Schoenberg transformations, mapping Euclidean configurations into Euclidean configurations, define in turn a transformed inertia, whose minimization produces robust location estimates. The procedure only depends upon Euclidean distances between observations, and applies equivalently to univariate and multivariate data. The choice of the family of transformations and their parameters defines a flexible location strategy, generalizing M-estimators. Two regimes of solutions are identified. Theoretical results on their existence and stability are provided, and illustrated on two data sets.
We define a new basis of cubic splines such that the coordinates of a natural cubic spline are sparse. We use it to analyse and to extend the classical Schoenberg and Reinsch result and to estimate a noisy cubic spline. We also discuss the choice of the smoothing parameter. All our results are illustrated graphically.
Robust estimation in generalized linear models based on the normal quantiles of the probability integral transformation
Instituto de Cálculo, University of Buenos Aires（计算研究所，布宜诺斯艾利斯大学）；CONICET（阿根廷国家科学研究委员会）；Instituto de Cálculo and Department of Mathematics, FCEN, University of Buenos Aires（计算研究所与数学系，理学院，布宜诺斯艾利斯大学）
Completely nonparametric transformation models with heteroscedastic errors are considered. Despite their flexibility, such models have rarely been used so far, since estimators of the model components have been missing and even identification of such models has not been clear until very recently. The results of Kloodt (2020) are used to construct the first two estimators of the transformation function in these models. While the first estimator converges to the true transformation function at a parametric rate, the second estimator can be obtained by an explicit formula and is less computationa…
We consider estimation in a class of semiparametric transformation models for right--censored data. These models gained much attention in survival analysis; however, most authors consider only regression models derived from frailty distributions whose hazards are decreasing. This paper considers estimation in a more flexible class of models and proposes conditional rank M-estimators for estimation of the Euclidean component of the model.
Transformation models provide a common tool for regression analysis of censored failure time data. The most common approach towards parameter estimation in these models is based on the nonparametric profile likelihood method. Several authors proposed also ad hoc M-estimators of the Euclidean component of the model. These estimators are usually simpler to impelement and many of them have good practical performance. In this paper we consider the form of the information bound for estimation if the Euclidean parameter of the model and propose a modification of inefficient M-estimators to one-step …
This paper proposes consistent estimators for transformation parameters in semiparametric models. The problem is to find the optimal transformation into the space of models with a predetermined regression structure like additive or multiplicative separability. We give results for the estimation of the transformation when the rest of the model is estimated non- or semi-parametrically and fulfills some consistency conditions. We propose two methods for the estimation of the transformation parameter: maximizing a profile likelihood function or minimizing the mean squared distance from independenc…
We consider partly linear transformation models applied to current status data. The unknown quantities are the transformation function, a linear regression parameter and a nonparametric regression effect. It is shown that the penalized MLE for the regression parameter is asymptotically normal and efficient and converges at the parametric rate, although the penalized MLE for the transformation function and nonparametric regression effect are only $n^{1/3}$ consistent. Inference for the regression parameter based on a block jackknife is investigated. We also study computational issues and demons…
Consider the semiparametric transformation model $Λ_{θ_o}(Y)=m(X)+ε$, where $θ_o$ is an unknown finite dimensional parameter, the functions $Λ_{θ_o}$ and $m$ are smooth, $ε$ is independent of $X$, and $\esp(ε)=0$. We propose a kernel-type estimator of the density of the error $ε$, and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of $θ_o$ and a nonparametric kernel estimator of $m$. The practical performance of the proposed density estimator is evaluated in a simulation study.
Analysis of survival data with biased samples caused by left-truncation or length-biased sampling has received extensive interest. Many inference methods have been developed for various survival models. These methods, however, break down when survival data are typically error-contaminated. Although error-prone survival data commonly arise in practice, little work has been available in the literature for handling length-biased data with measurement error. In survival analysis, the transformation model is one of the frequently used models. However, methods of analyzing the transformation model w…
In this paper we consider a heteroscedastic transformation model, where the transformation belongs to a parametric family of monotone transformations, the regression and variance function are modelled nonparametrically and the error is independent of the multidimensional covariates. In this model, we first consider the estimation of the unknown components of the model, namely the transformation parameter, regression and variance function and the distribution of the error. We show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and es…
A transformation relation between multivariate ARMA and CARMA processes is derived through a discretization procedure. This gives a direct relationship between the discrete time and continuous time analogues, serving as the basis for an estimation method for multivariate CARMA models. We will see that the autoregressive coefficients, making up the deterministic part of a multivariate CARMA model, are entirely given by the transformation relation. An Euler discretization convergence rate of jump diffusions is found for the case of small jumps of infinite variation. This substantiates applying t…
In a linear transformation model, there exists an unknown monotone nonlinear transformation function such that the transformed response variable and the predictor variables satisfy a linear regression model. In this paper, we present CENet, a new method for estimating the slope vector and simultaneously performing variable selection in the high-dimensional sparse linear transformation model. CENet is the solution to a convex optimization problem and can be computed efficiently from an algorithm with guaranteed convergence to the global optimum. We show that under a pairwise elliptical distribu…
Kernel estimation of a probability density function supported on the unit interval has proved difficult, because of the well known boundary bias issues a conventional kernel density estimator would necessarily face in this situation. Transforming the variable of interest into a variable whose density has unconstrained support, estimating that density, and obtaining an estimate of the density of the original variable through back-transformation, seems a natural idea to easily get rid of the boundary problems. In practice, however, a simple and efficient implementation of this methodology is far…
We consider semiparametric transformation models, where after pre-estimation of a parametric transformation of the response the data are modeled by means of nonparametric regression. We suggest subsequent procedures for testing lack-of-fit of the regression function and for significance of covariables, which - in contrast to procedures from the literature - are asymptotically not influenced by the pre-estimation of the transformation. The test statistics are asymptotically pivotal and have the same asymptotic distribution as in regression models without transformation. We show validity of a mu…
Copula modelling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily affected by boundary bias issues. In addition, most common copulas admit unbounded densities, and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimator is proposed. It is based on the idea of transforming the uniform marginals of the cop…
In this paper we deal with the estimation of population variance of the study variable y using auxiliary information on variable x. A family of ratio and product-type estimators are proposed using suitable transformation on both random variable x(auxiliary variable) and(study variable).Up to the first order of approximation the expression of mean square error and Bias term are obtained. An empirical study is carried out to illustrate the performance of the constructed estimator over others.
This paper investigates the statistical properties of non-linear transformations (NLT) of random variables, in order to establish useful tools for estimation and information theory. Specifically, the paper focuses on linear regression analysis of the NLT output and derives sufficient general conditions to establish when the input-output regression coefficient is equal to the \emph{partial} regression coefficient of the output with respect to a (additive) part of the input. A special case is represented by zero-mean Gaussian inputs, obtained as the sum of other zero-mean Gaussian random variabl…
In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (ϕ\circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability μ_s on [0,1] and a continuous by part function $ϕ:[0,1] \to \R$. We consider here $T_s:[0,1]\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorre…
A class of random non-stationary signals termed timbre x dynamics is introduced and studied. These signals are obtained by non-linear transformations of sta-tionary random gaussian signals, in such a way that the transformation can be approximated by translations in an appropriate representation domain. In such situations, approximate maximum likelihood estimation techniques can be de-rived, which yield simultaneous estimation of the transformation and the power spectrum of the underlying stationary signal. This paper focuses on the case of modulation and time warping of station-ary signals, a…
