Estimation Of Distribution Function Research Articles

Dealing with regression models’ endogeneity by means of an adjusted estimator for the Gaussian copula approach

AbstractEndogeneity in regression models is a key marketing research concern. The Gaussian copula approach offers an instrumental variable (IV)-free technique to mitigate endogeneity bias in regression models. Previous research revealed substantial finite sample bias when applying this method to regression models with an intercept. This is particularly problematic as models in marketing studies almost always require an intercept. To resolve this limitation, our research determines the bias’s sources, making several methodological advances in the process. First, we show that the cumulative distribution function estimation’s quality strongly affects the Gaussian copula approach’s performance. Second, we use this insight to develop an adjusted estimator that improves the Gaussian copula approach’s finite sample performance in regression models with (and without) an intercept. Third, as a broader contribution, we extend the framework for copula estimation to models with multiple endogenous variables on continuous scales and exogenous variables on discrete and continuous scales, and non-linearities such as interaction terms. Fourth, simulation studies confirm that the new adjusted estimator outperforms the established ones. Further simulations also underscore that our extended framework allows researchers to validly deal with multiple endogenous and exogenous regressors, and the interactions between them. Fifth, we demonstrate the adjusted estimator and the general framework’s systematic application, using an empirical marketing example with real-world data. These contributions enable researchers in marketing and other disciplines to effectively address endogeneity problems in their models by using the improved Gaussian copula approach.

Data-driven Approximation of Cumulative Distribution Function Using Particle Swarm Optimization based Finite Mixtures of Logistic Distribution

This paper proposes a data-driven approximation of the Cumulative Distribution Function using the Finite Mixtures of the Cumulative Distribution Function of Logistic distribution. Since it is not possible to solve the logistic mixture model using the Maximum likelihood method, the mixture model is modeled to approximate the empirical cumulative distribution function using the computational intelligence algorithms. The Probability Density Function is obtained by differentiating the estimate of the Cumulative Distribution Function. The proposed technique estimates the Cumulative Distribution Function of different benchmark distributions. Also, the performance of the proposed technique is compared with the state-of-the-art kernel density estimator and the Gaussian Mixture Model. Experimental results on κ−μ distribution show that the proposed technique performs equally well in estimating the probability density function. In contrast, the proposed technique outperforms in estimating the cumulative distribution function. Also, it is evident from the experimental results that the proposed technique outperforms the state-of-the-art Gaussian Mixture model and kernel density estimation techniques with less training data.

An improved class of predictive estimators of population distribution function in the presence of non-response

In the present work, we have proposed a novel general class of estimators for the estimation of population distribution function in two distinct situations of non-response. The estimation of distribution function (DF) may play an important role in environmental sciences such as in modelling the annual data of atmospheric NOx temporal concentration, drug removal rates from the body, to present the distribution of gene expression levels across cells etc. Similarly, it may also be useful in other environmental data like groundwater quality, rainfall, wind speed, river discharges, etc. for effective analysis leading to predictive modelling. In this study, it is shown that the suggested estimators of DF are among the best of all other considered estimators. Some estimators are also derived from the proposed family by choosing the suitable values of constants used. Theoretical comparisons of the proposed family of estimators with other family of estimators have been discussed. An empirical study based on two real data sets from literature has been conducted where it has been found that the performance of the proposed family of estimators is superior in terms of enhanced percentage relative efficiencies (PREs) than other family of estimators considered in this study. Further, a simulation study is also carried out which validates the competency behaviour of suggested estimators.

Fuzzy system reliability analysis based on new aging indexes

The concept of aging plays an important role in reliability analysis. The aim of this paper is providing the novel method to definition aging indexes based on fuzzy random variables and using the concept of α-pessimistic in both parametric and non-parametric approaches. In parametric approach, the concepts of survival function, hazard rate function and mean residual function were extended for fuzzy random variables. After that, we obtain and analyse these indexes for fuzzy exponential and fuzzy two-parameter Weibull distributions as two frequently used lifetime distribution in reliability theory. Moreover, with the assumption that the distribution of fuzzy observations is unknown (non-parametric approach), first an estimate for the fuzzy distribution function is presented, and then an estimate for the aging indices of the system is investigated. To do this, the concept of fuzzy empirical distribution function is investigated as an estimator of cumulative distribution function. Moreover, some properties of investigated estimator have been established.

Simple closed-form estimation of a binary latent variable model

Abstract This paper develops a closed-form nonparametric estimator of the conditional distribution function for a binary outcome variable given an unobserved latent variable. This type of function is commonly used in models that involve measurement error and dynamic models with agent-specific unobserved heterogeneity. This paper presents a consistent extremum sieve estimator with the following advantages: i) it has a closed-form expression for all the sieve coefficients; ii) it is computationally straightforward, equivalent to computing eigenvalues and eigenvectors of matrices without the use of iterative optimization algorithms. While as flexible as the sieve maximum likelihood estimator (MLE) previously proposed for this model, our estimator proves computationally simpler. The finite sample properties of the estimator are investigated through a Monte Carlo study, and the developed estimator is applied to a probit model to assess the targeting performance of a social welfare program.

Ultrasound normalized cumulative residual entropy imaging: Theory, methodology, and application

Background and Objective:Ultrasound information entropy imaging is an emerging quantitative ultrasound technique for characterizing local tissue scatterer concentrations and arrangements. However, the commonly used ultrasound Shannon entropy imaging based on histogram-derived discrete probability estimation suffers from the drawbacks of histogram settings dependence and unknown estimator performance. In this paper, we introduced the information-theoretic cumulative residual entropy (CRE) defined in a continuous distribution of cumulative distribution functions as a new entropy measure of ultrasound backscatter envelope uncertainty or complexity, and proposed ultrasound CRE imaging for tissue characterization. Methods:We theoretically analyzed the CRE for Rayleigh and Nakagami distributions and proposed a normalized CRE for characterizing scatterer distribution patterns. We proposed a method based on an empirical cumulative distribution function estimator and a trapezoidal numerical integration for estimating the normalized CRE from ultrasound backscatter envelope signals. We presented an ultrasound normalized CRE imaging scheme based on the normalized CRE estimator and the parallel computation technique. We also conducted theoretical analysis of the differential entropy which is an extension of the Shannon entropy to a continuous distribution, and introduced a method for ultrasound differential entropy estimation and imaging. Monte-Carlo simulation experiments were performed to evaluate the estimation accuracy of the normalized CRE and differential entropy estimators. Phantom simulation and clinical experiments were conducted to evaluate the performance of the proposed normalized CRE imaging in characterizing scatterer concentrations and hepatic steatosis (n = 204), respectively. Results:The theoretical normalized CRE for the Rayleigh distribution was π/4, corresponding to the case where there were ≥10 randomly distributed scatterers within the resolution cell of an ultrasound transducer. The theoretical normalized CRE for the Nakagami distribution decreased as the Nakagami parameter m increased, corresponding to that the ultrasound backscattered statistics varied from pre-Rayleigh to Rayleigh and to post-Rayleigh distributions. Monte-Carlo simulation experiments showed that the proposed normalized CRE and differential entropy estimators can produce a satisfying estimation accuracy even when the size of the test samples is small. Phantom simulation experiments showed that the proposed normalized CRE and differential entropy imaging can characterize scatterer concentrations. Clinical experiments showed that the proposed ultrasound normalized CRE imaging is capable to quantitatively characterize hepatic steatosis, outperforming ultrasound differential entropy imaging and being comparable to ultrasound Shannon entropy and Nakagami imaging. Conclusion:This study sheds light on the theory and methodology of ultrasound normalized CRE. The proposed ultrasound normalized CRE can serve as a new, flexible quantitative ultrasound envelope statistics parameter. The proposed ultrasound normalized CRE imaging may find applications in quantified characterization of biological tissues. Our code will be made available publicly at https://github.com/zhouzhuhuang.

Variational time reversal for free-energy estimation in nonequilibrium steady states.

Studying the structure of systems in nonequilibrium steady states necessitates tools that quantify population shifts and associated deformations of equilibrium free-energy landscapes under persistent currents. Within the framework of stochastic thermodynamics, we establish a variant of the Kawasaki-Crooks equality that relates nonequilibrium free-energy corrections in overdamped Langevin systems to heat dissipation statistics along time-reversed relaxation trajectories computable with molecular simulation. Using stochastic control theory, we arrive at a general variational approach to evaluate the Kawasaki-Crooks equality and use it to estimate distribution functions of order parameters in specific models of driven and active matter, attaining substantial improvement in accuracy over simple perturbative methods.

Merging Data with Modeling: An Example from Fatigue.

It is well known that errors are inevitable in experimental observations, but it is equally unavoidable to eliminate errors in modeling the process leading to the experimental observations. If estimation and prediction are to be done with reasonable accuracy, the accumulated errors must be adequately managed. Research in fatigue is challenging because modeling can be quite complex. Furthermore, experimentation is time-consuming, which frequently yields limited data. Both of these exacerbate the magnitude of the potential error. The purpose of this paper is to demonstrate a procedure that combines modeling with independent experimental data to improve the estimation of the cumulative distribution function (cdf) for fatigue life. Subsequently, the effect of intrinsic error will be minimized. Herein, a simplified fatigue crack growth modeling is used. The data considered are a well-known collection of fatigue lives for an aluminum alloy. For lower applied stresses, the fatigue lives can range over an order of magnitude and up to 107 cycles. For larger applied stresses, the scatter in the lives is considerably reduced. Consequently, modeling must encompass a variety of conditions. The primary conclusion of the effort is that merging independent experimental data with a reasonably acceptable model vastly improves the accuracy of the calibrated cdfs for fatigue life, given the loading conditions. This allows for improved life estimation and prediction. For the aluminum data, the calibrated cdfs are shown to be quite good by using statistical goodness-of-fit tests, stress-life (S-N) analysis, and confidence bounds estimated using the mean square error (MSE) method. A short investigation into the effect of sample size is also included. Thus, the proposed methodology is warranted.

Uncertain delayed production risk process and its application in the shortage for a production process system

Classical production scheduling based on stochastic process cannot deal with the cases that the estimated distribution function is not close enough to the real frequency. Unfortunately, since uncertainty is common for a production process system with machine breakdowns or innovative products, uncertainty theory proposed by Professor Baoding Liu in 2007 should be applied. To analyze the effect of delayed time existing in a production process, this paper first introduces uncertain delayed production risk process and proposes the concepts of delayed shortage index as well as delayed shortage time for measuring the function and its failure of the production process system. Moreover, some numerical examples are documented to illustrate the calculation for the delayed shortage index and the uncertainty distribution of the delayed shortage time.

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.

Estimation of Probability Distribution Function and Wind Energy Potential for Higher Heights

Estimation of Probability Distribution Function and Wind Energy Potential for Higher Heights