Estimation Of Cumulative 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.

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.

Estimation of the population distribution function using varied L ranked set sampling

A generalized ranked set sampling (RSS) plan has recently been provided in the literature called varied L RSS (VLRSS). It is shown that VLRSS encompasses several existing RSS variations and also it efficiently estimates the population mean. In this article, we extend the work and consider estimating the cumulative distribution function (CDF) using VLRSS. Three new CDF estimators are proposed and their asymptotic properties are also theoretically investigated. Taking into account the information supported by the unmeasured sampling units, we also propose a general class of CDF estimators. Using small Monte Carlo experiments, we study the behavior of the proposed CDF estimators with respect to the conventional CDF estimator under RSS. It is found that the conventional RSS-based CDF is outperformed by at least one of VLRSS-based CDF estimators in most of the considered cases. Finally, an empirical example is utilized to illustrate the potential application of the proposed estimators.

Efficient estimation of cumulative distribution functions of multiple failure modes using advanced generalized subset simulation

AbstractIt is important to estimate the cumulative distribution function (CDF) of a failure mode in a geotechnical system, because failure of the mode is often characterized by some empirical and/or semi‐empirical thresholds and geotechnical engineers sometimes might be interested in the probability of failure associated with various thresholds. The geotechnical systems also generally contain multiple failure modes. This requires an efficient estimation of cumulative distribution functions (CDFs) of multiple failure modes, which poses a significant challenge in application of conventional geotechnical reliability analysis methods. To address this challenge, this study proposes an advanced generalized subset simulation‐based method (AGSS) to efficiently estimate the CDFs of multiple failure modes of the geotechnical system. The proposed method utilizes the probability weights of samples generated in each sample sub‐space to efficiently estimate the CDFs, instead of using the conditional probability formula that may yield incorrect CDFs due to the sample “jumping” properties existing in generalized subset simulation (GSS). A cantilever retaining wall example and a parallel system example are employed to demonstrate the proposed method. It is found that the proposed method can effectively estimate the CDFs of multiple failure modes with high computational efficiency.

Nonparametric sequential estimation of the cumulative function by orthogonal series and application in reliability

This article proposes the orthogonal series method for the cumulative distribution function estimation based on a random number of observations Nt , which may be recorded in time We give the statistical properties of the estimator (bias, variance, mean square error, mean square integrated error, and the rate of convergence) and some asymptotic properties. We consider that Nt is independent of the observations and as The choice of the smoothing parameter (bandwidth) of the estimator is investigated by two methods: The Kronmal–Tarter method and a new technique proposed by Saadi et al. adapted to the random sample size case. A detailed study with a Dirichlet basis is presented. The obtained estimator is asymptotically unbiased and consistent. An application to the reliability field is investigated with simulations in order to study the behavior of the reliability and failure rate estimators, from different life distributions, namely: Lognormal, Gamma, and Weibull distributions. A real lifetime data sets are analyzed by our proposed approach. The results show that the proposed estimators are performant.

Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Noise robust and rotation invariant texture classification based on local distribution transform

Applying local binary pattern (LBP) to images with uniform distribution leads to generate discriminative features; however, the distribution of all images is not necessarily uniform. The distribution of an image can be uniformzed if it passes through its cumulative distribution function (CDF), while estimation of CDF is highly sensitive to additive noises. In this paper, we propose a novel transform, which locally uniformize all patches of an image and approximately estimate a robust CDF. The proposed local distribution transform (LDT) generates continuous values and by quantizing them into discrete values, a histogram of features is constructed. We have fused the LDT features to the features of rotation invariant LBP and local variance (VAR) in order to provide a rich set of robust-to-noise features, which can detect both uniform and non-uniform patterns. The performance of the proposed LDT-LBP_VAR is assessed over a wide range of datasets like Outex, UIUC, CUReT, Coral Reef, Virus and ORL. The datasets are also corrupted by additive Gaussian noise with different signal to noise ratio (SNR) and the empirical results demonstrate that the proposed hybrid features provide superior classification results (P < 0.05) to the plenty of advanced descriptors over the datasets in both noise-free and noisy conditions.

Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution

Let $Y$ , $X$ and $\varepsilon $ be continuous univariate random variables satisfying the model $Y = X + \varepsilon $ . Herein $X$ is of interest, $Y$ is a noisy version of $X$ , and $\varepsilon $ is a random noise independent of $X$ . This paper is devoted to a nonparametric estimation of cumulative distribution function $F_{X}$ of $X$ on the basis of independent random samples $(Y_{1}, \ldots , Y_{n})$ and $(\varepsilon '_{1}, \ldots , \varepsilon '_{m})$ drawn from the distributions of $Y$ and $\varepsilon $ , respectively. We provide an estimator for $F_{X}$ based on a direct inversion formula and the ridge-parameter regularization. Our estimator is shown to be mean consistency with respect to the mean squared error whenever the set of all zeros of the characteristic function of $\varepsilon $ has Lebesgue measure zero. We then derive some convergence rates of the mean squared error uniformly on a nonparametric class for $F_{X}$ and on some different regular classes for the density of $\varepsilon $ . A numerical example is performed to illustrate the efficiency of our method.

Efficient estimation of cumulative distribution function using moving extreme ranked set sampling with application to reliability

In this article, we consider the problem of estimating cumulative distribution function (CDF) and a reliability parameter using moving extreme ranked set sampling (MERSS). Two different CDF estimators are described and compared with their competitors in simple random sampling (SRS) and ranked set sampling (RSS). It turns out the CDF estimators in MERSS can be more efficient than their competitors in SRS and RSS at a point in a particular tail of the distribution when the quality of rankings is sufficiently good. Motivated by this efficiency gain, we develop some estimators for the stress-strength probability using MERSS. The suggested estimators are then compared with their counterparts in the literature via Monte Carlo simulation. Finally, a real dataset is used to show the applicability of the developed procedures.

Global correction of projection estimators under local constraint

This paper presents a general methodology for nonparametric estimation of a function s related to a nonnegative real random variable X, under a constraint of type \(s(0)=c\). When a projection estimator of the target function is available, we explain how to modify it in order to obtain an estimator which satisfies the constraint. We extend risk bounds from the initial to the new estimator, and propose and study adaptive procedures for both estimators. The example of cumulative distribution function estimation illustrates the method for two different models: the multiplicative noise model (\(Y=XU\) is observed, with U following a uniform distribution) and the additive noise model \((Y=X+V\) is observed where V is a nonnegative nuisance variable with known density).

Estimation of Probability Distribution and Its Application in Bayesian Classification and Maximum Likelihood Regression

Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.

An elaborate algorithm for analyzing the Borgonovo moment-independent sensitivity by replacing the probability density function estimation with the probability estimation

Borgonovo moment-independent sensitivity index (BMSI) was proposed to measure the sensitivity of model inputs according to the whole distribution of model output not only a specific moment. The main computational difficulty of the BMSI is to estimate the unconditional probability density function (PDF) and the conditional PDF of the model output. Generally, the estimation of cumulative distribution function (CDF) is easier than that of the PDF, but CDF-based method needs to calculate the extreme points of the differences between the unconditional PDF and the conditional PDF of model output. In addition, the computational cost of the existing CDF-based method also depends on the dimensionality of model inputs. To avoid these accessional computations, this paper derives a new formula by innovatively combining the law of total expectation in the successive intervals without overlapping and the Bayes theorem. The proposed new formula can obtain every input's BMSI only by one group of unconditional model inputs-output samples and does not need to estimate the PDF and the extreme points, which greatly reduces the computational difficulty of the BMSI by replacing the PDF estimation with the probability estimation. Four case studies are analyzed, and the results demonstrate the effectiveness of the proposed algorithm for estimating the BMSI.

Certain Bounds Related to Multi-Parameterized $k$ -Fractional Integral Inequalities and Their Applications

A $k$ -fractional integral identity with multiple parameters is investigated. Based on this identity, some estimation-type results related to $k$ -fractional integral inequalities for the first-order differentiable functions are obtained. These results are then applied to the estimation of cumulative distribution function and some other special means.

An advanced error correction methodology for merging in-situ observed and model-based soil moisture

In order to obtain an improved soil moisture (SM) dataset at large scale, an advanced SM merging methodology based on error correction methods was constructed to merge the model-based and in-situ observed SM data. The SM datasets in a 0–40 cm soil layer were derived from 10 km × 10 km Variable Infiltration Capacity (VIC) model and 797 in-situ stations, respectively. The merging methodology was conducted grid by grid, and mainly included two parts: bias correction and random error correction. Firstly, the bias correction was performed for the VIC simulations by applying the Cumulative Distribution Function (CDF) matching approach combined with the kriging technique. Secondly, the random error of the VIC simulations was corrected using an Optimal Interpolation (OI) technique based on a spatio-temporal correlation function which was proposed and constructed in this study. Through validations against in-situ observations, the merged SM was evaluated, and the merging errors in each step were analyzed and discussed. The results showed that the merged SM product was improved compared to the original SM data, both temporally and spatially. The SM merging methodology is effective and reliable in combining the accurate but sparse in-situ observations and the continuous VIC simulations. In addition, the spatial mismatch impact on the representativeness of in-situ stations was limited, and the merging errors were mainly produced in the CDF estimation process. The random error information in the spatial dimension exhibited a bigger impact on the random error correction comparing to that in the temporal dimension. This study provided strong encouragement for the efficient use of in-situ SM observations and provided valuable methods for combining multi-sources SM datasets.

Glivenko–Cantelli Theorem for the kernel error distribution estimator in the first-order autoregressive model

This paper considers the uniform strong consistency of the kernel estimator of the error cumulative distribution function (CDF) in the first-order autoregressive model. The classical Glivenko–Cantelli Theorem is extended to the residual based kernel smooth CDF estimator in the autoregressive model.

Histogram equalization with Bayesian estimation for noise robust speech recognition.

The histogram equalization approach is an efficient feature normalization technique for noise robust automatic speech recognition. However, it suffers from performance degradation when some fundamental conditions are not satisfied in the test environment. To remedy these limitations of the original histogram equalization methods, class-based histogram equalization approach has been proposed. Although this approach showed substantial performance improvement under noise environments, it still suffers from performance degradation due to the overfitting problem when test data are insufficient. To address this issue, the proposed histogram equalization technique employs the Bayesian estimation method in the test cumulative distribution function estimation. It was reported in a previous study conducted on the Aurora-4 task that the proposed approach provided substantial performance gains in speech recognition systems based on the acoustic modeling of the Gaussian mixture model-hidden Markov model. In this work, the proposed approach was examined in speech recognition systems with deep neural network-hidden Markov model (DNN-HMM), the current mainstream speech recognition approach where it also showed meaningful performance improvement over the conventional maximum likelihood estimation-based method. The fusion of the proposed features with the mel-frequency cepstral coefficients provided additional performance gains in DNN-HMM systems, which otherwise suffer from performance degradation in the clean test condition.

Gaussian process emulators for quantifying uncertainty in [formula omitted] spreading predictions in heterogeneous media

We explore the use of Gaussian process emulators (GPE) in the numerical simulation of CO2 injection into a deep heterogeneous aquifer. The model domain is a two-dimensional, log-normally distributed stochastic permeability field. We first estimate the cumulative distribution functions (CDFs) of the CO2 breakthrough time and the total CO2 mass using a computationally expensive Monte Carlo (MC) simulation. We then show that we can accurately reproduce these CDF estimates with a GPE, using only a small fraction of the computational cost required by traditional MC simulation. In order to build a GPE that can predict the simulator output from a permeability field consisting of 1000s of values, we use a truncated Karhunen-Loève (K-L) expansion of the permeability field, which enables the application of the Bayesian functional regression approach. We perform a cross-validation exercise to give an insight of the optimization of the experiment design for selected scenarios: we find that it is sufficient to use 100s values for the size of training set and that it is adequate to use as few as 15 K-L components. Our work demonstrates that GPE with truncated K-L expansion can be effectively applied to uncertainty analysis associated with modelling of multiphase flow and transport processes in heterogeneous media.

A note on estimating cumulative distribution functions by the use of convolution power kernels

Our paper investigates the nonparametric estimation of cumulative distribution functions of nonnegative valued random variables using convolution power kernels. Our proposed consistent estimator avoids boundary effects near the origin. We present its asymptotic properties and give a short simulation study.

Kernel estimation of cumulative distribution function of a random variable with bounded support

In the paper methods of reducing the so-called boundary effects, which appear in the estimation of certain functional characteristics of a random variable with bounded support, are discussed. The methods of the cumulative distribution function estimation, in particular the kernel method, as well as the phenomenon of increased bias estimation in boundary region are presented. Using simulation methods, the properties of the modified kernel estimator of the distribution function are investigated and an attempt to compare the classical and the modified estimators is made.

Kernel Estimation of Cumulative Distribution Function of a Random Variable with Bounded Support

Abstract In the paper methods of reducing the so-called boundary effects, which appear in the estimation of certain functional characteristics of a random variable with bounded support, are discussed. The methods of the cumulative distribution function estimation, in particular the kernel method, as well as the phenomenon of increased bias estimation in boundary region are presented. Using simulation methods, the properties of the modified kernel estimator of the distribution function are investigated and an attempt to compare the classical and the modified estimators is made.