Strong Uniform Consistency Rate Research Articles

On the behavior of the Lynden-Bell estimator in widely orthant-dependent model

. This article is devoted to investigating some asymptotic properties of the Lynden-Bell and the cumulative risk function estimators when the survival and the truncated times form widely orthant-dependent sequences. As main results, we establish strong uniform consistency rates for the proposed estimators. Then, as an application, we derive a strong uniform consistency rate for the hazard rate function estimator. The performances of the proposed estimators are illustrated via some simulations.

Strong Uniform Consistency Rates of Conditional Density Estimation in the Single Functional Index Model for Functional Data Under Random Censorship

Strong Uniform Consistency Rates of Conditional Density Estimation in the Single Functional Index Model for Functional Data Under Random Censorship

A kernel regression model for panel count data with time-varying coefficients

For the conditional mean function of panel count model with time-varying coefficients, we propose to use local kernel regression method for estimation. Partial log-likelihood with local polynomial is formed for estimation. Under some regularity conditions, strong uniform consistency rates are obtained for the local estimator. At target time point, we show that the local estimator converges in distribution to normal distribution. The baseline mean function estimator is also shown to be consistent. Simulation studies show that the time-varying coefficient estimator is close to the true value, the empirical coverage probabilities of the confidence interval is close to the nominal level. We also applied the proposed method to analyze a clinical study on childhood wheezing.

Strong uniform consistency rate of an M-estimator of regression function for incomplete data under α-mixing condition

In this paper, we propose a non parametric M-estimator of the regression function and we investigate its asymptotic properties, when the response variable is subject to both random left truncation and right censoring. In most works, non parametric M-estimation requires the use of an objective function ψ supposed to be bounded. Here the results hold with unbounded objective function. The strong uniform consistency rate is established under α-mixing dependence. A large simulation study with one and bi-dimensional regressor is conducted for fixed and local bandwidths to highlight the good behavior of our estimator.

Asymptotic Results for Truncated-censored and Associated Data

Left-truncation and right-censoring arise frequently when considering lifetime data. When both incompleteness conditions occur, a product-limit estimator was proposed and investigated in the independent case by Tsai et al. (Biometrika74, 883–886, 1987). In the presence of covariates, the conditional version was studied in the α-mixing setting by Liang et al. (Test21, 790–810, 2012). Our objective in the present paper is to derive strong uniform consistency rates for the cumulative hazard and the product-limit estimates when the lifetime observations form an associated sequence. Then, as an application we derive a strong uniform consistency rate for the kernel estimator of the hazard rate function considered by Uzunoḡullari and Wang (Biometrika79, 297–310, 1992) in the iid case.

Conditional Quantile Estimation for Truncated and Associated Data

In survival or reliability studies, it is common to have data which are not only incomplete but weakly dependent too. Random truncation and censoring are two common forms of such data when they are neither independent nor strongly mixing but rather associated. The focus of this paper is on estimating conditional distribution and conditional quantile functions for randomly left truncated data satisfying association condition. We aim at deriving strong uniform consistency rates and asymptotic normality for the estimators and thereby, extend to association case some results stated under iid and α-mixing hypotheses. The performance of the quantile function estimator is evaluated on simulated data sets.

Strong uniform consistency rates of conditional quantile estimation in the single functional index model under random censorship

AbstractThe main objective of this paper is to non-parametrically estimate the quantiles of a conditional distribution in the censorship model when the sample is considered as an -mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we estimate the quantiles by inverting this estimated cond-cdf and state the asymptotic properties when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. This approach can be applied in time series analysis.

Strong uniform consistency of a kernel conditional quantile estimator for censored and associated data

In survival or reliability studies, it is common to deal with data which are not only incomplete but weakly dependent too. Random right-censoring and random left-truncation are two common forms of such data when they are neither independent nor strongly mixing but rather associated. In this paper, we focus on kernel estimation of the conditional quantile function of a strictly stationary associated random variable $T$ given a $d$-dimensional vector of covariates $X$, under random right-censoring. As main results, we establish a strong uniform consistency rate for the estimator. Then the finite sample performance of the estimator is illustrated on a simulation study.

Strong uniform consistency rates of kernel estimators of cumulative distribution functions

ABSTRACTIn this paper we consider the uniform strong consistency, along with a rate, of the cumulative distribution function (CDF) estimator. We extend the extended Glivenko–Cantelli lemma (for empirical distribution function) in Fabian and Hannan (1985, pp. 80–83) to the kernel estimator of the CDF.

A Confidence Corridor for Expectile Functions

Let (X1; Y1), …, (Xn; Yn) be i.i.d. rvs and denote v(x) be the unknown τ - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x) – v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |vn(x) – v(x)|. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.

A note on estimating the conditional expectation under censoring and association: strong uniform consistency

Let $$\left\{ (X_{i},Y_{i}), i \ge 1 \right\} $$ be a strictly stationary sequence of associated random vectors distributed as (X, Y). This note deals with kernel estimation of the regression function $$r(x)=\mathbb {E}[Y|X=x]$$ in the presence of randomly right censored data caused by another variable C. For this model we establish a strong uniform consistency rate of the proposed estimator, say $$r_{n}(x)$$ . Simulations are drawn to illustrate the results and to show how the estimator behaves for moderate sample sizes.

Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data

In this paper, we investigate a nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the uniform almost complete convergence rates and the asymptotic normality of the conditional density kernel estimator, when the variables satisfy the strong mixing dependency, based on the single-index structure. The asymptotic \((1-\zeta )\) confidence intervals of conditional density function are given, for \(0 < \zeta < 1\). We further demonstrate the impact of this functional parameter to the conditional mode estimate. Simulation study is also presented. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.

Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the functional single-index model

Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the functional single-index model

Simultaneous confidence bands for expectile functions

Expectile regression, as a general M smoother, is used to capture the tail behaviour of a distribution. Let (X 1,Y 1),…,(X n ,Y n ) be i.i.d. rvs. Denote by v(x) the unknown τ-expectile regression curve of Y conditional on X, and by v n (x) its kernel smoothing estimator. In this paper, we prove the strong uniform consistency rate of v n (x) under general conditions. Moreover, using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1|v n (x)−v(x)|. According to the asymptotic theory, we construct simultaneous confidence bands around the estimated expectile function. Furthermore, we apply this confidence band to temperature analysis. Taking Berlin and Taipei as an example, we investigate the temperature risk drivers to these two cities.

A Confidence Corridor for Expectile Functions

Let (X1; Y1), ..., (Xn; Yn) be i.i.d. rvs and let v(x) be the unknown tau - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x)-v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup06x61 jvn(x)o€€€v(x)j. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.

Uniform law of the logarithm for the local linear estimator of the conditional distribution function

In this Note, we first recall the results of the behaviour of the nonparametric estimator of the conditional distribution function which we can find in the literature. We establish exact rate of strong uniform consistency for the local linear estimator of the conditional distribution function. Our methods of proofs are based upon modern empirical process theory in the spirit of the results of Einmahl and Mason (2000) [5] and Deheuvels and Mason (2004) [3].

UNIFORM BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATES OF M-REGRESSION AND ITS APPLICATION TO THE ADDITIVE MODEL

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Yi,Xi)}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

CONFIDENCE BANDS IN QUANTILE REGRESSION

Let (X1, Y1), …, (Xn, Yn) be independent and identically distributed random variables and let l(x) be the unknown p-quantile regression curve of Y conditional on X. A quantile smoother ln(x) is a localized, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) – l(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |ln(x) − l(x)|. The derived result helps in the construction of a uniform confidence band for the quantile curve l(x). This confidence band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.

The Stochastic Fluctuation of the Quantile Regression Curve

Let (X1, Y1), . . ., (Xn, Yn) be i.i.d. rvs and let l(x) be the unknown p-quantile regression curve of Y on X. A quantile-smoother ln(x) is a localised, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) - l(x)}. Using strong approximations of the empirical process and extreme value theory allows us to consider the asymptotic maximal deviation sup06x61 |ln(x)-l(x)|. The derived result helps in the construction of a uniform confidence band for the quantile curve l(x). This confidence band can be applied as a model check, e.g. in econometrics. An application considers a labour market discrimination effect.

Uniform limit laws of the logarithm for estimators of the additive regression function in the presence of right censored data

It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function \citeDebbarhViallon3. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.