The receiver operating characteristic (ROC) curve describes the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a variable measured on a continuous scale. The data consist of a training set of m responses $X_1, \dots, X_m$ from healthy individuals and n responses $Y_1, \dots, Y_n$ from diseased individuals. The responses are assumed i.i.d. from unknown distributions F and G, respectively. We consider estimation of the…
This paper derives higher-order terms in the double-saddlepoint expansion of Skovgaard for a unidimensional conditional cumulative distribution function. Expansions for continuous and lattice random variables are derived. Results are applied to the sufficient statistic in logistic regression.
We give a stochastic expansion for the empirical distribution function $\hat{F}_n$ of residuals in a p-dimensional linear model. This expansion holds for p increasing with n. It shows that, for high-dimensional linear models, $\hat{F}_n$ strongly depends on the chosen estimator $\hat{\theta}$ of the parameter $\theta$ of the linear model. In particular, if one uses an ML-estimator $\hat{\theta}_{ML}$ which is ML motivated by a wrongly specified error distribution function G,…
Direct use of the empirical quantile function provides a standard distribution-free approach to constructing confidence intervals and confidence bands for population quantiles. We apply this method to construct confidence intervals and confidence bands for regression quantiles and to construct prediction intervals based on sample regression quantiles. Comparison of the direct method with the studentization and the bootstrap methods are discussed. Simulation results show that…
SUMMARY It is shown that an asymptotically precise one-term correction to the asymptotic distribution function of the classical Cramér-von Mises statistic approximates the exact distribution function remarkably closely for sample sizes as small as 7 or even smaller. This correction can be quickly evaluated, and hence it is suitable for the computation of practically exact p-values when testing simple goodness of fit. Similar findings hold for Watson's rotationally invariant…
Frank A. Cowell · Maria-Pia Victoria-Feser·刊期 1996-01 · 卷 64（1）·Econometrica已编入正刊
不平等测度的稳健性AI 解读
Inequality measures are often used to summarize information about empirical income distributions. However the resulting picture of the distribution and of changes in the distribution can be severely distorted if the data are contaminated. The nature of this distortion will in general depend upon the underlying properties of the inequality measure. This issue is investigated theoretically using a technique based on the influence function, and the magnitude of the effect is…
Linear models with error components are widely used to analyse panel data. Some applications of these models require knowledge of the probability densities of the error components. Existing methods handle this requirement by assuming that the densities belong to known parametric families of distributions (typically the normal distribution). This paper shows how to carry out nonparametric estimation of the densities of the error components, thereby avoiding the assumption…
A new transformation model for two survival curves with possibly nonproportional hazards is presented. The model assumes the existence of an unknown transformation of the survival curves into Weibull distributions with different shape parameters. The partial likelihood approach fails in the model. A new method of estimation called the empirical process approach is proposed. This approach is based on the strong approximation of the Kaplan-Meier product-limit estimators and…
Various smoothing methods for quantile density estimation are unified into a generalized kernel smoothing. Based on a stochastic upper bound of the derivatives sequence for a sequence of smoothed Brownian bridges, uniform in-probability consistency of generalized kernel quantile density estimators on any closed subinterval of the open unit interval is derived.
Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index $\beta$ will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative $\beta$ the new estimator…
Consider an iid sample $Y_1, \dots, Y_n$ of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution $H_{\beta}, \beta \epsilon \mathbb{R}$. We investigate the testing problem $\beta = \beta_0$ against a sequence $\beta = \beta_n$ of contiguous 0 n alternatives, based on the point processes $N_n$ of the exceedances among $Y_i$ over a sequence of thresholds $t_n$. It turns out that…
In a nonrandom sample or a nonrandomized comparison of treated and control groups, methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection. Quantiles of a single distribution and for a shift in two distributions are both discussed. These confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized…
In a nonrandom sample or a nonrandomized comparison of treated and control groups, methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection. Quantiles of a single distribution and for a shift in two distributions are both discussed. These confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized…
Equilibrium search models generalize the one-sided models used earlier by endogenizing the wage offer distribution. Attempts to estimate versions of these equilibrium n search models have been unsuccessful, ironically because the predicted wage distribution did not fit the wage data. In this paper we extend the applicability of search models by introducing firm heterogeneity to account for the shape of the wage density. The resulting estimation problem is difficult, but we…
SUMMARY Two types of semiparametric model are proposed for comparing two samples with possibly heterogeneous treatment effects. One model is a two-sample location-scale model which assumes that the Q-Q-plot of the two distributions involved is linear; the other is the class of two-sample transformation models which assume parametric forms for a probability plot—the receiver operating characteristic curve. The empirical process approach is used to construct strong…
The order of the Kolmogorov-Smirnov distance for the bootstrap of $U$-quantiles is considered. We observe that the order of the bootstrap of $U$-quantiles depends on the order of the local variance of the first term of the Hoeffding decomposition at the $U$-quantile. This order can be smaller than the order of the bootstrap of quantiles: $U$-quantiles can be smoother than quantiles.
It is shown that, in the nonparametric setting for the so-called Wicksell problem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than $n^{-1/2}\sqrt{\log n}$. We present an isotonic estimator of the distribution function which attains this rate and derive its asymptotic (normal) distribution. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator.
Abstract This paper explores the relationship between conventional models for binary response such as the probit and logit, and the proportional hazard (PH) and related specifications for grouped duration data. I outline a general class of hazard models for grouped duration data based upon the choice of period‐specific distribution functions, facilitating a thorough analysis of the implications of various specifications and consideration of various issues of model…
Motivated by some problems arising from multiclinic trials, we consider stratified two-sample designs. Nonparametric effects are defined and nonparametric hypotheses are formulated in a design where treatment, centers (strata), and interactions are assumed to be fixed factors. The interpretation of the nonparametric effects and hypotheses is analyzed in two classes of semiparametric models: the linear models and models with Lehmann alternatives. The case where centers and…
A new model for the analysis of lifetime data in the presence of a covariate is derived based on physical and statistical considerations. The model depends on five parameters that have clear physical interpretations. Two methods for the estimation of the parameters and of the quantiles are presented: one based on the order statistics and the other a regression estimator. The quantile estimators as well as the estimators of four of the five parameters are given in closed…
We consider the problem of interval estimation of probabilities for randomly truncated data. The best-known method is based on the normal approximation of the product-limit estimator and has a drawback that it may produce intervals containing impossible values outside the range [0, 1]. Its small-sample performance is also not satisfactory. In this article we investigate an alternative approach that derives confidence intervals directly from a conditional nonparametric…
Some new diagnostic measures in discriminant analysis are proposed. They can be expressed in terms of the two fundamental influence statistics in discriminant analysis: d i 2 and ψ i . A theorem on the asymptotic distributions of the fundamental statistics is derived. Based on the theorem, the proposed measures can be shown to be asymptotically distributed as functions of independent chi-squared and standard normal random variables. Critical values and expected quantiles of…
Anthony F. Shorrocks·刊期 1995-09 · 卷 63（5）·Econometrica已编入正刊
再论森贫困指数AI 解读
A modification of the official Sen poverty index yields an index which is not only continous in individual incomes and consistent with the transfer axiom, but also admits a geometric interpretation in terms of the area beneath a graph called the inverse generalized Lorenz curve for normalized poverty gaps by Jenkins and Lambert (1993) or the poverty gap profile by Shorrocks (1994).
This Monte Carlo study examines several estimation procedures of the asymptotic covariance matrix in the quantile and censored quantile regression models: design matrix bootstrap, error bootstrapping, order statistic, sigma bootstrap, homoskedastic kernel, and heteroskedastic kernel. The Monte Carlo samples are drawn from two alternative data sets: (a) the unaltered Current Population Survey (CPS) for 1987 and (b) this CPS data with independence between error term and…
The receiver operating characteristic (ROC) curve describes the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a variable measured on a continuous scale. The data consist of a training set of m responses X1,…,Xm from healthy individuals and n responses Y1,…,Yn from diseased individuals. The responses are assumed i.i.d. from unknown distributions F and G, respectively. We consider estimation of the…
We give a stochastic expansion for the empirical distribution function F^n of residuals in a p-dimensional linear model. This expansion holds for p increasing with n. It shows that, for high-dimensional linear models, F^n strongly depends on the chosen estimator θ^ of the parameter θ of the linear model. In particular, if one uses an ML-estimator θ^ML which is ML motivated by a wrongly specified error distribution function G,…
Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index β will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative β the new estimator…
Consider an iid sample Y1,…,Yn of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution Hβ,βϵR. We investigate the testing problem β=β0 against a sequence β=βn of contiguous 0 n alternatives, based on the point processes Nn of the exceedances among Yi over a sequence of thresholds tn. It turns out that…
The Asymptotic Accuracy of the Bootstrap of U-Quantiles
The order of the Kolmogorov-Smirnov distance for the bootstrap of U-quantiles is considered. We observe that the order of the bootstrap of U-quantiles depends on the order of the local variance of the first term of the Hoeffding decomposition at the U-quantile. This order can be smaller than the order of the bootstrap of quantiles: U-quantiles can be smoother than quantiles.
It is shown that, in the nonparametric setting for the so-called Wicksell problem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than n−1/2logn. We present an isotonic estimator of the distribution function which attains this rate and derive its asymptotic (normal) distribution. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator.