Pilot Estimator Research Articles

Bootstrap bandwidth selection for the pair correlation function of inhomogeneous spatial point processes

This work focuses on kernel estimation of the pair correlation function (PCF) for inhomogeneous spatial point processes. We propose a bootstrap bandwidth selector based on minimizing the mean integrated squared error (MISE). The variance term is estimated by nonparametric bootstrap, and the bias by a plug-in approach using a pilot estimator of the PCF. Kernel estimators of the PCF also require a pilot estimator of the first-order intensity. We test the performance of the bandwidth selector and the role of the pilot intensity estimator in a simulation study. The bootstrap bandwidth selector is competitive with cross-validation procedures, but the contribution of the bandwidth parameter to the goodness-of-fit of the kernel PCF estimator is minor in comparison with that of the pilot intensity function. The data-based kernel intensity estimator leads to biased kernel PCF estimators, while both kernel and parametric covariate-based intensities provide accurate estimators of the PCF.

Pilot estimators for a kind of sparse covariance matrices with incomplete heavy-tailed data

<abstract><p>This paper investigates generalized pilot estimators of covariance matrix in the presence of missing data. When the random samples have only bounded fourth moment, two kinds of generalized pilot estimators are provided, the generalized Huber estimator and the generalized truncated mean estimator. In addition, we construct thresholding generalized pilot estimator for a kind of sparse covariance matrices and establish the convergence rates in terms of probability under spectral and Frobenius norms respectively. Moreover, the convergence rates in sense of expectation are also given under an extra condition. Finally, simulation studies are conducted to demonstrate the superiority of our method.</p></abstract>

Surprise sampling: Improving and extending the local case-control sampling

Fithian and Hastie (2014) proposed a new sampling scheme called local case-control (LCC) sampling that achieves stability and efficiency by utilizing a clever adjustment pertained to the logistic model. It is particularly useful for classification with large and imbalanced data. This paper proposes a more general sampling scheme based on a working principle that data points deserve higher sampling probability if they contain more information or appear "surprising" in the sense of, for example, a large error of pilot prediction or a large absolute score. Compared with the relevant existing sampling schemes, as reported in Fithian and Hastie (2014) and Ai, et al. (2018), the proposed one has several advantages. It adaptively gives out the optimal forms to a variety of objectives, including the LCC and Ai et al. (2018)'s sampling as special cases. Under same model specifications, the proposed estimator also performs no worse than those in the literature. The estimation procedure is valid even if the model is misspecified and/or the pilot estimator is inconsistent or dependent on full data. We present theoretical justifications of the claimed advantages and optimality of the estimation and the sampling design. Different from Ai, et al. (2018), our large sample theory are population-wise rather than data-wise. Moreover, the proposed approach can be applied to unsupervised learning studies, since it essentially only requires a specific loss function and no response-covariate structure of data is needed. Numerical studies are carried out and the evidence in support of the theory is shown.

Nonparametric tests for circular regression

No matter the nature of the response and/or explanatory variables in a regression model, some basic issues such as the existence of an effect of the predictor on the response, or the assessment of a common shape across groups of observations, must be solved prior to model fitting. This is also the case for regression models involving circular variables (supported on the unit circumference). In that context, using kernel regression methods, this paper provides a flexible alternative for constructing pilot estimators that allow to construct suitable statistics to perform no-effect tests and tests for equality and parallelism of regression curves. Finite sample performance of the proposed methods is analysed in a simulation study and illustrated with real data examples.

On the ‘optimal’ density power divergence tuning parameter

The density power divergence, indexed by a single tuning parameter α, has proved to be a very useful tool in minimum distance inference. The family of density power divergences provides a generalized estimation scheme which includes likelihood-based procedures (represented by choice for the tuning parameter) as a special case. However, under data contamination, this scheme provides several more stable choices for model fitting and analysis (provided by positive values for the tuning parameter α). As larger values of α necessarily lead to a drop in model efficiency, determining the optimal value of α to provide the best compromise between model-efficiency and stability against data contamination in any real situation is a major challenge. In this paper, we provide a refinement of an existing technique with the aim of eliminating the dependence of the procedure on an initial pilot estimator. Numerical evidence is provided to demonstrate the very good performance of the method. Our technique has a general flavour, and we expect that similar tuning parameter selection algorithms will work well for other M-estimators, or any robust procedure that depends on the choice of a tuning parameter.

Non‐Parametric Spectral Density Estimation Under Long‐Range Dependence

One major aim of time series analysis, particularly in the physical and geo‐sciences, is the estimation of the spectral density function. With weakly dependent time processes, non‐parametric, kernel‐based methods are available for spectral density estimation, which involves smoothing the periodogram by a kernel function. However, a similar non‐parametric approach is presently unavailable for strongly, or long‐range, dependent processes. In particular, as the spectral density function under long‐range dependence commonly has a pole at the origin, kernel‐based methods developed for weakly dependent processes (i.e., with bounded spectral densities) do not apply readily for long‐range dependence without suitable modification. To address this, we propose a non‐parametric kernel‐based method for spectral density estimation, which is valid under both weak and strong dependence. Based on the initial or pilot estimator of the long‐memory parameter, the method involves a frequency domain transformation to dampen the dependence in periodogram ordinates and mimic kernel‐based estimation under weak dependence. Under mild assumptions, the proposed non‐parametric spectral density estimator is shown to be uniformly consistent, and general expressions are provided for rates of estimation error and optimal kernel bandwidths. The method is investigated through simulation and illustrated through data examples, which also consider bandwidth selection.

Semiparametric estimation of the single-index varying-coefficient model

ABSTRACTIn this paper, we consider the choice of pilot estimators for the single-index varying-coefficient model, which may result in radically different estimators, and develop the method for estimating the unknown parameter in this model. To estimate the unknown parameters efficiently, we use the outer product of gradient method to find the consistent initial estimators for interest parameters, and then adopt the refined estimation method to improve the efficiency, which is similar to the refined minimum average variance estimation method. An algorithm is proposed to estimate the model directly. Asymptotic properties for the proposed estimation procedure have been established. The bandwidth selection problem is also considered. Simulation studies are carried out to assess the finite sample performance of the proposed estimators, and efficiency comparisons between the estimation methods are made.

An $\{\ell_{1},\ell_{2},\ell_{\infty}\}$-regularization approach to high-dimensional errors-in-variables models

Several new estimation methods have been recently proposed for the linear regression model with observation errors in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar{\delta}$, and (2) zero-mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar{\delta}$, while the second assumption has been essentially applied when an estimator for the second moment of the observational error is available. This work proposes and studies two new estimators which, compared to other procedures for regression models with errors in the design, exploit an additional $\ell_{\infty}$-norm regularization. The first estimator is applicable when both (1) and (2) hold but does not require an estimator for the second moment of the observational error. The second estimator is applicable under (2) and requires an estimator for the second moment of the observation error. Importantly, we impose no assumption on the accuracy of this pilot estimator, in contrast to the previously known procedures. As the recent proposals, we allow the number of covariates to be much larger than the sample size. We establish the rates of convergence of the estimators and compare them with the bounds obtained for related estimators in the literature. These comparisons show interesting insights on the interplay of the assumptions and the achievable rates of convergence.

Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation

We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations. We indicate how our results can be extended to joint regression models.

Constructed Data Pilot-Assisted Channel Estimators for Mobile Environments

Nowadays, more and more communication systems are deployed under mobile environments with very high velocity. This paper focuses on the channel estimation problem in vehicular scenarios, which is very challenging in view of the extremely time-varying characteristics of mobile channels. Specifically, we propose a novel channel estimator named constructed data pilot (CDP) estimator for the current communication standards by fully exploiting the channel correlation characteristics across two concatenated symbols. On the basis of the CDP estimator, we further resort to two efficient techniques to improve its performance over the entire signal-to-noise-ratio (SNR) region. For the first technique, the time-variant mobile channel is modeled as a first-order Markov process so that the exact autocorrelation value of the two adjacent symbols can be derived. For the second technique, the SNR is estimated and serves as a priori information. Simulation results reveal that our proposed channel estimators outperform existing alternatives with lower computational complexity.

Semiparametric Gaussian Copula Models: Geometry and Efficient Rank-Based Estimation

For multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator is proposed for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize adaptivity of the model with respect to the unknown marginal distributions and of efficiency of the pseudo-likelihood estimator. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator with respect to our one-step estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations.

Estimating a Cosmological Mass Bias Parameter with Bootstrap Bandwidth Selection

SummaryWe focus on selecting optimal bandwidths for non-parametric estimation of the two-point correlation function of a point pattern. We obtain these optimal bandwidths by using a bootstrap approach to select a bandwidth that minimizes the integrated squared error. The variance term is estimated by using a non-parametric spatial bootstrap, whereas the bias term is estimated with a plug-in approach using a pilot estimator of the two-point correlation function based on a parametric model. The choice of parametric model for the pilot estimator is very flexible. Depending on applications, parametric statistical point models, physical models or functional models can be used. We also explore the use of the procedure for selecting adaptive optimal bandwidths. We investigate the performance of the bandwidth selection procedure by using a simulation study. In our data example, we apply our method to a Sloan Digital Sky Survey galaxy cluster catalogue by using a pilot estimator based on the power law functional model in cosmology. The resulting non-parametric two-point correlation function estimate is then used to estimate a cosmological mass bias parameter that describes the relationship between the galaxy mass distribution and the underlying matter distribution.

Improving Sheather and Jones’ bandwidth selector for difficult densities in kernel density estimation

Kernel density estimation is a widely used statistical tool and bandwidth selection is critically important. The Sheather and Jones’ (SJ) selector [A reliable data-based bandwidth selection method for kernel density estimation, J. R. Stat. Soc. Ser. B 53 (1991), pp. 683–690] remains the best available data-driven bandwidth selector. It can, however, perform poorly if the true density deviates too much in shape from normal. This paper first develops an alternative selector by following ideas in Park and Marron [On the use of pilot estimators in bandwidth selection, Nonparametr. Stat. 1 (1992), pp. 231–240] to reduce the impact of the normal reference density. The selector can bring drastic improvement to less smooth densities that the SJ selector has difficulty with, but may be slightly worse off otherwise. We then propose to combine the alternative selector and SJ selector by using the smaller of the two bandwidths, which has the effect of automatically picking the better one for a particular density. In our extensive simulation, study using the 15 benchmark densities in Marron and Wand [Exact mean integrated squared error, Ann. Statist. 20 (1992), pp. 712–736], the combined selector significantly improves the SJ selector for 5 difficult densities and retains the superior performance of the SJ selector for the other 10. A ready-to-use R function is provided.

Asymptotic unbiased density estimators

This paper introduces a computationally tractable density estimator that has the same asymptotic variance as the classical Nadaraya-Watson density estimator but whose asymptotic bias is zero. We achieve this result using a two stage estimator that applies a multiplicative bias correction to an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator.

Modifying the kernel distribution function estimator towards reduced bias

We explore the convergence rates of a kernel-based distribution function estimator with variable bandwidth. As in density estimation, a considerable bias reduction from O(h 2) to O(h 4) can be obtained by replacing the bandwidth h by h/f 1/2(X i ). We show that the necessary replacement of f 1/2 by some pilot estimator , depending on a second bandwidth g, has no penalizing effect on bias and variance, provided we undersmooth with the pilot bandwidth g, that is g/h→0 in a certain way. Owing to the considerable bias reduction, a simple plug-in normal reference bandwidth selector works effectively in practice. Distribution function estimators with good convergence properties and with simple bandwidth selectors are desirable for repetitive use in smoothed bootstrap algorithms.

Semi-parametric estimation of partially linear single-index models

One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root- n consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth are constrained to a certain speed, which is rarely determinable in practical situations. In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions from Carroll et al. [Generalized partially linear single-index models, J. Amer. Statist. Assoc. 92 (1997) 477–489]: no root- n pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).

Estimation in additive cox models by marginal integration

Assuming an additive model on the covariate effect in proportional hazards regression, we consider the estimation of the component functions. The estimator is based on the marginal integration method. Then we use a new kind of nonparametric estimator as the pilot estimator of the marginal integration. The pilot estimator is constructed by an analogy to the two-sample problems and by appealing to the principles of local partial likelihood and local linear fitting. We derive the asymptotic distribution of the marginal integration estimator of the component functions. The result of a simulation study is also given.

Choosing a robustness tuning parameter

A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L 2 distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g ( · ) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g ( · ) , we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β , which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g ( · ) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.

Rate optimal estimation with the integration method in the presence of many covariates

For multivariate regressors, integrating the Nadaraya–Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.