Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Summary Given a random sample of sizen from a densityf ₀ on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ ₀=−f ₀ ^′ /f⁰. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫[ψ ²−2ψ′]dF_n where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF _n is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the caseJ(ψ)=∫[d ²ψ/dx²]² is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators. This research was supported by the Office of Naval Research under Grant Number N00014-82-C-0062.     

Asymptotic expansion for distribution function of moment estimator for the extreme-value index

Asymptotic expansion for distribution function of the moment estimator for the extreme-value index is obtained under reasonable conditions of second order regular variation.     

The circle as an attractor of an iterated function system on the plane

We give an iterated function system (IFS) on the plane with the circle as attractor. In doing this, we also give a sufficient condition for radially contracting functions on the plane (or on Rn) to be a contraction. A counterexample shows that radial contractiveness is not enough to be a contraction.     

长方体上均匀分布的密度函数

讨论了长方体上均匀分布密度函数问题,得到了长方体体积的估计量、估计量的点估计及估计量的密度函数.     

A note on the universal consistency of the kernel distribution function estimator

The problem of universal consistency of data driven bandwidth selectors for the kernel distribution estimator is analyzed. We provide a uniform in bandwidth result for the kernel estimate of a continuous distribution function. Our smoothness assumption is minimal in the sense that if the true distribution function has some discontinuity then the kernel estimate is no longer consistent.     

Computing the probability density function of the stable Paretian distribution

The practical implementation of a stable Paretian model is a nontrivial task, because—with the exception of a few special cases—its probability density function cannot be expressed analytically. Here, we present an algorithm for calculating the probability density function of the asymmetric stable Paretian distribution. Due to the use of the Fast Fourier Transform, the algorithm is computationally efficient, easily implemented, and of similar accuracy as existing algorithms.     

双边删失场合生存函数的非参数Bayes估计的弱收敛性

本文针对双边删失数据讨论了生存函数的非参数Bayes估计问题,运用Fred-holm积分的一些经典结果,文中证明了该估计弱收敛的一个Gaussian过程。     

Conformal iterated function systems with applications to the geometry of continued fractions

In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.

     

Asymptotics for a weighted least squares estimator of the disease onset distribution function for a survival-sacrifice model

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. A weighted least squares estimator for the distribution function of the disease onset was proposed. Asymptotic properties of that estimator are established here. We demonstrate its strong uniform consistency. A minimax lower bound for the estimation of the disease onset distribution is obtained, as well as the local asymptotic distribution for their estimator.     

A Graphical Method of Exploring the Mean Structure in Longitudinal Data Analysis

Abstract

In a longitudinal study, individuals are observed over some period of time. The investigator wishes to model the responses over this time as a function of various covariates measured on these individuals. The times of measurement may be sparse and not coincident across individuals. When the covariate values are not extensively replicated, it is very difficult to propose a parametric model linking the response to the covariates because plots of the raw data are of little help. Although the response curve may only be observed at a few points, we consider the underlying curve y(t). We fit a regression model y(t) = x ^Tβ(t) + ε(t) and use the coefficient functions β(t) to suggest a suitable parametric form. Estimates of y(t) are constructed by simple interpolation, and appropriate weighting is used in the regression. We demonstrate the method on simulated data to show its ability to recover the true structure and illustrate its application to some longitudinal data from the Panel Study of Income Dynamics.     

Mixed simulation-state estimation of water distribution systems based on a least squares loop flows state estimator

This paper presents combined simulation and state estimation algorithm for water distribution systems based on the loop corrective flows and the variation of nodal demands as independent variables and it optimizes the Least Squares (LS) criterion. The combination of the two algorithms for simulation and state estimation is based on the delimitation of regions in the water network that are state estimated while for the remaining parts of the water network the simulation task is realized. The sizes of the respective delimitations can be based either on the hydraulic or topological distances from the real pressure measurements, flow measurements or measured nodal consumptions. The delimitations are realized through modifications of the inverse of the upper form tree incidence matrix which is used in order to construct the respective state estimated or simulated water network areas: the simulated nodes and pipes have the corresponding incidence columns zeroed in the inverse of the upper form tree incidence matrix while the state estimated nodes and pipes keep the values of their incidence described in the corresponding columns of the inverse of the upper form tree incidence matrix. The combined novel algorithm can be also applied to regions of water distribution systems which contain low pipe flows so that to avoid any convergence problems in the numerical algorithm. It results an efficient and effective novel mixed simulation-state estimation which is implemented on realistic water distribution systems.     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

A new idea to construct the fractal interpolation function

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures.     

Estimation of a change-point in the mean function of functional data

The paper develops a comprehensive asymptotic theory for the estimation of a change-point in the mean function of functional observations. We consider both the case of a constant change size, and the case of a change whose size approaches zero, as the sample size tends to infinity. We show how the limit distribution of a suitably defined change-point estimator depends on the size and location of the change. The theoretical insights are confirmed by a simulation study which illustrates the behavior of the estimator in finite samples.     

A note on the conditional density estimate in the single functional index model

In this paper, we consider the estimation of the conditional density of a scalar response variable Y, given a Hilbertian random variable X when the observations are linked with a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional mode estimate. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is also discussed but not tackled.     

Bandwidth selection for the local polynomial estimator under dependence: A simulation study

Summary  Seven of the most popular methods for bandwidth selection in regression estimation are compared by means of a thorough simulation study, when the local polynomial estimator is used and the observations are dependent. The study is completed with two plug-in bandwidths for the generalized local polynomial estimator proposed by Vilar-Fernandez & Francisco-Fernández (2002).     

A further study on value distribution of the Riemann zeta‐function

The paper concerns the uniqueness problem of Riemann zeta‐function. It is showed that the Riemann zeta‐function is uniquely determined in terms of the preimages of three complex values except possibly a set G with , where G is called an exceptional set.     

A study of the effect of the loss function on Bayes Estimate,posterior risk and hazard function for Lindley distribution

In this paper, mathematical properties of Lindley distribution via Bayesian approach are derived under different loss functions. These properties include: Bayes Estimators, posterior risks and failure rate function for simulation scheme. Elicitation of hyperparameters is also discussed. A real life application to waiting time data at the bank is also described for the developed procedures (also using WinBUGS). Results are compared on the basis of posterior risk.