Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

Semiparametric efficient adaptive estimation of asymmetric GARCH models

In this paper we derive a semiparametric efficient adaptive estimator of an asymmetric GARCH model. Applying some general results from Drost et al. [1997. The Annals of Statistics 25, 786–818], we first estimate the unknown density function of the disturbances by kernel methods, then apply a one-step Newton–Raphson method to obtain a more efficient estimator than the quasi-maximum likelihood estimator. The proposed semiparametric estimator is adaptive for parameters appearing in the conditional standard deviation model with respect to the unknown distribution of the disturbances.     

Control variate method for stationary processes

The sample mean is one of the most natural estimators of the population mean based on independent identically distributed sample. However, if some control variate is available, it is known that the control variate method reduces the variance of the sample mean. The control variate method often assumes that the variable of interest and the control variable are i.i.d. Here we assume that these variables are stationary processes with spectral density matrices, i.e. dependent. Then we propose an estimator of the mean of the stationary process of interest by using control variate method based on nonparametric spectral estimator. It is shown that this estimator improves the sample mean in the sense of mean square error. Also this analysis is extended to the case when the mean dynamics is of the form of regression. Then we propose a control variate estimator for the regression coefficients which improves the least squares estimator (LSE). Numerical studies will be given to see how our estimator improves the LSE.     

Robust estimation for the order of finite mixture models

In this paper, we study a robust and efficient estimation procedure for the order of finite mixture models based on the minimizing a penalized density power divergence estimator. For this task, we use the locally conic parametrization approach developed by Dacunha-Castelle and Gassiate (ESAIM Probab Stat 285–317, 1997a; Ann Stat 27:1178–1209, 1999), and verify that the minimizing a penalized density power divergence estimator is consistent. Simulation results are provided for illustration.     

Goodness-of-fit tests in linear EV regression with replications

This paper proposes a goodness-of-fit test for checking the adequacy of parametric forms of the regression error density functions in linear errors-in-variables regression models. Instead of assuming the distribution of the measurement error to be known, we assume that replications of the surrogates of the latent variables are available. The test statistic is based upon a weighted integrated squared distance between a nonparametric estimator and a semi-parametric estimator of the density functions of certain residuals. Under the null hypothesis, the test statistic is shown to be asymptotically normal. Consistency and local power results of the proposed test under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed test is evaluated via simulation studies. A real data example is also included to demonstrate an application of the proposed test.     

On efficiency of the ratio estimator

Summary The exact mean square error for the ratio estimator of a finite population total based on simple random sampling without replacement is shown to have an expected value less than that of the variance of the ratio estimator based on Midzuno’s scheme, under a usual super-population model.     

On kernel nonparametric regression designed for complex survey data

In this article, we consider nonparametric regression analysis between two variables when data are sampled through a complex survey. While nonparametric regression analysis has been widely used with data that may be assumed to be generated from independently and identically distributed (iid) random variables, the methods and asymptotic analyses established for iid data need to be extended in the framework of complex survey designs. Local polynomial regression estimators are studied, which include as particular cases design-based versions of the Nadaraya–Watson estimator and of the local linear regression estimator. In this paper, special emphasis is given to the local linear regression estimator. Our estimators incorporate both the sampling weights and the kernel weights. We derive the asymptotic mean squared error (MSE) of the kernel estimators using a combined inference framework, and as a corollary consistency of the estimators is deduced. Selection of a bandwidth is necessary for the resulting estimators; an optimal bandwidth can be determined, according to the MSE criterion in the combined mode of inference. Simulation experiments are conducted to illustrate the proposed methodology and an application with the Canadian survey of labour and income dynamics is presented.     

Asymptotics of SIMEX-based variance estimation

In this paper, we study the asymptotic properties of simulation extrapolation (SIMEX) based variance estimation that was proposed by Wang et al. (J R Stat Soc Series B 71:425–445, 2009). We first investigate the asymptotic normality of the parameter estimator in general parametric variance function and the local linear estimator for nonparametric variance function when permutation SIMEX (PSIMEX) is used. The asymptotic optimal bandwidth selection with respect to approximate mean integrated squared error (AMISE) for nonparametric estimator is also studied. We finally discuss constructing confidence intervals/bands of the parameter/function of interest. Other than applying the asymptotic results so that normal approximation can be used, we recommend a nonparametric Monte Carlo algorithm to avoid estimating the asymptotic variance of estimator. Simulation studies are carried out for illustration.     

Über die Konsistenz einer Schätzung mehrdimensionaler Dichten auf der Basis trigonometrischer Reihen

Summary This paper describes nonparametric estimates of multivariate densities based on orthogonal series, a method evaluated byencov [1962] for univariate density functions. Starting with the sample step function we show another way to get this estimator. Further we give a simple expression for mean integrated square error. Choosing the special case of trigonometric series this estimator is shown to be consistent.

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Diese Arbeit wurde im Rahmen eines Forschungsauftrages der Fraunhofer-Gesellschaft zur Förderung der Angewandten Forschung eV (München) erstellt.     

Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE

In generalized autoregressive conditional heteroskedastic (GARCH) models, the standard identifiability assumption that the variance of the iid process is equal to 1 can be replaced by an alternative moment assumption. We show that, for estimating the original specification based on the standard identifiability assumption, efficiency gains can be expected from using a quasi-maximum likelihood (QML) estimator based on a non Gaussian density and a reparameterization based on an alternative identifiability assumption. A test allowing to determine whether a reparameterization is needed, that is, whether the more efficient QMLE is obtained with a non Gaussian density, is proposed.     

Spline regression for hazard rate estimation when data are censored and measured with error

In this paper, we study an estimation problem where the variables of interest are subject to both right censoring and measurement error. In this context, we propose a nonparametric estimation strategy of the hazard rate, based on a regression contrast minimized in a finite‐dimensional functional space generated by splines bases. We prove a risk bound of the estimator in terms of integrated mean square error and discuss the rate of convergence when the dimension of the projection space is adequately chosen. Then we define a data‐driven criterion of model selection and prove that the resulting estimator performs an adequate compromise. The method is illustrated via simulation experiments that show that the strategy is successful.     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

Bootstrap refinements for QML estimators of the GARCH(1,1) parameters

This paper reconsiders a block bootstrap procedure for Quasi Maximum Likelihood estimation of GARCH models, based on the resampling of the likelihood function, as proposed by Gonçalves and White [2004. Maximum likelihood and the bootstrap for nonlinear dynamic models. Journal of Econometrics 119, 199–219]. First, we provide necessary conditions and sufficient conditions, in terms of moments of the innovation process, for the existence of the Edgeworth expansion of the GARCH(1,1) estimator, up to the k$k$-th term. Second, we provide sufficient conditions for higher order refinements for equally tailed and symmetric test statistics. In particular, the bootstrap estimator based on resampling the likelihood has the same higher order improvements in terms of error in the rejection probabilities as those in Andrews [2002. Higher-order improvements of a computationally attractive k$k$-step bootstrap for extremum estimators. Econometrica 70, 119–162].     

Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

Relative error prediction in nonparametric deconvolution regression model

In this paper, we studied an alternative estimator of the regression function when the covariates are observed with error. It is based on the minimization of the relative mean squared error. We obtain expressions for its asymptotic bias and variance together with an asymptotic normality result. Our technique is illustrated on simulation studies. Numerical results suggest that the studied estimator can lead to tangible improvements in prediction over the usual kernel deconvolution regression estimator, particularly in the presence of several outliers in the dataset.     

THE ROLE OF CONDITIONAL HETEROSKEDASTICITY IN IDENTIFYING AND ESTIMATING LINEAR TRIANGULAR SYSTEMS,WITH APPLICATIONS TO ASSET PRICING MODELS THAT INCLUDE A MISMEASURED FACTOR

A new estimator is proposed for linear triangular systems, where identification results from the model errors following a bivariate and diagonal GARCH(1,1) process with potentially time‐varying error covariances. This estimator applies when traditional instruments are unavailable. I demonstrate its usefulness on asset pricing models like the capital asset pricing model and Fama–French three‐factor model. In the context of a standard two‐pass cross‐sectional regression approach, this estimator improves the pricing performance of both models. Set identification bounds and an associated estimator are also provided for cases where the conditions supporting point identification fail. Copyright © 2013 John Wiley & Sons, Ltd.     

Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.