Estimation Of Distribution Function Research Articles

Calibrated Empirical Neutrosophic Cumulative Distribution Function Estimation for Both Symmetric and Asymmetric Data

The traditional stratification weight is widely used in survey sampling for estimation under stratified random sampling (StRS). A neutrosophic calibration approach is proposed under neutrosophic statistics for the first time with the aim of improving conventional stratification weight. This addresses the challenge of estimating the empirical cumulative distribution function (CDF) of a finite population using the neutrosophic technique. The neutrosophic technique extends traditional statistics, dealing with indeterminate, vague, and uncertain values. Thus, using additional information, we are able to obtain an effective estimate of the neutrosophic CDF. The suggested estimator yields an interval range in which the population empirical CDF is likely to exist rather than a single numerical value. The proposed family of neutrosophic estimators will be defined under suitable calibration constraints. A simulation study is also computed in order to assess the effectiveness of the suggested and adapted neutrosophic estimators using real-life symmetric and asymmetric datasets.

Nonparametric estimation of the copula function with bivariate twice censored data

In this work, we are interested in the nonparametric estimation of the copula function in the presence of bivariate twice censored data. Assuming that the copula functions of the right and the left censoring variables are known, we propose an estimator of the joint distribution function of the variables of interest, then we derive an estimator of their copula function. Using a representation of the proposed estimator of the joint distribution function as a sum of independent and identically distributed variables, we establish the weak convergence of the empirical copula that we introduce.

Multidimensional credibility: A new approach based on joint distribution function

Abstract In the traditional multidimensional credibility models developed by Jewell ((1973) Operations Research Center, pp. 73–77.), the estimation of the hypothetical mean vector involves complex matrix manipulations, which can be challenging to implement in practice. Additionally, the estimation of hyperparameters becomes even more difficult in high-dimensional risk variable scenarios. To address these issues, this paper proposes a new multidimensional credibility model based on the conditional joint distribution function for predicting future premiums. First, we develop an estimator of the joint distribution function of a vector of claims using linear combinations of indicator functions based on past observations. By minimizing the integral of the expected quadratic distance function between the proposed estimator and the true joint distribution function, we obtain the optimal linear Bayesian estimator of the joint distribution function. Using the plug-in method, we obtain an explicit formula for the multidimensional credibility estimator of the hypothetical mean vector. In contrast to the traditional multidimensional credibility approach, our newly proposed estimator does not involve a matrix as the credibility factor, but rather a scalar. This scalar is composed of both population information and sample information, and it still maintains the essential property of increasingness with respect to the sample size. Furthermore, the new estimator based on the joint distribution function can be naturally extended and applied to estimate the process covariance matrix and risk premiums under various premium principles. We further illustrate the performance of the new estimator by comparing it with the traditional multidimensional credibility model using bivariate exponential-gamma and multivariate normal distributions. Finally, we present two real examples to demonstrate the findings of our study.

The Law of the Iterated Logarithm for Lp-Norms of Kernel Estimators of Cumulative Distribution Functions

In this paper, we consider the strong convergence of Lp-norms (p≥1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

Calibration estimation of distribution function based on multidimensional scaling of auxiliary information

The distribution function is a functional parameter of great interest in many research areas, such as medicine or economics. Among other properties, it facilitates the estimation of parameters such as quantiles. Accordingly, techniques are needed to estimate this function efficiently. Survey statisticians have access to large, high-dimension databases and use them to optimise the estimates obtained. One way to incorporate auxiliary information in the estimation stage is through the calibration method, which was initially designed to estimate totals and means and consists of adjusting new sample weights in order to reduce the variance of estimators. However, calibration techniques may be subject to over-calibration, i.e. the loss of efficiency when high-dimension auxiliary data sets are incorporated. Although alternative approaches have been proposed, in which the calibration method incorporates auxiliary information in the estimation of the distribution function, these alternatives do not seek to incorporate qualitative auxiliary information, which must be introduced in the usual way through dummy variables. However, this workaround can greatly increase the dimension of the auxiliary information, producing either over-calibration or even incompatible calibration constraints. In this article, we propose adapting the calibration method through multidimensional scaling, in order to incorporate quantitative and qualitative information, thus avoiding the negative consequences of over-calibration in the estimation of the distribution function.

DYNAMICS OF PLASMA IN A DUAL-FREQUENCY CAPACITIVE DISCHARGE IN AN AR/XE MIXTURE

The spatiotemporal dynamics of the plasma of a symmetrical dual-frequency 81 MHz/1.76 MHz capacitive discharge under the in uence of a low-frequency eld of 1.76 MHz has been studied.Using the method of phase-resolved optical emission spectroscopy, the dynamics of the radiation intensity of argon and xenon in plasma was obtained.The electron energy distribution function at the center of the discharge was measured using a Langmuir probe and the electron density was measured using a microwave probe. The main result is the dynamics of the intensity ratios of the selected argon and xenon lines depending on di erent conditions: at pressures of 40, 200 and 400 mTorr, LF voltage amplitude of 100, 200 and 400 V, input power at 81 MHz of 3 W and 15 W. The dynamics of high-energy electrons is studied based on a two-temperature approximation of the electron energy distribution function.

Generation of normal distributions revisited

AbstractNormally distributed random numbers are commonly used in scientific computing in various fields. It is important to generate a set of random numbers as close to a normal distribution as possible for reducing initial fluctuations. Two types of samples from a uniform distribution are examined as source samples for inverse transform sampling methods. Three types of inverse transform sampling methods with new approximations of inverse cumulative distribution functions are also discussed for converting uniformly distributed source samples to normally distributed samples.

Multivariate Density Estimation by Neural Networks.

We propose nonparametric methods to obtain the Probability Density Function (PDF) to assess the properties of the underlying data generating process (DGP) without imposing any assumptions on the DGP, using neural networks (NNs). The proposed NN has advantages compared to well-known parametric and nonparametric density estimators. Our approach builds on literature on cumulative distribution function (CDF) estimation using NN. We extend this literature by providing analytical derivatives of this obtained CDF. Our approach hence removes the numerical approximation error in differentiating the CDF output, leading to more accurate PDF estimates. The proposed solution applies to any NN model, i.e., for any number of hidden layers or hidden neurons in the multilayer perceptron (MLP) structure. The proposed solution applies the PDF estimation by NN to continuous distributions as well as discrete distributions. We also show that the proposed solution to obtain the PDF leads to good approximations when applied to correlated variables in a multivariate setting. We test the performance of our method in a large Monte Carlo simulation using various complex distributions. Subsequently, we apply our method to estimate the density of the number of vehicle counts per minute measured with road sensors for a time window of 24 h.

Quantile difference estimation with censoring indicators missing at random.

In this paper, we define estimators of distribution functions when the data are right-censored and the censoring indicators are missing at random, and establish their strong representations and asymptotic normality. Besides, based on empirical likelihood method, we define maximum empirical likelihood estimators and smoothed log-empirical likelihood ratios of two-sample quantile difference in the presence and absence of auxiliary information, respectively, and prove their asymptotic distributions. Simulation study and real data analysis are conducted to investigate the finite sample behavior of the proposed methods.

Retrieving land surface reflectance anisotropy with Sentinel-3 observations and prior BRDF model constraints

Sentinel-3 supports governmental policies and European programmes such as the Copernicus Global Land Service with daily monitoring of the land surface by measuring the Earth's surface with its multi-band wide-swath visible and near-infrared radiometers. To fully exploit Sentinel-3 data sets, performing a correction of directional effects caused by the land surface reflectance anisotropy properties is mandatory. This correction requires first an estimation of the bidirectional reflectance distribution function (BRDF) to be properly achieved. The quality and robustness of global estimates of key land surface variables derived from space-borne sensors depend clearly on an accurate assessment of the spectral BRDF.Here, we present an algorithm aiming at harnessing the Sentinel-3's wide imaging swath to retrieve the land surface BRDF, approximating it via a kernel-driven semi-empirical model which can then be used to normalise the observations to a common Sun-sensor geometry. Our algorithm, named ReBeLS (Regularised BRDF inversion for Land Surface), uses a temporally regularised BRDF inversion approach and, together with prior knowledge of the land surface BRDF obtained from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensors, assimilates Sentinel-3 surface reflectance observations. We focus on the results derived from observations provided by the Sentinel-3 Ocean and Land Colour Instrument (OLCI) sensors; however, the algorithm can also be applied to reflectance observations acquired by the Sea and Land Surface Temperature Radiometer (SLSTR). The retrieved BRDF model parameters reproduce within uncertainties the directional signature as seen by the Visible Infrared Imaging Radiometer Suite (VIIRS) sensor. The generated Sentinel-3 normalised surface reflectance products are a significant advance compared with time series of directional surface reflectance and maximum value composite images, showing a statistically significant (p≪0.05) correlation (r>0.9) with VIIRS nadir BRDF-adjusted surface reflectance. Furthermore, the ReBeLS Sentinel-3 BRDF products offer enough flexibility to allow users to compute spectral surface albedos or to normalise surface reflectance to any Sun-sensor configuration.

Value‐at‐Risk under Measurement Error

We propose a method for estimating Value‐at‐Risk that corrects for the effect of measurement errors in stock prices. We show that the presence of measurement errors might pose serious problems for estimating risk measures. In particular, when stock prices are contaminated, existing estimators of Value‐at‐Risk are inconsistent and might lead to an underestimation of risk, which can result in extreme leverage ratios within the held portfolios. Using a Fourier transform and a deconvolution kernel estimator of the probability distribution function of actual latent prices, we derive a robust estimator of Value‐at‐Risk in the presence of measurement errors. Monte Carlo simulations and real data analysis illustrate satisfactory performance of the proposed method.

Extropy based inaccuracy measure in order statistics

In this paper, we provide a measure based on inaccuracy between distributions of the ith order statistic and the parent random variable. This measure characterizes the distribution function of parent random variable uniquely. We demonstrate that the extropy of the parent random variable is the average of the accuracy measure. It is also shown that the measure of inaccuracy defined is invariant under scale but not under location transformation. Nonparametric estimators for the proposed measures are also obtained. A Monte Carlo simulation study is performed to verify the performance of the suggested estimators. Simulation results show that the estimator based on the reflection boundary technique for probability density function estimation and the empirical method for cumulative distribution function estimation has the best performance among estimators. Also, a real dataset is considered to show an application of the proposed estimators on model selection.

Polar Sea Ice Detection Using a Rotating Fan Beam Scatterometer

Scatterometers are dedicated to monitoring sea surface wind vectors, but they also provide valuable data for polar applications. As a new type of scatterometer, the rotating fan beam scatterometer delivers a higher diversity of incidence angles and more azimuth sampling. The paper takes the first rotating fan beam scatterometer, the China France Oceanography Satellite scatterometer (CSCAT), as an example to explore the effectiveness of this new type of scatterometer in polar sea ice detection. In this paper, a Bayesian method with consideration of geometric characteristics of CSCAT is developed for sea ice detection. The implementation of this method includes the definition of CSCAT backscatter space, an estimation of the sea ice Physical Model Function (GMF), a calculation of the sea ice backscatter distance to the sea ice GMF, a probability distribution function (PDF) estimation of the square distance to the GMF (sea ice GMF and wind GMF), and a calculation of the sea ice Bayesian posterior probability. This algorithm was used to generate a daily CSCAT polar sea ice mask during the CSCAT mission period (2019–2022) (by setting a 55% threshold on the Bayesian posterior probability). The sea ice masks were validated against passive microwaves by quantitatively comparing the sea ice edges and extents. The validation suggests that the CSCAT sea ice edge and extent show good agreement with the sea ice concentration distribution (i.e., sea ice concentration ≥ 15%) of the Advanced Microwave Scanning Radiometer 2 (AMSR2). The average Euclidean distance of the sea ice edges was basically less than 12.5 km, and the deviation of the sea ice extents was less than 0.3 × 106 km2.

Estimation of uncertainty distribution function by the principle of least squares

In order to estimate the unknown parameters in an uncertainty distribution function, this article uses the principle of least squares that minimizes the sum of the squared deviations between the uncertainty distribution and the empirical distribution of the observed data. After that, the principle of least squares is applied to determining the uncertain disturbance term of uncertain regression model and uncertain time series model, and estimating the unknown parameters in uncertain differential equation. Finally, in order to illustrate the proposed method, some real-world examples are provided, including PetroChina stock price, electricity price, grain yield, China’s population, and beef price.

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

In this paper, we propose to study the asymptotic properties of some conditional functional parameters, such as the distribution function, the density, and the hazard function, for an explanatory variable with values in a Hilbert space (infinite dimension) and a response variable real in a quasi-associated dependency framework. We consider the non parametric estimation of the conditional distribution function by the kernel method in the presence of the quasi-associated dependence, and we establish under general hypotheses the almost complete convergence with speed of the estimator built in the associated case. The estimation of the conditional hazard function will be conducted by utilizing the two outcomes of the conditional distribution function and the conditional density. We establish the asymptotic normality of the kernel estimator as the conditional risk function of a properly normalized functional. We explicitly give the asymptotic variance. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data. All the statistical analyses were performed using R software.

A GPU-accelerated viewer for HEALPix maps

HEALPix by Górskiet al. (2005) is a de-facto standard for Cosmic Microwave Background (CMB) data storage and analysis, and is widely used in current and upcoming CMB experiments. Almost all the datasets in Legacy Archive for Microwave Background Data Analysis (LAMBDA) use HEALPix as a format of choice. Visualizing the data plays important role in research, and several toolsets were developed to do that for HEALPix maps, most notably original Fortran facilities and Python integration with healpy. With the current state of GPU performance, it is now possible to visualize extremely large maps in real time on a laptop or a tablet. HEALPix Viewer described here is developed for macOS, and takes full advantage of GPU acceleration to handle extremely large datasets in real time. It compiles natively on Intel and Arm64 architectures, and uses Metal framework for high-performance GPU computations. The aim of this project is to reduce the effort required for interactive data exploration, as well as time overhead for producing publication-quality maps. Drag and drop integration with Keynote and Powerpoint makes creating presentations easy. The main codebase is written in Swift, a modern and efficient compiled language, with high-performance computing parts delegated entirely to GPU, and a few inserts in C interfacing to cfitsio library for I/O. Graphical user interface is written in SwiftUI, a new declarative UI framework based on Swift. Most common spherical projections and colormaps are supported out of the box, and the available source code makes it easy to customize the application and to add new features if desired. On a M1 Max laptop, an nside=8192 maps are processed in real time, with geometry effects rendered at 60fps in full resolution with no appreciable load to the machine. Main user-facing delays are limited to CPU-bound cfitsio load times, and sorting needed to construct Cumulative Distribution Function (CDF) estimators for statistical analysis (hidden in background queue). Overall performance improves on the current Python software stack by a factor of 3–180x depending on the task at hand.

Non parametric multivariate distribution estimation under right censoring

A non parametric estimator of the joint distribution function of a positive bivariate random vector is introduced. The case where one of the two variables is subject to right censoring is considered. To construct the proposed estimator, Poisson distributions are used for smoothing the empirical estimator of Stute (1993). The strong uniform convergence is established. Also, by stating the asymptotic i.i.d. representation, the asymptotic bias, variance, and normality are deduced. The smooth estimator is applied for analyzing survival data from patients with advanced lung cancer.

Interdisciplinary Fundamental Concepts in STEM: Solid State Physics and COVID-19 Pandemic Evolution

The rapid development of computer-aided design tools, such as MATLAB or Octave, or Mathematica enabled students to solve many complicated problems focusing less on underlying STEM-related concepts that are interdisciplinary. Therefore, it is important to demonstrate how it could be done using specific examples that could be linked to different subjects or even to their everyday life experience. This paper reports on using the COVID-19 pandemic evolution model (Shur, 2022) in my class on the physics of advanced semiconductors devices. I use this model to show how the concepts, such as the Born-Oppenheimer approximation and Fermi-Dirac distribution function could be used in a completely different STEM field. In solid-state physics, the Born-Oppenheimer approximation is used to separate rapid electronic motion, relevant to the electronic states and much slower nuclei motion (since nuclei are thousands of times heavier than electrons). Likewise, the COVID-19 model uses a relatively fast pandemic evolution growth or decay constant, a slow function of time itself. In solid-state physics, the Fermi-Dirac distribution function describes the transition from the occupied electronic states to empty electron states with the temperature determining the transition interval. The COVID-19 model uses the generalized Fermi-distribution function to describe the mitigation measures that determine the transition from a high to a lower infection rate. A more accurate COVID-19 evolution model requires a generalized Fermi-Dirac function that accounts for a slow variation of the effect of the mitigation measures with time. In turn, this generalization could be used in solid-state physics to describe the electron temperature increase in the electric field.

Reliability of Ensemble Climatological Forecasts

AbstractEnsemble climatological forecasts play a critical part in benchmarking the predictive performance of hydroclimatic forecasts. Accounting for the skewness and censoring characteristics of hydroclimatic variables, ensemble climatological forecasts can be generated by the log, Box‐Cox and log‐sinh transformations, by the combinations of the Bernoulli distribution with the Gaussian, Gamma, log‐normal, generalized extreme value, generalized logistic and Pearson type III distributions and by the non‐parametric resampling, empirical cumulative distribution function and kernel density estimation methods. This paper is concentrated on the reliability of the 12 types of ensemble climatological forecasts. Specifically, mathematical formulations are presented and large‐sample tests are devised to verify the forecast reliability for the Multi‐Source Weighted‐Ensemble Precipitation version 2 across the globe. Climatological forecasts of monthly precipitation over 18,425 grid cells are generated for 30 years under leave‐one‐year‐out cross validation, leading to 6,633,000 (12 × 18425 × 30) sets of ensemble climatological forecasts. The results point out that the reliability of climatological forecasts considerably varies across the 12 methods, particularly in regions with high hydroclimatic variability. One observation is that climatological forecasts tend to deviate from the distributions of observations when there is inadequate flexibility to fit precipitation data. Another observation is that ensemble spreads can be overly wide when there exist overfits of sample‐specific noises in cross validation. Through the tests of global precipitation, the robustness of the log‐sinh transformation and the Bernoulli‐Gamma distribution is highlighted. Overall, the investigations can serve as a guidance on the uses of transformations, distributions and non‐parametric methods in generating climatological forecasts.

Drell–Yan lepton-pair production: qT resummation at N4LL accuracy

We consider Drell–Yan lepton pairs produced in hadronic collisions. We present high-accuracy QCD predictions for the transverse-momentum (qT) distribution and fiducial cross sections in the small qT region. We resum to all perturbative orders the logarithmically enhanced contributions up to the next-to-next-to-next-to-next-to-leading logarithmic (N4LL) accuracy and we include the hard-virtual coefficient at the next-to-next-to-next-to-leading order (N3LO) (i.e. O(αS3)) with an approximation of the N4LO coefficients. The massive axial-vector and vector contributions up to three loops have also been consistently included. The resummed partonic cross section is convoluted with approximate N3LO parton distribution functions. We show numerical results at LHC energies of resummed qT distributions for Z/γ⁎,W± production and decay, including the W± and Z/γ⁎ ratio, estimating the corresponding uncertainties from missing higher orders corrections and from incomplete or missing perturbative information coefficients at N4LL and N4LO. Our resummed calculation has been encoded in the public numerical program DYTurbo.