Estimation Of Distribution Function Research Articles

Nonparametric analysis of recurrent gap time data

Recurrent event data are frequently encountered in longitudinal studies. In many applications, the times between successive recurrent events (gap times) are often of interest and lead to problems that have received much attention recently. In this article, using the approach of inverse probability-of-censoring weights (IPCW), we propose nonparametric estimators for the estimation of the bivariate distribution and survival functions for gap times of recurrent event data. We also consider the estimation of Kendall’s tau for two gap times by expressing it as an integral functional of the bivariate survival function. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate their finite sample performance.

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.

An empirical saddlepoint approximation based method for smoothing survival functions under right censoring

AbstractThe Kaplan–Meier (KM) estimator is ubiquitously used for estimating survival functions, but it provides only a discrete approximation at the observation times and does not deliver a proper distribution if the largest observation is censored. Using KM as a starting point, we devise an empirical saddlepoint approximation‐based method for producing a smooth survival function that is unencumbered by choice of tuning parameters. The procedure inverts the moment generating function (MGF) defined through a Riemann–Stieltjes integral with respect to an underlying mixed probability measure consisting of the discrete KM mass function weights and an absolutely continuous exponential right‐tail completion. Uniform consistency, and weak and strong convergence results are established for the resulting MGF and its derivatives, thus validating their usage as inputs into the saddlepoint routines. Relevant asymptotic results are also derived for the density and distribution function estimates. The performance of the resulting survival approximations is examined in simulation studies, which demonstrate a favourable comparison with the log spline method (Kooperberg & Stone, 1992) in small sample settings. For smoothing survival functions we argue that the methodology has no immediate competitors in its class, and we illustrate its application on several real data sets. The Canadian Journal of Statistics 47: 238–261; 2019 © 2019 Statistical Society of Canada

Estimation of population distribution function involving measurement error in the presence of non response

This article addresses the problem of estimating population distribution function for simple random sampling in the presence of non response and measurement error together. We suggest a general class of estimators for estimating the cumulative distribution function using the auxiliary information. The expressions for the bias and mean squared error are derived up to the first order of approximation. The performance of the proposed class of estimators is compared with considered estimators both theoretically and numerically. A real data set is used to support the theoretical findings.

A bootstrap approach for bandwidth selection in estimating conditional efficiency measures

Conditional efficiency measures are needed when the production process does not depend only on the inputs and outputs, but may be influenced by external factors and/or environmental variables (Z). They are estimated by means of a nonparametric estimator of the conditional distribution function of the inputs and outputs, conditionally on values of Z. For doing this, smoothing procedures and smoothing parameters, the bandwidths, are involved. So far, Least Squares Cross Validation (LSCV) methods have been used, which have been proven to provide bandwidths with optimal rates for estimating conditional distributions. In efficiency analysis, the main interest is in the estimation of the conditional efficiency score, which typically depends on the boundary of the support of the distribution and not on the full conditional distribution. In this paper, we show indeed that the rate for the bandwidths which is optimal for estimating conditional distributions, may not be optimal for the estimation of the efficiency scores. We propose hence a new approach based on the bootstrap which overcomes these difficulties. We analyze and compare, through Monte Carlo simulations, the performances of LSCV techniques with our bootstrap approach in finite samples. As expected, our bootstrap approach shows generally better performances and is more robust to the various Monte Carlo scenarios analyzed. We also illustrate our methodology through an empirical example using an US Aggressive-Growth Mutual Funds data set.

Exact mean integrated squared error and bandwidth selection for kernel distribution function estimators

An exact, closed form, and easy to compute expression for the mean integrated squared error (MISE) of a kernel estimator of a normal mixture cumulative distribution function is derived for the class of arbitrary order Gaussian-based kernels. Comparisons are made with MISE of the empirical distribution function, the infeasible minimum MISE, and the uniform kernel. A simple plug-in method of simultaneously selecting the optimal bandwidth and kernel order is proposed based on a non asymptotic approximation of the unknown distribution by a normal mixture. A simulation study shows that the method provides a viable alternative to existing bandwidth selection procedures.

Nonparametric estimation of multivariate distribution function for truncated and censored lifetime data

A number of models for generating statistical data in various fields of insurance, including life insurance, pensions, and general insurance have been considered. It is shown that the insurance statistics data, as a rule, are truncated and censored, and often multivariate. We propose a non-parametric estimation of the distribution function for multivariate truncated-censored data in the form of a quasi-empirical distribution and a simple iterative algorithm for its construction. To check the accuracy of the proposed evaluation of the distribution function for truncated-censored data, simulation studies have been conducted, which showed its high efficiency. The proposed estimates have been tested for many years by the IAAC Group of Companies in the actuarial valuation of corporate social liabilities according to IAS 19 Employee Benefits. Apart from insurance, some results of the work can be used, for example in medicine, biology, demography, mathematical theory of reliability, etc.

A Novel Approximation for Computation Bivariate Distribution Functions in Polygonal Area

Generally bivariate probability density function defined in a rectangular area is used to calculate the cumulative distribution function from the bivariate probability density function. However, definition limits of the probability density functions being non-rectangular are in existence in practice. In this paper, primarily arbitrary non-rectangular areas are defined by applying a polygonal approach. The polygonal area obtained as a result of this approach constitutes boundaries of the probability density function. Thus, the bivariate piecewise probability density function can be defined in an arbitrary area. Then the cumulative distribution function is calculated in the obtained area. Two types of approaches are used for these calculations. The first approach is applied to take integral analytically of bivariate continuous probability density function in the polygonal area. The second approach is developed a numerical method since the explicit integral of the selected probability density function cannot be found.

On the estimation of the quantile density function by orthogonal series

The classical estimator of a quantile density function by orthogonal series depends on the empirical distribution function estimator Hn. The fact that Hn is a step function even when the underlying cumulative distribution function is continuous, has called for the need (in certain areas of application like estimating the quantile density function ) for smooth estimators of Hn. The present work has two goals. The first one is to introduce a new technique for estimating by orthogonal series for any orthonormal system in , a smooth nonparametric estimators of and are proposed. Asymptotic properties of the proposed estimators are studied. The second is to introduce a new method for selection of a smoothing parameter. A simulation study is done to compare the performances of the new approach with the (Chesneau et al 2016) one, when comparing mean integrated square error of the two estimators.

An alternative distribution function estimation method using rational Bernstein polynomials

This paper gives a general method for nonparametric distribution function estimation using the rational Bernstein polynomials as an alternative to the current estimators. The proposed new method is compared with Bernstein polynomials and empirical distribution function methods by simulation studies. The new method guarantees monotone nondecreasing function by applying linear constraints on the coefficients of the rational Bernstein basis functions and smooth the empirical distribution function. Furthermore, as a special case, it reduces to Bernstein polynomial estimator method. Some theoretical properties of the new estimator are investigated. Simulation study shows that the proposed estimator is preferable to the Bernstein polynomials and empirical distribution function estimator methods.

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.

ТОЧНОСТЬ РАСЧЕТА ПОКАЗАТЕЛЯ ХЕРСТА И ВЕРОЯТНОСТИ ОЖИДАНИЯ ОБСЛУЖИВАНИЯ ПАКЕТОВ САМОПОДОБНОГО ТРАФИКА

Estimation of the service quality (QoS) characteristics in a single-channel system with an infinite queue of the packet network often reduces to determining the Hurst exponent of self-similar traffic, after which on the Norros formula calculates the average number of packets in the system and all other characteristics. The Hurst exponent can be determined by R/S-statistics method based on actual measurements of traffic characteristics or from probability distribution functions describing this traffic. However, the well-known formulas for calculating the Hurst exponent, which show its dependence on the parameters of the probability distribution of traffic, are not accurate. A method is proposed for improving the accuracy of calculating QoS characteristics in a packet communication network by more accurately finding the traffic self-similarity coefficient or the Hurst exponent depending on the parameters of the probability distribution function of the time interval between packets. In case when, in traffic, the time interval between packets is described by Pareto or Weibull distributions, new formulas have been obtained for calculating the traffic self-similarity factor based on the shape parameter of these distributions. After a more accurate determination of the Hurst exponent in this way, the average number of packets in the system is then calculated using the Norros formula, and then from the proposed approximation of the system state distribution function, service waiting probability of packet to calculate. With an increase in the accuracy of calculating the Hurst exponent, the accuracy of calculating the service quality characteristics also increases. Simulation modeling confirmed the validity of these methods for calculating QoS characteristics in a system with self-similar traffic. In this case, the discrepancy between the results of modeling and calculation does not exceed 5% when the system load changes in the range of 0.3 < ρ < 1 and the value of the Hurst exponent is 0.5 < H < 0.9.

Landau damping and frequency-shift of (0, 0, 2) scissors mode in a disc-shaped Bose-Einstein condensate

应用哈特里-福克-博戈留波夫平均场理论近似和基于托马斯-费米近似的解析方法, 研究碟形玻色-爱因斯坦凝聚体中(0, 0, 2)剪刀模的朗道阻尼和频移, 计算阻尼系数和频移大小以及它们的温度依赖. 计算中, 在集体激发本征频移微扰关系中考虑元激发弛豫及其弛豫之间的正交关系以获得阻尼和频移的计算公式, 把凝聚体基态波函数取为高斯分布函数的一级近似以消除托马斯-费米近似中三模耦合矩阵元的发散. 采用与相关实验研究相同的粒子数、囚禁频率和各向异性参量, 理论计算结果与相关实验测量结果相符合. 由于理论的复杂性和计算的困难性, 在大多数基于平均场理论的单分量和两分量玻色-爱因斯坦凝聚集体激发阻尼和频移的研究中采用半经典近似, 把准粒子激发能谱看成是连续的来积分计算各个准粒子跃迁对阻尼和频移的贡献, 而本文和本文前期工作按分立的准粒子激发频谱计算阻尼或频移, 并在研究过程中提出了考虑元激发弛豫及弛豫之间正交关系的改进方法, 希望这种方法对今后的工作有一定参考价值.

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Interval Estimation of Value-at-Risk Based on Nonparametric Models

Value-at-Risk (VaR) has become the most important benchmark for measuring risk in portfolios of different types of financial instruments. However, as reported by many authors, estimating VaR is subject to a high level of uncertainty. One of the sources of uncertainty stems from the dependence of the VaR estimation on the choice of the computation method. As we show in our experiment, the lower the number of samples, the higher this dependence. In this paper, we propose a new nonparametric approach called maxitive kernel estimation of the VaR. This estimation is based on a coherent extension of the kernel-based estimation of the cumulative distribution function to convex sets of kernel. We thus obtain a convex set of VaR estimates gathering all the conventional estimates based on a kernel belonging to the above considered convex set. We illustrate this method in an empirical application to daily stock returns. We compare the approach we propose to other parametric and nonparametric approaches. In our experiment, we show that the interval-valued estimate of the VaR we obtain is likely to lead to more careful decision, i.e., decisions that cannot be biased by an arbitrary choice of the computation method. In fact, the imprecision of the obtained interval-valued estimate is likely to be representative of the uncertainty in VaR estimate.

Approximate Methods for Inverting Generating Functions from the Pál-Bell Equations for Low Source Problems

A number of approximate probability distribution functions (pdf’s) for the neutron density are examined with reference to low source startup. The most accurate method for determining the safe source strength, to reduce the likelihood of a rogue transient during startup, is that arising from the Pál-Bell equations. When these equations are extended to include space and energy dependence the numerical work becomes extensive. A pdf is developed which gives results that compare favorably with those from the exact solution but requires very much less numerical work. The method is applicable to space- and energy-dependent problems. Extensive numerical examples are given of the new method and of others which have been proposed over the years. In addition, we explore other approximations, unrelated to the generating function, that can lead to substantial computational savings. We have additionally described the principles behind, and provided a simple review of, the low source algorithm from which anyone unfamiliar with low source concepts can benefit.

Estimation of the error distribution function for partial linear single-index models

We consider the estimation of the error distribution function of partial linear single-index models. The estimation methods for the error distribution function based on the classical empirical distribution function as well as empirical likelihood method are discussed, the latter method allows for incorporation of additional information on the error distribution function into estimation. We show weak convergence of the corresponding empirical processes to Gaussian processes and compare both approaches with the asymptotic theory and by means of simulation studies.

A Tractable Approach to Joint Transmission in Multiuser Visible Light Communication Networks

In this paper, an analytical model for the coverage analysis of multiuser visible light communication (VLC) networks is presented, taking into account the cooperation among access points (APs). Specifically, the cooperation is realized through joint transmission (JT), where the coordinated APs jointly transmit data to users in a noncoherent or coherent manner to reduce the inter-cell interference and enhance the useful signal power. Using a second-order moment matching approach, we find approximate distribution functions of the signal power and interference, and derive tractable results for the network coverage probability based on the signal-to-interference-plus-noise ratio (SINR). For both noncoherent and coherent JT, the derived coverage probabiklity are further simplified into closed forms when the communication link is interference-limited. We validate the derived results through Monte Carlo simulations and apply them to study behaviors and trends of the network under various parameter settings. Results show that JT can improve the coverage performance of the network and coherent JT can provide higher performance gains than its noncoherent counterpart, at the cost of higher implementation complexities.

Synthesis, characterization and application of surface-modified biochar synthesized from rice husk, an agro-industrial waste for the removal of hexavalent chromium from drinking water at near-neutral pH

This study reports the synthesis, characterization and applicability of surface-modified biochar, an eco-friendly and economic adsorbent for the removal of highly toxic hexavalent chromium from drinking water at a pH suitable for drinking water. The acid modification of the pristine biochar prepared by the pyrolysis of rice husk substantially enhanced the carbon content percentage by 25.29% and active sites by 10 times in the acid-modified biochar and produced a BET surface area of 142.78 ± 2.57 m2 g−1. Furthermore, the alteration of the surface of pristine biochar resulted in the substantial increase in C=C, C=O, C–O, phenolic and alcoholic –OH bonds on the surface of the biochar which further facilitated the adsorption of Cr. The batch adsorption study was carried out to comprehend the process of adsorption. The Freundlich isotherm model has been found to be describing the adsorption process most precisely. The maximum adsorption capacity has been found to be 4109 μg g−1 at 30 °C and 4536 μg g−1 at 50 °C from the analysis of approximate adsorption site energy distribution function. The analysis of approximate adsorption site energy distribution function further showed the availability of more energetic sites to the adsorbate Cr(VI) oxyanions at higher temperatures. Further, the endothermic nature of the adsorption apart from the experimental outcomes has been evinced from the D–R isotherm model and from the values of ∆G° and ∆H°. The negative values of ∆G° at all the experimental temperatures indicate the spontaneity of the adsorption process. Furthermore, the values of ∆G° more than − 20 kJ mole−1 indicate the physisorption of Cr(VI) oxyanions on the adsorbent surface.

Fiber orientation distribution function from non-negative sparse recovery with quantitative analysis of local fiber orientations and tractography using DW-MRI datasets

Diffusion weighted MRI (DW-MRI) is the unique non-invasive imaging modality capable of estimating in vivo the structure of the white matter. In this paper, we propose, evaluate and validate a new DW-MRI method to model and recover high quality tractogram even with multiple fiber populations in a voxel and from a limited number of acquisitions.Our method relies on the estimation of the Fiber Orientation Distribution (FOD) function, parameterized as a non-negative sum of rank-1 tensors and the use of a non-negative sparse recovery scheme to efficiently recover the tensors, and their number. Each fiber population of a voxel is characterized by the orientation and the weight of a rank-1 tensor.Using both deterministic and probabilistic tractography algorithms, we show that our method is able to accurately reconstruct narrow crossing fibers and obtain a high quality connectivity reconstruction even from a limited number of acquisitions. To this end, a validation scheme based on the connectivity recovered from tractography is developed to quantitatively evaluate and analyze the performance of our method. The tractometer tool is used to quantify the tractography obtained from a simulated DW-MRI dataset including a high angular resolution dataset of 60 gradient directions and a dataset of 30 gradient directions, each of them corrupted with Rician noise of SNR 10 and 20. The performance of our FOD model and its impact on the tractography results are also demonstrated and illustrated on in vivo DW-MRI datasets with high and low angular resolutions.