Preliminary Estimator Research Articles

Efficient Estimation for Models With Nonlinear Heteroscedasticity

We study efficient estimation for models with nonlinear heteroscedasticity. In two-step quantile regression for heteroscedastic models, motivated by several undesirable issues caused by the preliminary estimator, we propose an efficient estimator by constrainedly weighting information across quantiles. When the weights are optimally chosen under certain constraints, the new estimator can simultaneously eliminate the effect of preliminary estimator as well as achieve good estimation efficiency. When compared to the Cramér-Rao lower bound, the relative efficiency loss of the new estimator has a conservative upper bound, regardless of the model design structure. The upper bound is close to zero for practical situations. In particular, the new estimator can asymptotically achieve the optimal Cramér-Rao lower bound if the noise has either a symmetric density or the asymmetric Laplace density. Monte Carlo studies show that the proposed method has substantial efficiency gain over existing ones. In an empirical application to GDP and inflation rate modeling, the proposed method has better prediction performance than existing methods.

On multi-step estimation of delay for SDE

We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLEs for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. the parameter, and therefore the numerical calculation of the MLEs encounter certain difficulties. We propose One-step and Two-step MLEs, whose calculation has no such problems and provide an estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First, we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and a modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.

Preliminary test estimation in uniformly locally asymptotically normal models

AbstractPreliminary test estimation is a methodology that combines goodness‐of‐fit testing and estimation. It is a classical procedure when it is suspected that the parameter to be estimated satisfies some prespecified constraints. In the present paper, we establish general results on the asymptotic behavior of preliminary test estimators. More precisely, we show that, in uniformly locally asymptotically normal (ULAN) models, a general asymptotic theory can be derived for preliminary test estimators based on estimators admitting generic Bahadur‐type representations. This allows for a detailed comparison between classical estimators and preliminary test estimators in ULAN models. Our results, that, in standard linear regression models, are shown to reduce to some classical results, are also illustrated in more modern and involved setups, such as the multisample one where m covariance matrices are to be estimated when it is suspected that these matrices might be equal, might be proportional, or might share a common “scale”. Simulation results confirm our theoretical findings and an illustration on a real data example is provided.

IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison

The Interpolated Discrete Fourier Transform (IpDFT) is one of the most popular algorithms for Phasor Measurement Units (PMUs), due to its quite low computational complexity and its good accuracy in various operating conditions. However, the basic IpDFT algorithm can be used also as a preliminary estimator of the amplitude, phase, frequency and rate of change of frequency of voltage or current AC waveforms at times synchronized to the Universal Coordinated Time (UTC). Indeed, another cascaded algorithm can be used to refine the waveform parameters estimation. In this context, the main novelty of this work is a fair and extensive performance comparison of three different state-of-the-art IpDFT-tuned estimation algorithms for PMUs. The three algorithms are: (i) the so-called corrected IpDFT (IpDFTc), which is conceived to compensate for the effect of both the image of the fundamental tone and second-order harmonic; (ii) a frequency-tuned version of the Taylor Weighted Least-Squares (TWLS) algorithm, and (iii) the frequency Down-Conversion and low-pass Filtering (DCF) technique described also in the IEEE/IEC Standard 60255-118-1:2018. The simulation results obtained in the P Class and M Class testing conditions specified in the same Standard show that the IpDFTc algorithm is generally preferable under the effect of steady-state disturbances. On the contrary, the tuned TWLS estimator is usually the best solution when dynamic changes of amplitude, phase or frequency occur. In transient conditions (i.e., under the effect of amplitude or phase steps), the IpDFTc and the tuned TWLS algorithms do not clearly outperform one another. The DCF approach generally returns the worst results. However, its actual performances heavily depend on the adopted low-pass filter.

Estimation and inference for distribution and quantile functions in endogenous treatment effect models

Given a standard endogenous treatment effect model, we propose nonparametric estimation and inference procedures for the distribution and quantile functions of the potential outcomes among compliers, as well as the local quantile treatment effect function. The preliminary distribution function estimator is a weighted average of indicator functions, but is not monotonically increasing in general. We therefore propose a simple monotonizing method for proper distribution function estimation, and obtain the quantile function estimator by inversion. Our monotonizing method is an alternative to Chernozhukov et al. (2010) and is arguably preferable when the outcome has unbounded support. We show that all the estimators converge weakly to Gaussian processes at the parametric rate, and propose a multiplier bootstrap for uniform inference. Our uniform results thus generalize the pointwise theory developed by Frölich and Melly (2013). Monte Carlo simulations and an application to the effect of fertility on family income distribution illustrate the use of the methods. All results extend to the subpopulation of treated compliers as well.

On strong consistency and asymptotic normality of one-step Gauss-Newton estimators in ARMA time series models

ABSTRACT One-step Gauss-Newton estimators for causal and invertible autoregressive moving-average (ARMA) time series models with an unknown mean are considered. A proof of the strong consistency and asymptotic normality of these estimators is provided. In a simulation study, their empirical properties are illustrated in ARMA models and compared to a other estimators: the estimators based on the innovations algorithm procedure, the estimators based on the spectral methods of [Krampe J, Kreiss J-P, Paparoditis E. Estimated Wold representation and spectral-density-driven bootstrap for time series. J R Stat Soc: Ser B (Statist Method). 2018;80:703–726.], and the one-step Gauss-Newton estimators using these methods as preliminary estimators. In the order-one moving-average model, we included the estimator based on the method of moments. The sample mean is used as a preliminary estimator to estimate the mean. These estimators are compared empirically to the maximum Gaussian likelihood estimator with respect to exact biases and mean squared errors.

Multivariate extremes over a random number of observations

AbstractThe classical multivariate extreme‐value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max‐stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max‐stable distributions. We also connect the extremal dependence structure of classical max‐stable distributions and that of our new family of max‐stable distributions. Using an inversion method, we derive a semiparametric composite‐estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large‐sample theory of the composite‐estimator and illustrate its finite‐sample performance via a simulation study.

Development of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of Kumaraswamy-G Distributions Based on Progressive Type-II Censoring

Classical estimators and preliminary test estimators (PTEs) of the powers of model parameter and the two measures of reliability, namely $$R\left( t \right) = P(X > t)$$ and $$P = P(X > Y)$$ of Kumaraswamy-G distribution, are developed under progressive type-II censoring. The preliminary test confidence intervals (PTCIs) are also developed based on uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). Merits of PTEs and PTCIs are established. Simulation results are presented, and real-life data set is also analysed.

Robust estimation for semi-functional linear regression models

Semi-functional linear regression models postulate a linear relationship between a scalar response and a functional covariate, and also include a non-parametric component involving a univariate explanatory variable. It is of practical importance to obtain estimators for these models that are robust against high-leverage outliers, which are generally difficult to identify and may cause serious damage to least squares and Huber-type M-estimators. For that reason, robust estimators for semi-functional linear regression models are constructed combining B-splines to approximate both the functional regression parameter and the nonparametric component with robust regression estimators based on a bounded loss function and a preliminary residual scale estimator. Consistency and rates of convergence for the proposed estimators are derived under mild regularity conditions. The reported numerical experiments show the advantage of the proposed methodology over the classical least squares and Huber-type M-estimators for finite samples. The analysis of real examples illustrates that the robust estimators provide better predictions for non-outlying points than the classical ones, and that when potential outliers are removed from the training and test sets both methods behave very similarly.

Robust method of estimating of odd ratios in case-control association studies

In case-control studies, association of a disease with a genetic trait can be studied in terms of Single Nucleotide Polymorphisms (SNP). Prospective and retrospective likelihoods are the two common approaches used to study this association. In this paper, a method based on Preliminary test, has been proposed, which is efficient than prospective approach by exploiting model assumptions of Hardy Weinberg Equilibrium (HWE) and robust against failure of model assumptions as compared to retrospective approach (2015-16). In this proposed approach, a Preliminary Test Estimator (PTE) has been suggested based on both prospective and retrospective approach for estimating of association of a disease with genetic marker. The proposed PTE has been empirically evaluated through a simulation study. The Preliminary Test Estimator developed was found to be robust against the deviation from HWE.

Identifying latent group structures in nonlinear panels

We propose a procedure to identify latent group structures in nonlinear panel data models where some regression coefficients are heterogeneous across groups but homogeneous within a group and the group number and membership are unknown. To identify the group structures, we consider the order statistics for the preliminary unconstrained consistent estimators of the regression coefficients and translate the problem of classification into the problem of break detection. Then we extend the sequential binary segmentation algorithm of Bai (1997) for break detection from the time series setup to the panel data framework. We demonstrate that our method is able to identify the true latent group structures with probability approaching one and the post-classification estimators are oracle-efficient. The method has the advantage of more convenient implementation compared with some alternative methods, which is a desirable feature in nonlinear panel applications. To improve the finite sample performance, we also consider an alternative version based on the spectral decomposition of certain estimated matrix and link the group identification issue to the community detection problem in the network literature. Simulations show that our method has good finite sample performance. We apply this method to explore how individuals’ portfolio choices respond to their financial status and other characteristics using the Netherlands household panel data from year 1993 to 2015, and find three latent groups.

Improved estimation in elliptical linear mixed measurement error models

In this paper, we propose a set of improved estimators for the fixed effect parameters in the linear mixed models when the covariates are measured with additive errors and prior information for the parameters is available. When it is suspected that the parameter vector may be the null-vector with some degree of uncertainty, we define improved estimators which including the preliminary test estimator, the Stein-type estimator and the postive-rule Stein-type estimator. It is assumed that the measurement error distributed according to the law belonging to the class of elliptically contoured distributions. The asymptotic properties of resulting estimators such as the asymptotic distributional quadratic biases and the asymptotic distributional quadratic risks are examined. A simulation study is also performed to illustrate the finite sample performance of the proposed procedures and finally Boston housing data set is analysed.

Performance of Preliminary Test Estimators for Error Variance Based on W, LR and LM Tests

The performances of preliminary test estimators for error variance based on W, LR and LM tests in a normal linear model are considered in this paper. Firstly, the risks of the proposed estimators are derived and compared by theoretical analysis and numerical calculation, respectively. The results show that their risks are related to the equality constraint error and the critical value of test. Moreover, the minimum value of the risks can be achieved when the critical value of test equals to one. Secondly, the superiority conditions of the proposed estimators are discussed. Finally, the results are illustrated by a simulation example.

Comparative Study of LASSO, Ridge Regression, Preliminary Test and Stein-type Estimators for the Sparse Gaussian Regression Model

This paper compares the performance characteristics of penalty estimators, namely, LASSO and ridge regression (RR), with the least squares estimator (LSE), restricted estimator (RE), preliminary test estimator (PTE) and the Stein-type estimators. Under the assumption of orthonormal design matrix of a given regression model, we find that the RR estimator dominates the LSE, RE, PTE, Stein-type estimators and LASSO estimator uniformly, while, similar to Hansen (2013), neither LASSO nor LSE, PTE and Stein-Type estimators dominates the other. Our conclusions are based on the analysis of L_2-risks and relative risk efficiencies (RRE) together with the RRE related tables and graphs.

A Vetting Protocol for the Analytical Procedures Platform for the AP-Phase of PCAOB Audits

Study Context AS5[2017], issued by the Public Company Accounting Oversight Board, requires the use of Analytical Procedures [AP] at the Planning and Substantive Phases of Assurance Audits of firms traded on active exchanges. Logically, an aspect of this requirement is satisfied by using a Panel of the Client’s data at the Planning Phase to forecast the Client’s YE-closing values and then at the Substantive Phase to dispose the directional difference between the: [Actual Client’s YE-value and the AP-Forecasted YE-value]—the Disposition Phase. Research Focus To date, neither the PCAOB nor the AICPA have suggested a pilot-test paradigm to vet the AP-forecasting Protocol under consideration. To address this lacuna, we detail an AP: Decision Support System [AP:DSS] that offers to the Audit InCharge a two-stage pre-analysis AP-vetting [Pilot-Test] platform that employs False Negative [FN] and False Positive [FP] Profilers. In inferential analyses, the FP-Risk is usually benchmarked using the FN-Risk. Deliverables A comprehensive AP-vetting model is offered and illustrated using: (i) a preliminary estimator of a reasonable sample size, (ii) two Standard Forecasting Models: The Excel versions of the OLS Linear Two-parameter and the Moving Average Models, and (iii) a Benchmarking protocol. Unique in this AP:DSS vetting protocol is that the FP-risk is contexted by the FN-risk from the independent benchmark domain. This duality enhances the inferential impact of the vetting protocol as it uses separate variable sets. The AP:DSS is available at no cost as an e-Download.

A random‐effects model for clustered circular data

AbstractThis article considers a circular regression model for clustered data, where both the cluster effects and the regression errors have von Mises distributions. It involvesβ, a vector of parameters for the fixed effects, and two concentration parameters for the error distribution. A measure of intra‐cluster circular correlation and a predictor for an unobserved cluster random effect are studied. Preliminary estimators for the vectorβand the two concentration parameters are proposed, and their performance is compared with that of the maximum likelihood estimators in a simulation study. A numerical example investigating the factors impacting the orientation taken by a sand hopper when released is presented.The Canadian Journal of Statistics47: 712–728; 2019 © 2019 Statistical Society of Canada

Preliminary Test Estimators and Confidence Intervals for the Parametric Functions of the Moore and Bilikam Family of Lifetime Distributions Based on Records

We consider two measures of reliability functions namely R(t)=P(X>t) and P=P(X>Y) for the Moore and Bilikam (1978) family of lifetime distributions which covers fourteen distributions as specific cases. For record data from this family of distributions, preliminary test estimators (PTEs) and preliminary test confidence interval (PTCI) based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), empirical Bayes estimator (EBE) are obtained for the parameter. The bias and mean square error (MSE) (exact and asymptotic) of the proposed estimators are derived to study their relative efficiency and through simulation studies we establish that PTEs perform better than ordinary UMVUE, MLE and EBE. We also obtain the coverage probability (CP) and the expected length of the PTCI of the parameter and establish that the confidence intervals based on MLE are more precise. An application of the ordinary preliminary test estimator is also considered. To the best of the knowledge of the authors, no PTEs have been derived for R(t) and P based on records and thus we define improved PTEs based on MLE and UMVUE of R(t) and P. A comparative study of different methods of estimation done through simulations establishes that PTEs perform better than ordinary UMVUE and MLE.

Establishment of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of an Exponentiated Distribution Based on Type-II Censoring

The present paper has developed the preliminary test estimators (PTEs) of the model parameter raised to certain power, σ p , and the two measures of reliability, namely, the reliability function, R(t ) and the reliability of an item or a system, P of an exponentiated distribution, under Type- II censoring, based on their uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). The preliminary test confidence intervals (PTCIs) are also developed for σ, R(t ) and P based on their UMVUEs and MLEs. Further, the paper has derived expression for coverage probability of the PTCI of the model parameter, σ. Merits of the proposed PTEs are also established through analysis of simulated numerical data.

Improved estimators for stress-strength reliability using record ranked set sampling scheme

In this paper, the problem of estimating the stress-strength reliability based on upper record ranked set sampling is studied. In the exponential parent distribution, improved estimation strategies including the preliminary test estimator (PTE) and Stein-type shrinkage estimator (SE) are developed using the maximum likelihood estimator (MLE) and a prior information about R. Next, asymptotic distributional properties of the proposed estimator are exactly derived. Then we compare the proposed estimators in terms of asymptotic relative efficiency criteria. Furthermore, we construct some bootstrap confidence intervals (CI) in order to achieve more accurate inferences. The results show that our proposed improved estimators are superior to the MLE in wide range of parameter space. Specifically, the proposed shrinkage preliminary test estimator (SPTE) outperforms the MLE uniformity for small values of type one error. Performance analysis is also investigated through some numerical endeavors.

Generalized M-estimators for high-dimensional Tobit I models

This paper develops robust confidence intervals in high-dimensional and left-censored regression. Type-I censored regression models, where a competing event makes the variable of interest unobservable, are extremely common in practice. In this paper, we develop smoothed estimating equations that are adaptive to censoring level and are more robust to the misspecification of the error distribution. We propose a unified class of robust estimators, including one-step Mallow’s, Schweppe’s, and Hill-Ryan’s estimator that are adaptive to the left-censored observations. In the ultra-high-dimensional setting, where the dimensionality can grow exponentially with the sample size, we show that as long as the preliminary estimator converges faster than $n^{-1/4}$, the one-step estimators inherit asymptotic distribution of fully iterated version. Moreover, we show that the size of the residuals of the Bahadur representation matches those of the pure linear models – that is, the effects of censoring disappear asymptotically. Simulation studies demonstrate that our method is adaptive to the censoring level and asymmetry in the error distribution, and does not lose efficiency when the errors are from symmetric distributions.