paths. Experiments on the RESIDE dataset demonstrate that the proposed method achieves a 2.26 dB improvement in the peak signal-to-noise ratio compared to that obtained with the TaylorV2 expansion attention mechanism, with an improvement of 0.88 dB in heavily hazy regions (fog concentration > 0.8). Ablation studies validate the effectiveness of Cauchy distribution convolution in handling dense fog and conventional lighting conditions. This study provides a new theoretical perspective for modeling in computer vision tasks, introducing a novel attention mechanism and multi-path encoding approach.

Nonparametric regression is a standard statistical tool with increased importance in the Big Data era. Boundary points pose additional difficulties but local polynomial regression can be used to alleviate them. Local linear regression, for example, is easy to implement and performs quite well both at interior and boundary points. Estimating the conditional distribution function and/or the quantile function at a given regressor point is immediate via standard kernel methods but problems ensue if local linear methods are to be used. In particular, the distribution function estimator is not guaranteed to be monotone increasing, and the quantile curves can “cross.” In the article at hand, a simple method of correcting the local linear distribution estimator for monotonicity is proposed, and its good performance is demonstrated via simulations and real data examples. Supplementary materials for this article are available online.

The odd exponentiated Rayleigh-G family: A flexible statistical framework for reliability and radiation modeling