Asymptotic Mean Square Error Research Articles

A Class of Semiparametric Tail Index Estimators and Its Applications

A new class of semiparametric estimators of the tail index is proposed. These estimators are based on a rather general class of semiparametric statistics. Their asymptotic normality is proved. The new estimators are compared with several other recently introduced estimators of the tail index in terms of the asymptotic mean-square error. An algorithm to calculate the new estimators is developed and then applied to several real data sets.

Tail Index Estimation: Quantile Driven Threshold Selection

The selection of upper order statistics in tail estimation is notoriously difficult. Most methods are based on asymptotic arguments, like minimizing the asymptotic mse, that do not perform well in finite samples. Here we advance a data driven method that minimizes the maximum distance between the fitted Pareto type tail and the observed quantile. To analyse the finite sample properties of the metric we organize a horse race between the other methods. In most cases the finite sample based methods perform best. To demonstrate the economic relevance of choosing the proper methodology we use daily equity return data from the CRSP database and find economic relevant variation between the tail index estimates.

Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response

Based on an incomplete observed sample (Xt,Yt,ζt)0≤t≤T, where ζt=1 if Yt is observed at time t and ζt=0 otherwise, the asymptotic mean square error of the regression estimator m̂T(x),x∈Rd, is showed to satisfy the following inequality: limT→∞T4∕(d+4)E(m̂T(x)−m(x))2<c where c is an explicit constant. A simulation study is performed to illustrate the effect of the sampling mesh as well as the missing at random mechanism on the consistency of the regression estimator for regular and irregular continuous time processes.

Further Results on the Cramér–Rao Bound for Sparse Linear Arrays

Sparse linear arrays, such as co-prime and nested arrays, can identify up $O(M^2)$ uncorrelated sources with only $O(M)$ sensors by utilizing their difference coarray model. In our previous work, we derived closed-form asymptotic mean-squared error (MSE) expressions for two coarray based MUSIC algorithms and analyzed the Cramer-Rao bound (CRB) in high signal-to-noise ratio (SNR) regions, under the assumption of uncorrelated sources. In this paper, we provide further analysis of the CRB presented in our previous work. We first establish the connection between two CRBs, the CRB derived with the assumption that the sources are uncorrelated, and the classical stochastic CRB derived without this assumption. We show that they are asymptotically equal in high SNR regions for uncorrelated sources. Next, we analyze the behavior of the former CRB for co-prime and nested arrays with a large number of sensors. We investigate the effect of configuration parameters on this CRB. We show that, for co-prime and nested arrays, this CRB can decrease at a rate of $O(M^{-5})$ for large values of $M$ , while this rate is only $O(M^{-3})$ for an $M$ -sensor ULA. This finding analytically demonstrates that co-prime and nested arrays can achieve better asymptotic estimation performance when the number of sensors is a limiting factor. We also show that for a fixed aperture, co-prime and nested arrays require more snapshots to achieve the same performance as ULAs, showing the trade-off between the number of spatial samples and the number of temporal samples. Finally, we demonstrate our analytical results with numerical experiments.

MSE based optimization of the measure-transformed MUSIC algorithm

Abstract Recently, we developed a robust generalization of the MUltiple SIgnal Classification (MUSIC) algorithm. This generalization, called measure-transformed MUSIC (MT-MUSIC), operates by applying a transform to the probability measure (distribution) of the data. The considered transform is structured by a non-negative data-weighting function, called MT-function, that, when properly chosen, mitigates the effect of non-Gaussian heavy-tailed noise that produces outliers. In this paper, we characterize the asymptotic mean-squared-error (MSE) performance of the MT-MUSIC algorithm. Under some mild regularity conditions, we show that the MT-MUSIC estimator is asymptotically normal and unbiased, and obtain an analytic expression for the asymptotic MSE matrix. We go on to develop a strongly consistent estimator for the asymptotic MSE matrix that is constructed via the same sequence of samples being used for implementation of the MT-MUSIC. This paves the way for development of a data-driven procedure for optimal selection of the MT-function parameters that minimizes an empirical estimate of the asymptotic average root MSE (RMSE). The performance advantage of the proposed MSE based optimization of the MT-MUSIC is illustrated in simulation examples.

New kernel estimators of the hazard ratio and their asymptotic properties

We propose a kernel estimator of a hazard ratio that is based on a modification of Cwik and Mielniczuk (Commun Stat-Theory Methods 18(8):3057–3069, 1989)’s method. A naive nonparametric estimator is Watson and Leadbetter (Sankhyā: Indian J Stat Ser A 26(1):101–116, 1964)’s one, which is naturally given by the kernel density estimator and the empirical distribution estimator. We compare the asymptotic mean squared error (AMSE) of the hazard estimators, and then, it is shown that the asymptotic variance of the new estimator is usually smaller than that of the naive one. We also discuss bias reduction of the proposed estimator and derived some modified estimators. While the modified estimators do not lose nonnegativity, their AMSE is small both theoretically and numerically.

Performance Analysis of Coarray-Based MUSIC in the Presence of Sensor Location Errors

Sparse linear arrays, such as co-prime and nested arrays, can resolve more uncorrelated sources than the number of sensors by applying the MUtiple SIgnal Classification (MUSIC) algorithm to their difference coarray model. We aim at statistically analyzing the performance of the MUSIC algorithm applied to the difference coarray model, namely, the coarray-based MUSIC, in the presence of sensor location errors. We first introduce a signal model for sparse linear arrays in the presence of deterministic unknown location errors. Based on this signal model, we derive a closed-form expression of the asymptotic mean-squared error of a commonly used coarray-based MUSIC algorithm, SS-MUSIC, in the presence of small sensor location errors. We show that the sensor location errors introduce a constant bias that depends on both the physical array geometry and the coarray geometry, which cannot be mitigated by only increasing the signal-to-noise ratio. We next give a short extension of our analysis to cases when the sensor location errors are stochastic and investigate the Gaussian case. Finally, we derive the Cramer–Rao bound for joint estimation of direction-of-arrivals and sensor location errors for sparse linear arrays, which can be applicable even if the number of sources exceeds the number of sensors. Numerical simulations show good agreement between empirical results and our theoretical results.

A generalized focused information criterion for GMM

SummaryThis paper proposes a criterion for simultaneous generalized method of moments model and moment selection: the generalized focused information criterion (GFIC). Rather than attempting to identify the “true” specification, the GFIC chooses from a set of potentially misspecified moment conditions and parameter restrictions to minimize the mean squared error (MSE) of a user‐specified target parameter. The intent of the GFIC is to formalize a situation common in applied practice. An applied researcher begins with a set of fairly weak “baseline” assumptions, assumed to be correct, and must decide whether to impose any of a number of stronger, more controversial “suspect” assumptions that yield parameter restrictions, additional moment conditions, or both. Provided that the baseline assumptions identify the model, we show how to construct an asymptotically unbiased estimator of the asymptotic MSE to select over these suspect assumptions: the GFIC. We go on to provide results for postselection inference and model averaging that can be applied both to the GFIC and various alternative selection criteria. To illustrate how our criterion can be used in practice, we specialize the GFIC to the problem of selecting over exogeneity assumptions and lag lengths in a dynamic panel model, and show that it performs well in simulations. We conclude by applying the GFIC to a dynamic panel data model for the price elasticity of cigarette demand.

Performance analysis of G-MUSIC based DOA estimator with random linear array: A single source case

We analyze the performance of direction-of-arrival (DOA) estimation based on the G-MUSIC algorithm for a single source using a random linear array, i.e., only a random subset of sensor array is in operation. The classical asymptotic analysis assumes that the number of snapshots tends to infinity whereas the number of sensors remains fixed. In contrast, this work studies the scenario where both quantities tend to infinity at the same rate. This general asymptotic regime provides a more accurate description of the practical situation where these two quantities are finite and comparable. We prove the consistency of the DOA estimator and derive an analytic expression for the mean square error (MSE) performance in the general asymptotic situation. The asymptotic MSE turns out to be a function of signal-to-noise ratio, probability of operating sensors and ratio of number of sensors to number of snapshots. Numerical results are presented to confirm the effectiveness of our theoretical calculations.

Automatic Optimal Batch Size Selection for Recursive Estimators of Time-Average Covariance Matrix

ABSTRACTThe time-average covariance matrix (TACM) , where Γk is the auto-covariance function, is an important quantity for the inference of the mean of an -valued stationary process (d ⩾ 1). This article proposes two recursive estimators for Σ with optimal asymptotic mean square error (AMSE) under different strengths of serial dependence. The optimal estimator involves a batch size selection, which requires knowledge of a smoothness parameter , for some β. This article also develops recursive estimators for ϒβ. Combining these two estimators, we obtain a fully automatic procedure for optimal online estimation for Σ. Consistency and convergence rates of the proposed estimators are derived. Applications to confidence region construction and Markov chain Monte Carlo convergence diagnosis are discussed. Supplementary materials for this article are available online.

Optimal Shrinkage of Singular Values

We consider the recovery of low-rank matrices from noisy data by shrinkage of singular values, in which a single, univariate nonlinearity is applied to each of the empirical singular values. We adopt an asymptotic framework, in which the matrix size is much larger than the rank of the signal matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. For a variety of loss functions, including Mean Square Error (MSE) - (square Frobenius norm), the nuclear norm loss and the operator norm loss, we show that in this framework there is a well-defined asymptotic loss that we evaluate precisely in each case. In fact, each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities. We provide a general method for evaluating these optimal nonlinearities, and demonstrate our framework by working out simple, explicit formulas for the optimal nonlinearities in the Frobenius, nuclear and operator norm cases. For example, for a square low-rank n-by-n matrix observed in white noise with level σ, the optimal nonlinearity for MSE loss simply shrinks each data singular value y to √(y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -4nσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) (or to 0 if y <; 2√nσ). This optimal value y to nonlinearity guarantees an asymptotic MSE of 2nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding, providing guarantees of 3nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and 6nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively. Our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision. As an example, we compute optimal shrinkers for the Schatten-p norm loss, for any p > 0.

Combining reliability functions of a Weibull distribution

In this article, a large sample pooling procedure is considered for the reliability function of a Weibull distribution. Asymptotic properties of shrinkage estimation procedures based on the preliminary test are developed. It is shown that the proposed estimator has substantially smaller asymptoticmean squared error (AMSE) than the usual maximumlikelihood (ML) estimator inmost of the parameter space. Analytic AMSE expressions of the proposed estimators are obtained and the dominance picture of the estimators is presented by comparing them. It is shown that the suggested estimators yield a wider dominance range over theML estimator than the usual pretest estimator and give a meaningful size of the pretest. To appraise the small sample performance of the estimators, detailed Monte-Carlo simulation studies are also carried out.

Using invalid instruments on purpose: Focused moment selection and averaging for GMM

In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a novel moment selection procedure for GMM–the Focused Moment Selection Criterion (FMSC)–in which moment conditions are chosen not based on their validity but on the MSE of their associated estimator of a user-specified target parameter. The FMSC mimics the situation faced by an applied researcher who begins with a set of relatively mild “baseline” assumptions and must decide whether to impose any of a collection of stronger but more controversial “suspect” assumptions. When the (correctly specified) baseline moment conditions identify the model, the FMSC provides an asymptotically unbiased estimator of asymptotic MSE, allowing us to select over the suspect moment conditions. I go on to show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose simulation-based procedures for inference that can be applied to a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedures perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria and income per capita.

Coarrays, MUSIC, and the Cramér–Rao Bound

Sparse linear arrays, such as co-prime arrays and nested arrays, have the attractive capability of providing enhanced degrees of freedom. By exploiting the coarray structure, an augmented sample covariance matrix can be constructed and MUSIC (MUtiple SIgnal Classification) can be applied to identify more sources than the number of sensors. While such a MUSIC algorithm works quite well, its performance has not been theoretically analyzed. In this paper, we derive a simplified asymptotic mean square error (MSE) expression for the MUSIC algorithm applied to the coarray model, which is applicable even if the source number exceeds the sensor number. We show that the directly augmented sample covariance matrix and the spatial smoothed sample covariance matrix yield the same asymptotic MSE for MUSIC. We also show that when there are more sources than the number of sensors, the MSE converges to a positive value instead of zero when the signal-to-noise ratio (SNR) goes to infinity. This finding explains the "saturation" behavior of the coarray-based MUSIC algorithms in the high SNR region observed in previous studies. Finally, we derive the Cram\'er-Rao bound (CRB) for sparse linear arrays, and conduct a numerical study of the statistical efficiency of the coarray-based estimator. Experimental results verify theoretical derivations and reveal the complex efficiency pattern of coarray-based MUSIC algorithms.

Model averaging in semiparametric estimation of treatment effects

Choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects. This paper proposes a data-driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the ATT. The proposed procedures minimize the estimated MSE of the ATT estimator in a local asymptotic framework. We formulate model averaging as a statistical decision problem in a limit experiment, and derive an averaging scheme that is Bayes optimal with respect to a given prior. The averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of Bayes asymptotic MSE. Our Monte Carlo studies illustrate the size of the MSE gains. We apply the averaging procedure to evaluate the effect of a labor market program.

SMOOTHED ESTIMATING EQUATIONS FOR INSTRUMENTAL VARIABLES QUANTILE REGRESSION

The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth h. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some h &gt; 0, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal h also minimizes the higher-order type I error of a SEE-based χ2 test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes simpler and more reliable, especially with (more) endogenous regressors. Monte Carlo simulations demonstrate all of these superior properties in finite samples, and we apply our estimator to JTPA data. Smoothing the estimating equations is not just a technical operation for establishing Edgeworth expansions and bootstrap refinements; it also brings the real benefits of having more precise estimators and more powerful tests.

Searching for long-range dependence in real effective exchange rate: towards parity?

Abstract After the widespread adoption of flexible exchange rate regime since 1973 the volatility of the exchange rate has increased, as a consequence of greater trade openness and financial integration. As a result, it has become difficult to find evidence of the purchasing power parity hypothesis (PPP). This study investigates the possibility of a fall in the persistence of the real exchange rate as a consequence of the financial and commercial integration by employing monthly real effective exchange rate dataset provided by the International Monetary Fund (IMF). Beginning with an exploratory data analysis in the frequency domain, the fractional coefficient d was estimated employing the bias-reduced estimator on a sample of 20 countries over the period ranging from 1975 to 2011. As the main novelty, this study applies a bias-reduced log-periodogram regression estimator instead of the traditional method proposed by GPH which eliminates the first and higher orders biases by a data-dependent plug-in method for selecting the number of frequencies to minimize asymptotic mean-squared error (MSE). Additionally, this study also estimates a moving window of fifteen years to observe the path of the fractional coefficient in each country. No evidence was found of a statistically significant change in the persistence of the real exchange rate.

A Covariate Selection Criterion for Estimation of Treatment Effects

We study how to select or combine estimators of the average treatment effect (ATE) and the average treatment effect on the treated (ATT) in the presence of multiple sets of covariates. We consider two cases: (1) all sets of covariates satisfy the unconfoundedness assumption and (2) some sets of covariates violate the unconfoundedness assumption locally. For both cases, we propose a data-driven covariate selection criterion (CSC) to minimize the asymptotic mean squared errors (AMSEs). Based on our CSC, we propose new average estimators of ATE and ATT, which include the selected estimators based on a single set of covariates as a special case. We derive the asymptotic distributions of our new estimators and propose how to construct valid confidence intervals. Our Monte Carlo simulations show that in finite samples, our new average estimators achieve substantial efficiency gains over the estimators based on a single set of covariates. We apply our new estimators to study the impact of inherited control on firm performance.

Fisher information versus signal-to-noise ratio for a split detector

We study the problem of estimating the magnitude of a Gaussian beam displacement using a two-pixel or ``split'' detector. We calculate the maximum likelihood estimator and compute its asymptotic mean-squared error via the Fisher information. Although the signal-to-noise ratio is known to be simply related to the Fisher information under idealized detection, we find the two measures of precision differ markedly for a split detector. We show that a greater signal-to-noise ratio ``before'' the detector leads to a greater information penalty, unless adaptive realignment is used. We find that with an initially balanced split detector, tuning the normalized difference in counts to $0.884753...$ gives the highest posterior Fisher information, which provides an improvement by at least a factor of about 2.5 over operating in the usual linear regime. We discuss the implications for weak-value amplification, a popular probabilistic signal amplification technique.

When are Direct Multi‐step and Iterative Forecasts Identical?

Although both direct multi‐step‐ahead forecasting and iterated one‐step‐ahead forecasting are two popular methods for predicting future values of a time series, it is not clear that the direct method is superior in practice, even though from a theoretical perspective it has lower mean squared error (MSE). A given model can be fitted according to either a multi‐step or a one‐step forecast error criterion, and we show here that discrepancies in performance between direct and iterative forecasting arise chiefly from the method of fitting, and is dictated by the nuances of the model's misspecification. We derive new formulas for quantifying iterative forecast MSE, and present a new approach for assessing asymptotic forecast MSE. Finally, the direct and iterative methods are compared on a retail series, which illustrates the strengths and weaknesses of each approach. Copyright © 2015 John Wiley &amp; Sons, Ltd.