Estimation Of Distribution Function Research Articles

Estimation of finite population distribution function of sensitive variable*

The finite population proportion of a sensitive characteristic is estimated indirectly by using Randomized Response (RR) Techniques (RRT’s) pioneered by Warner (1965) followed by several other RRT’s in the literature. The existing literature contains several RRT’s for estimating the finite population mean of the sensitive quantitative variable. However, there might be a situation when the population proportion bearing the value of the stigmatizing variable below a threshold is of more concern than the exact population mean. The problem hence reduces to the estimation of the finite population distribution function of a quantitative sensitive variable. Following Chaudhuri and Saha (2004), a logistic regression approach has been used to estimate the finite population proportion bearing value of the stigmatizing variable below a threshold. As an alternative to this method, this article also attempts to provide suitable modifications for sensitive variables, in the estimation of distribution function proposed by Chaudhuri and Shaw (2020), when the variable of interest is innocuous. Numerical results based on a simulated population present interesting finding on the proposed methodologies.

A phase II trial of bevacizumab and rucaparib in recurrent carcinoma of the cervix or endometrium.

5527 Background: Treatment options for patients with recurrent cervical and endometrial cancer remain limited. Even with optimum care, median survival has stalled at 12-17 months. The PARP inhibitor rucaparib has demonstrated activity in both BRCA wild-type and mutant cancers. Furthermore, preclinical studies suggest a synergistic effect of PARP inhibitors and antiangiogenic agents. We hypothesized that the combination of rucaparib and the VEGF inhibitor bevacizumab would yield a clinically-significant anti-cancer effect in patients with persistent or recurrent cervical or endometrial carcinoma. Methods: NCT03476798 is a phase II trial of adults with histologically-documented carcinoma of the cervix or endometrium. Patients with evaluable lesions who had undergone at least one prior line of systemic therapy, had adequate performance status and organ function, with a life expectancy of at least three months were eligible. Biopsies were obtained prior to treatment initiation for assessment of baseline tumor biomarkers, including ARID1A mutation status. Each cycle comprised 21 days. Rucaparib was administered orally at 600 mg, twice daily. Bevacizumab was administered by IV at 15 mg/kg on day 1 of each cycle. The primary objective was to estimate the proportion of patients with persistent or recurrent cervical or endometrial cancer who survive progression-free for at least six months (PFS6). Kaplan-Meier analysis was used to estimate progression-free survival. Results: There were 28 evaluable patients; six had cervical and 22 had endometrial cancer. Median age was 60.5 years (range, 30-74). Self-reported patient races were White (82.1%), Black (10.7%), and Native American (7.1%). Self-identified Hispanic or Latina patients comprised 3.6% of the cohort. Twenty-two of 28 patients had progressive disease by six months [survival distribution function estimate = 0.214 (lower CI, 0.087; upper CI, 0.378)]. Of the six patients who achieved PFS6, one had cervical and five had endometrial cancer. Six patients had a mutation in the ARID1A gene and those patients achieved PFS6 at a rate of 66.7%. Conclusions: The study hypothesis was evaluated in a two-stage design, and the interim analysis occurred once 28 evaluable patients were enrolled. In order to move on to the second stage, at least seven patients needed to remain progression-free at six months, but only six did. Thus, the study was ended after the interim analysis. The combination of rucaparib and bevacizumab did not provide the expected clinical benefit in this cohort of patients, but may warrant further exploration in patients with ARID1A mutations. Clinical trial information: NCT03476798.

Smooth kNN Local Linear Estimation of the Conditional Distribution Function

Previous works were dedicated to the functional k-Nearest Neighbors (kNN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of (Xi,Yi)i=1,…,n of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of Yi given functional input variable Xi. Precisely, we combine the kNN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.

Simultaneous inference for Kendall’s tau

We introduce multiple contrast tests and simultaneous confidence intervals for rank correlation measures in general multivariate factorial designs. To this end, we derive the unconditional asymptotic joint sampling distribution of multiple correlation coefficients under the null and arbitrary alternatives. We neither require distributions to be discrete nor continuous and adjust for ties using a normalized version of the bivariate distribution function and scale point estimators appropriately to obtain Kendall’s τA and τB, Somers’ D, and Goodman and Kruskal’s γ. Simulation studies for a range of scenarios indicate that the proposed methods control the family wise error rate in the strong sense even when sample sizes are rather small. A case study on the iris flower data set demonstrates how to perform inference in practice.

A Study on Modelling of Bivariate Competing Risks with Archimedean Copulas

In this study we consider Archimedean copula functions to obtain estimates of cause-specific distribution functions in bivariate competing risks set up. We assume that two failure times of the same group are dependent and this dependency can be modeled by an Archimedean copula. Based on the Archimedean copula which gives best fit to the competing risk data with independent censoring we obtain the estimates of cause specific sub distributions.

Nonparametric inference for P(X < Y ) with paired variables

We propose two classes of nonparametric point estimators of θ = P(X < Y ) in the case where (X, Y ) are paired, possibly dependent, absolutely continuous random variables. The proposed estimators are based on nonparametric estimators of the joint density of (X, Y ) and the distri bution function of Z = Y − X. We explore the use of several density and distribution function estimators and characterise the convergence of the re sulting estimators of θ. We consider the use of bootstrap methods to obtain confidence intervals. The performance of these estimators is illustrated us ing simulated and real data. These examples show that not accounting for pairing and dependence may lead to erroneous conclusions about the rela tionship between X and Y .

Gradient based empirical cumulative distribution function approximation for robust aerodynamic design

A robust optimization approach based on the use of the conditional value at risk function is presented, together with an application to a robust transonic aerodynamic design problem of the central section of a Blended Wing-Body configuration. The conditional value at risk is estimated using an approach based on the empirical cumulative probability distribution function. The quantities of interest of the risk function are the aerodynamic characteristics of the airfoil, namely lift, drag, and pitching moment coefficients, computed solving the Reynolds-averaged Navier-Stokes equations with the open-source fluid-dynamic solver SU2. Conditional value at risk computation is costly, so techniques and methods for the reduction of the computational cost are introduced. In particular, the empirical cumulative distribution function is approximated with a first-order series expansion using efficiently calculated gradients from SU2 discrete adjoint solver.

Enhanced estimation of population distribution function in the presence of non-response

This paper addresses the problem of estimation of the population distribution function in the presence of non response which may be useful for data analysis in socio-economic surveys. We have proposed some enhanced difference and ratio type estimators using the information on two auxiliary variables in two well defined situations of non response. We derive the expressions of bias and mean square error of the proposed estimators to the first order approximation. In the empirical study it has been shown that the proposed estimators perform better than other estimators which are constructed using one auxiliary variable. A simulation study is also carried out in support of this work.

Determining the proton content with a quantum computer

We present a first attempt to design a quantum circuit for the determination of the parton content of the proton through the estimation of parton distribution functions (PDFs), in the context of high energy physics (HEP). The growing interest in quantum computing and the recent developments of new algorithms and quantum hardware devices motivates the study of methodologies applied to HEP. In this work we identify architectures of variational quantum circuits suitable for PDFs representation (qPDFs). We show experiments about the deployment of qPDFs on real quantum devices, taking into consideration current experimental limitations. Finally, we perform a global qPDF determination from collider data using quantum computer simulation on classical hardware and we compare the obtained partons and related phenomenological predictions involving hadronic processes to modern PDFs.

Boundary-free kernel-smoothed goodness-of-fit tests for data on general interval

We propose kernel-type smoothed Kolmogorov-Smirnov and Cramér-von Mises tests for data on general interval, using bijective transformations. Though not as severe as in the kernel density estimation, utilizing naive kernel method directly to those particular tests will result in boundary problem as well. This happens mostly because the value of the naive kernel distribution function estimator is still larger than 0 (or less than 1) when it is evaluated at the boundary points. This situation can increase the errors of the tests especially the second-type error. In this article, we use bijective transformations to eliminate the boundary problem. Some numerical studies illustrating the estimator and the tests’ performances will be presented in the last part of this article.

Sharp minimax distribution estimation for current status censoring with or without missing

Nonparametric estimation of the cumulative distribution function and the probability density of a lifetime X modified by a current status censoring (CSC), including cases of right and left missing data, is a classical ill-posed problem with biased data. The biased nature of CSC data may preclude us from consistent estimation unless the biasing function is known or may be estimated, and its ill-posed nature slows down rates of convergence. Under a traditionally studied CSC, we observe a sample from $(Z,\Delta )$ where a continuous monitoring time $Z$ is independent of $X$, $\Delta :=I(X\leq Z)$ is the status, and the bias of observations is created by the density of $Z$ which is estimable. In presence of right or left missing, we observe corresponding samples from $(\Delta Z,\Delta )$ or $((1-\Delta )Z,\Delta )$; the data are again biased but now the density of $Z$ cannot be estimated from the data. As a result, to solve the estimation problem, either the density of $Z$ must be known (like in a controlled study) or an extra cross-sectional sampling of $Z$, which is typically simpler than an underlying CSC study, be conducted. The main aim of the paper is to develop for this biased and ill-posed problem the theory of efficient (sharp-minimax) estimation which is inspired by known results for the case of directly observed $X$. Among interesting aspects of the developed theory: (i) While sharp-minimax analysis of missing CSC may follow the classical Pinsker’s methodology, analysis of CSC requires a more complicated estimation procedure based on a special smoothing in both frequency and time domains; (ii) Efficient estimation requires solving an old-standing problem of approximating aperiodic Sobolev functions; (iii) If smoothness of the cdf of $X$ is known, then its rate-minimax estimation is possible even if the density of $Z$ is rougher. Real and simulated examples, as well as extensions of the core models to dependent $X$ and Z and case-control CSC, are presented.

On Some Smooth Estimators of the Quantile Function for a Stationary Associated Process

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

The strong uniform consistency of kernel estimator of a smooth distribution function in censored linear regression

ABSTRACT This paper considers the kernel estimation of a smooth error distribution function in linear regression with right-censored data. Based on the Kaplan–Meier estimator of the residual distribution, we define the kernel-smoothed estimator of the error distribution function. Under the appropriate conditions, uniformly strong consistency of the proposed estimator is obtained.

Refined descriptive sample inference in the estimation of the cumulative distribution function

In this paper, we compare the estimators of the cumulative distribution function obtained by the refined descriptive (RD) sample and simple random (SR) sample, and show that the one obtained by RD sample is more efficient than that obtained by SR sample. For this purpose, we have compared the probability structure of RD sample and SR sample where the latter is generated using the principle of the RD sampling method. Simulation results of different distributions are given to validate the proposed theoretical comparison.

Approximation of Multiparticle Distribution Function in Micromagnetic Modeling

We propose the new system of equations for magnetodynamics. It is based on BBGKY hierarchy and uses a new approximations of mulit-particles distribution functions, which takes nearest-neighbour correlations into account. Thus we call the new system of equations correlational magnetodynamics system (CMD). It consists of well known Landau–Lifshitz–Bloch(LLB) equation complemented with an equation for two-particles correlations. Compared to traditional LLB equation, numerical simulations with CMD produces results that are closer to atomistic simulations.

Estimation of finite population distribution function with dual use of auxiliary information under non-response.

In this paper, we propose two new families of estimators for estimating the finite population distribution function in the presence of non-response under simple random sampling. The proposed estimators require information on the sample distribution functions of the study and auxiliary variables, and additional information on either sample mean or ranks of the auxiliary variable. We considered two situations of non-response (i) non-response on both study and auxiliary variables, (ii) non-response occurs only on the study variable. The performance of the proposed estimators are compared with the existing estimators available in the literature, both theoretically and numerically. It is also observed that proposed estimators are more precise than the adapted distribution function estimators in terms of the percentage relative efficiency.

Applications of Meijer’s Factorization Theorems in Performance Analyses of All-Optical Multi-Hop FSO Systems

The use of bivariate Fox H-functions (BFHFs) in performance analyses of wireless communication systems has gained considerable attention in past few decades. However, the non-existence of robust built-in routines for evaluating such functions in standard computing systems poses numerous challenges in numerical experiments and simulations. Motivated by the apparent need to circumvent these difficulties in performance analyses of cooperative wireless communications, this work presents an alternative method for obtaining the exact, approximate and asymptotic BFHF-free cumulative distribution function (CDF) of the end-to-end (e2e) signal-to-noise ratio (SNR) of multi-hop amplify-and-forward (AF) relaying wireless communication systems. As an illustration, the e2e performance analysis of an all-optical dual-hop free-space optical (FSO) transmission system over Gamma-Gamma turbulence in the presence of pointing errors is revisited. Specifically, new mathematical formulae for the statistical characteristics of the e2e SNR for systems with AF fixed-gain relaying as well as channel-state-information(CSI)-assisted using heterodyne detection (HD) or intensity modulation with direct detection (IM/DD) are derived in terms of mathematically malleable and uniformly convergent infinite series of weighted Meijer G-functions. The usefulness of the derived CDFs is illustrated through derivation of traditional system performance metrics. The accuracy of the derived analytical formulae is verified via Monte Carlo simulations in MATLAB®. Finally, based on results observed in this paper, useful expansions of common BFHFs in terms of easily computable univariate hypergeometric functions are proposed.

Estimation and inference for distribution and quantile functions in endogenous treatment effect models

Given a standard endogenous treatment effect model, we propose nonparametric estimation and inference procedures for the distribution and quantile functions of the potential outcomes among compliers, as well as the local quantile treatment effect function. The preliminary distribution function estimator is a weighted average of indicator functions, but is not monotonically increasing in general. We therefore propose a simple monotonizing method for proper distribution function estimation, and obtain the quantile function estimator by inversion. Our monotonizing method is an alternative to Chernozhukov et al. (2010) and is arguably preferable when the outcome has unbounded support. We show that all the estimators converge weakly to Gaussian processes at the parametric rate, and propose a multiplier bootstrap for uniform inference. Our uniform results thus generalize the pointwise theory developed by Frölich and Melly (2013). Monte Carlo simulations and an application to the effect of fertility on family income distribution illustrate the use of the methods. All results extend to the subpopulation of treated compliers as well.

Estimating General Parameters from Non-Probability Surveys Using Propensity Score Adjustment

This study introduces a general framework on inference for a general parameter using nonprobability survey data when a probability sample with auxiliary variables, common to both samples, is available. The proposed framework covers parameters from inequality measures and distribution function estimates but the scope of the paper is broader. We develop a rigorous framework for general parameter estimation by solving survey weighted estimating equations which involve propensity score estimation for units in the non-probability sample. This development includes the expression of the variance estimator, as well as some alternatives which are discussed under the proposed framework. We carried a simulation study using data from a real-world survey, on which the application of the estimation methods showed the effectiveness of the proposed design-based inference on several general parameters.

Dimension-reduced semiparametric estimation of distribution functions and quantiles with nonignorable nonresponse

To estimate distribution functions and quantiles of a response variable when the data having nonignorable nonresponse and the dimension of covariate is not low, this article assumes that the propensity follows a general semiparametric model, but the distribution of the response variable and related covariates is unspecified. To address the identifiability problem, an instrumental covariate, which is related to the response variable but unrelated to the propensity given the response variable and other covariates, is used to construct sufficient instrumental estimating equations. Three different semiparametric estimation methods are developed based on inverse probability weighting, mean imputation, and augmented inverse probability weighting. Furthermore to improve the efficiency and alleviate the curse of dimensionality, the sufficient dimension reduction technique is employed to produce efficient kernel estimation, and a class of dimension-reduced estimators for distribution functions and quantiles is proposed. Consistency and asymptotic normality of the proposed estimators are established. It can be shown that these estimators are asymptotically equivalent. The finite-sample performance of the estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.