On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator

We consider the problem of the construction of the goodness-of-fit test in the case of continuous time observations of a diffusion process with small noise. The null hypothesis is parametric and we use a minimum distance estimator of the unknown parameter. We propose an asymptotically distribution free test for this model.     

On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes

We consider the problem of the construction of the asymptotically distribution free test by the observations of ergodic diffusion process. It is supposed that under the basic hypothesis the trend coefficient depends on a finite-dimensional parameter and we study the Cramér-von Mises type statistics. The underlying statistics depends on the deviation of the local time estimator from the invariant density with parameter replaced by the maximum likelihood estimator. We propose a linear transformation which yields the convergence of the test statistics to an integral of the Wiener process. Therefore the test based on this statistics is asymptotically distribution free.     

A new instrumental variable estimation for diffusion processes

We consider the problem of parametric inference from continuous sample paths of the diffusion processes {x(t)} generated by the system of possibly nonstationary and/or nonlinear Ito stochastic differential equations. We propose a new instrumental variable estimator of the parameter whose pivotal statistic has a Gaussian distribution for all possible values of parameter. The new estimator enables us to construct exact level-α confidence intervals and tests for the parameter in the possibly non-stationary and/or nonlinear diffusion processes. Applications to several non-stationary and/or nonlinear diffusion processes are considered as examples. This work was supported by Korea Research Foundation Grant (KRF-2001-015-DP0057).     

On score-functions and goodness-of-fit tests for stochastic processes

The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.     

A preliminary test procedure for the scale parameter of exponential distribution when the selection parameter is unknown

Summary A preliminary test estimator is considered for the scale parameter of the two-parameter exponential distribution with unknown selection parameter, where the distribution does not satisfy the regularity condition of Wilks' theorem—the density is not differentiable. A method of specifying the level of significance of the preliminary test based on is proposed AIC. This work was partly supported by Scientific Research Fund No. 58450058 from the Ministry of Education of Japan. The Institute of Statistical Mathematics     

Two tests for heteroscedasticity in nonparametric regression

In this paper, two new tests for heteroscedasticity in nonparametric regression are presented and compared. The first of these tests consists in first estimating nonparametrically the unknown conditional variance function and then using a classical least-squares test for a general linear model to test whether this function is a constant. The second test is based on using an overall distance between a nonparametric estimator of the conditional variance function and a parametric estimator of the variance of the model under the assumption of homoscedasticity. A bootstrap algorithm is used to approximate the distribution of this test statistic. Extended versions of both procedures in two directions, first, in the context of dependent data, and second, in the case of testing if the variance function is a polynomial of a certain degree, are also described. A broad simulation study is carried out to illustrate the finite sample performance of both tests when the observations are independent and when they are dependent.     

A generalization of the growth curve model which allows missing data

This study presents methods for estimating and testing hypotheses about linear functions of the unknown parameters in a generalization of the growth curve model which allows missing data. The estimators proposed are best asymptotically normal (BAN). A testing method for large samples is described which uses a test criterion given in general form by Wald. The asymptotic null distribution of the test statistic is a central chi-square variable. A BAN estimator of a linear vector function of the unknown parameters of the expectation model and consistent estimators of the variance-covariance parameters are required for computation.     

An efficient estimator for the expectation of a bounded function under the residual distribution of an autoregressive process

Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simple efficient estimator. It is adaptive with respect to the autoregression parameter.     

双边定数截尾下Pareto分布的可靠性分析

基于双边定数截尾样本,选取未知参数的先验分布为无信息先验和Gamma分布,分别在平方损失和LINEX损失下,研究了Pareto分布的形状参数和可靠性指标(可靠度和失效率)的Bayes估计.为了研究估计的精度,采用Monte-Carlo模拟的方法给出了数值检验的例子.结果表明在LINEX损失下并选用Gamma先验分布时,参数的Bayes估计是最优的.     

不同损失下Lomax分布形状参数的Bayes估计

本文研究了两参数Lomax分布形状参数的Bayes估计问题.当尺度参数已知时,给出了在几种不同损失函数下形状参数的Bayes估计表达式,并运用随机模拟方法对各个估计进行了比较.     

熵损失函数下两参数Lomax分布形状参数的Bayes估计

在熵损失函数下,讨论了两参数Lomax分布形状参数的Bayes估计和可容许估计.并讨论了一类(cT+d)~(-1)形式估计的可容许性和不可容许性.     

Sharp Error Estimate for Maximum Likelihood Estimator of Nonstationary Diffusion Processes

We consider the maximum likelihood estimator of the unknown parameter in a class of nonstationary diffusion processes. We give further a precise estimate for the error of the estimator.     

Testing for small bias of tail index estimators

We determine the joint asymptotic normality of kernel and weighted least-squares estimators of the upper tail index of a regularly varying distribution when each estimator is a bivariate function of two parameters: the tuning parameter is motivated by possible underlying second-order behavior in regular variation, while no such behavior is assumed, and the fraction parameter determines that upper portion of the sample on which the estimator is based. Under the hypothesis that the scaled asymptotic biases of the estimators vanish uniformly in the parameter points considered, these results imply joint asymptotic normality for deviations of ratios of the estimators from 1, which in turn yield asymptotic chi-square tests for checking the small-bias hypothesis, equivalent to the constructibility of asymptotic confidence intervals. The test procedure suggests adaptive choices of the tuning and fraction parameters: data-driven (t)estimators.     

Parametric tail copula estimation and model testing

Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.     

Sensitivity of Bayes decisions for the success probability for samples from a nonbinomial population

Summary A family of generalised negative binomial distributions is employed to investigate inference robustness of the Bayes estimator of the unknown parameter of the binomial distribution. A zone of sensitivity for the test of significance is constructed to forewarn the pro-Jeffreys Bayesians against indiscriminate choice of the probability in favour of the null hypothesis. A few selected tables are presented to illustrate the effect of relaxation of the ‘binomiality’ assumption.     

Convergence rates of MLE in a partly linear model

This paper considers the estimation for a partly linear model with case 1 interval censored data. We assume that the error distribution belongs to a known family of scale distributions with an unknown scale parameter. The sieve maximum likelihood estimator (MLE) for the model’s parameter is shown to be strongly consistent, and the convergence rate of the estimator is obtained and discussed.     

Estimation bayésienne pour un processus de Poisson d'intensité discontinue

We consider the problem of parameter estimation by n independent observations of inhomogeneous Poisson process of discontinuous intensity. The unknown parameter is two dimensional with the first component-frequency (amplitude) and the second component-frequency of periodic Poisson process. We show that the estimator of the first component is asymptotically normal with the rate √n and the estimator of the second component has nondegenerate distribution with the rate n.     

Empirical bayesian test of the smoothness

In the context of adaptive nonparametric curve estimation a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes. These classes are often parametrized by a quantity representing the smoothness of the signal. It has already been realized by many that the problem of estimating the smoothness is not sensible. What can then be inferred about the smoothness? The paper attempts to answer this question. We consider implications of our results to hypothesis testing about the smoothness and smoothness classification problem. The test statistic is based on the empirical Bayes approach, i.e., it is the marginalized maximum likelihood estimator of the smoothness parameter for an appropriate prior distribution on the unknown signal.     

Statistical Inference in Single-Index and Partially Nonlinear Models

A finite series approximation technique is introduced. We first applythis approximation technique to a semiparametric single-index model toconstruct a nonlinear least squares (LS) estimator for an unknown parameterand then discuss the confidence region for this parameter based on theasymptotic distribution of the nonlinear LS estimator. Meanwhile, acomputational algorithm and a small sample study for this nonlinear LSestimator are developed. Additionally, we apply the finite seriesapproximation technique to a partially nonlinear model and obtain some newresults.     

Risk measures, distortion parameters, and their empirical estimation

Risk measures are of considerable current interest. Among other uses, they allow an insurer to calculate a risk-loaded premium for a random loss. However, the premium principle in use by the insurer may be, at least in part, based on considerations other than risk. It is then important to quantify the degree to which the premium compensates the insurer for the risk associated with the loss. This can be done by choosing a suitable risk measure and solving for the parameter that leads to the insurer’s premium. When the loss distribution is unknown, this becomes a statistical estimation problem.In this paper, we investigate the nonparametric estimation of the parameter associated with a distortion-based risk measure. It is assumed that the premium principle is known, but no information is assumed about the loss distribution, and therefore empirical estimators are used. We explore the asymptotic properties of the resulting estimator of the risk measure parameter in general and for three well-known risk measures in particular: the proportional hazards transform, the Wang transform, and the conditional tail expectation.