First-order Bias Research Articles

Solar cycle variations in the powers and damping rates of low-degree solar acoustic oscillations

Helioseismology uses the Sun’s natural resonant oscillations to study the solar interior. The properties of the solar oscillations are sensitive to the Sun’s magnetic activity cycle. Here we examine variations in the powers, damping rates, and energy supply rates of the most prominent acoustic oscillations in unresolved, Sun-as-a-star data, obtained by the Birmingham Solar Oscillations Network (BiSON) during solar cycles 22, 23, and the first half of 24. The variations in the helioseismic parameters are compared to the 10.7cm flux, a well-known global proxy of solar activity. As expected the oscillations are most heavily damped and the mode powers are at a minimum at solar activity maximum. The 10.7cm flux was linearly regressed using the fractional variations of damping rates and powers observed during cycle 23. In general, good agreement is found between the damping rates and the 10.7cm flux. However, the linearly regressed 10.7cm flux and fractional variation in powers diverge in cycles 22 and 24, indicating that the relationship between the mode powers and the 10.7cm flux is not consistent from one cycle to the next. The energy supply rate of the oscillations, which is usually approximately constant, also decreases at this time. We have determined that this discrepancy is not because of the first-order bias introduced by an increase in the level of background noise or gaps in the data. Although we cannot categorically rule out an instrumental origin, the divergence observed in cycle 24, when the data were of high quality and the data coverage was over 80%, raises the possibility that the effect may be solar in origin.

An improved nonparametric lower bound of species richness via a modified good-turing frequency formula.

It is difficult to accurately estimate species richness if there are many almost undetectable species in a hyper-diverse community. Practically, an accurate lower bound for species richness is preferable to an inaccurate point estimator. The traditional nonparametric lower bound developed by Chao (1984, Scandinavian Journal of Statistics 11, 265-270) for individual-based abundance data uses only the information on the rarest species (the numbers of singletons and doubletons) to estimate the number of undetected species in samples. Applying a modified Good-Turing frequency formula, we derive an approximate formula for the first-order bias of this traditional lower bound. The approximate bias is estimated by using additional information (namely, the numbers of tripletons and quadrupletons). This approximate bias can be corrected, and an improved lower bound is thus obtained. The proposed lower bound is nonparametric in the sense that it is universally valid for any species abundance distribution. A similar type of improved lower bound can be derived for incidence data. We test our proposed lower bounds on simulated data sets generated from various species abundance models. Simulation results show that the proposed lower bounds always reduce bias over the traditional lower bounds and improve accuracy (as measured by mean squared error) when the heterogeneity of species abundances is relatively high. We also apply the proposed new lower bounds to real data for illustration and for comparisons with previously developed estimators.

II.1) Improving the Finite-Sample Estimators of Cointegrated Panel Models

Though ordinary least square (OLS) estimates are super-consistent with cointegrated variables, their finite-T bias can be large in the presence of endogenous feedback. Fully modified OLS (FMOLS) are parsimonious tools to measure the cointegrating [long-run] slope between integrated variables in the presence of endogenous feedback, and correct the first-order OLS bias to the extent necessary to provide a nuisance parameter-free asymptotic distribution.Yet because FMOLS rely on a first-step OLS estimator that is biased, and has weak power and size, FMOLS also has poor finite-T properties. I show that FMOLS asymptotically leave an O(h/T) fraction of the OLS bias, where h is the selected bandwidth.I also propose an improved estimator, which corrects the first-order conditional bias and removes the residual O(h/T) FMOLS bias.I establish the maximal speed at which N can grow simultaneously to T for the tests statistics of panel group-mean of time-series estimators to be (asymptotically) nuisance parameter-free. My improved estimator permits analysing wider panels.In the scenarios reviewed by previous research, my finite-T FMOLS has lower bias and RMSE than the `asymptotic' FMOLS for all values of T and N.

Jackknife estimation with a unit root

We study jackknife estimators in a first-order autoregression with a unit root. Non-overlapping sub-sample estimators have different limit distributions, so the jackknife does not fully eliminate first-order bias. We therefore derive explicit limit distributions of the numerator and denominator to calculate the expectations that determine optimal jackknife weights. Simulations show that the resulting jackknife estimator produces substantial reductions in bias and RMSE.

Bias Reduction in Nonlinear and Dynamic Panels in the Presence of Cross-Section Dependence, with a GARCH Panel Application

Nonlinear and dynamic panel models that contain individual-specific parameters are well-known to suffer from the incidental parameter bias. This bias and methods for correcting it have been studied extensively for panels with time-series dependence. However, a general analysis under cross-section dependence is missing. This paper extends the literature to dependence across both dimensions in large-N large-T panels. Our analysis is based on the integrated likelihood method which nests many common estimation approaches. We show that the composition of the first-order bias changes only under strong cross-section dependence, but not under weak or cluster-type dependence. Using bias correction techniques, we also propose a novel estimation approach for GARCH-type financial volatility modelling in small samples. Simulation analysis and a forecast exercise suggest that this method achieves success with as little as 150 time-series observations, which is much less than what is required by standard volatility estimation methods. We also use this approach to analyse the volatility characteristics of monthly hedge fund returns.

Performance Predictions for Parameter Estimators That Minimize Cost-Functions Using Wirtinger Calculus With Application to CM Blind Equalization

We present calculations of the first-order bias and first-order variance of parameters which are estimated batch-wise from the global minimum of a cost-function of complex-valued signals embedded in zero-mean Gaussian noise. The derivation involves the calculation of the multidimensional Taylor series of the non-analytic cost-function up to third order using the elegant Wirtinger calculus. Whereas closed-form expressions for the variance can be obtained straightforwardly from a second-order Taylor series, and have been presented in various other contexts, an exact expression for the bias cannot be derived, in general. In this paper, we propose approximate expressions for the first-order bias and confirm them in a comparison of results from extensive analytical calculations with results from Monte Carlo (MC) simulations for the statistical efficiency of a batch-processing blind equalizer using the constant-modulus (CM) criterion. We study the equalization of independent and identically distributed (i.i.d.) random symbols and obtain asymptotic (for large batch size) expressions for the averages of the bias and variance over zero-mean random (real-valued) signals of binary phase shift keying (BPSK), and (complex-valued) signals of M-ary PSK modulation (M >; 2) . Finally, we compare the statistical efficiency of the CM estimator with the one of the maximum likelihood (ML) blind estimation of the path parameters and equalized symbols with CM constraint.

A Simple Correction to Remove the Bias of the Gini Coefficient due to Grouping

We propose a first-order bias correction term for the Gini index to reduce the bias due to grouping. It depends on only the number of individuals in each group and is derived from a measurement error framework. We also provide a formula for the remaining second-order bias. Both Monte Carlo and EU and U.S. empirical evidence show that the first-order correction reduces a considerable share of the bias, but that some remaining second-order bias is increasing in the variance. We propose a procedure that addresses the remaining second-order bias by using additional information.

General second-order covariance of Gaussian maximum likelihood estimates applied to passive source localization in fluctuating waveguides

A method is provided for determining necessary conditions on sample size or signal to noise ratio (SNR) to obtain accurate parameter estimates from remote sensing measurements in fluctuating environments. These conditions are derived by expanding the bias and covariance of maximum likelihood estimates (MLEs) in inverse orders of sample size or SNR, where the first-order covariance term is the Cramer-Rao lower bound (CRLB). Necessary sample sizes or SNRs are determined by requiring that (i) the first-order bias and the second-order covariance are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. An analytical expression is provided for the second-order covariance of MLEs obtained from general complex Gaussian data vectors, which can be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) it allows for both the mean and variance of the measurement to be functions of the estimation parameters. Here, conditions are derived to obtain accurate source localization estimates in a fluctuating ocean waveguide containing random internal waves, and the consequences of the loss of coherence on their accuracy are quantified.

Finite-sample properties of the maximum likelihood estimator for the binary logit model with random covariates

We examine the finite sample properties of the maximum likelihood estimator for the binary logit model with random covariates. Previous studies have either relied on large-sample asymptotics or have assumed non-random covariates. Analytic expressions for the first-order bias and second-order mean squared error function for the maximum likelihood estimator in this model are derived, and we undertake numerical evaluations to illustrate these analytic results for the single covariate case. For various data distributions, the bias of the estimator is signed the same as the covariate’s coefficient, and both the absolute bias and the mean squared errors increase symmetrically with the absolute value of that parameter. The behaviour of a bias-adjusted maximum likelihood estimator, constructed by subtracting the (maximum likelihood) estimator of the first-order bias from the original estimator, is examined in a Monte Carlo experiment. This bias-correction is effective in all of the cases considered, and is recommended for use when this logit model is estimated by maximum likelihood using small samples.

A generic algorithm for reducing bias in parametric estimation

A general iterative algorithm is developed for the computation of reduced-bias parameter estimates in regular statistical models through adjustments to the score function. The algorithm unifies and provides appealing new interpretation for iterative methods that have been published previously for some specific model classes. The new algorithm can usefully be viewed as a series of iterative bias corrections, thus facilitating the adjusted score approach to bias reduction in any model for which the first- order bias of the maximum likelihood estimator has already been derived. The method is tested by application to a logit-linear multiple regression model with beta-distributed responses; the results confirm the effectiveness of the new algorithm, and also reveal some important errors in the existing literature on beta regression.

Performance Bound Approximation for Bearing Estimation with Bias Correction

Array-based bearing estimation often displays a threshold behavior, that is, below certain signal-to-noise ratio (SNR) the estimation mean-square error (MSE) increases dramatically. The error increase is known to be largely attributed to sidelobe ambiguities in signal field correlation along with estimation bias at low SNR. This paper investigates the bias-related contribution from the perspective of local performance bounds. The first-order bias of the maximum likelihood estimate is first derived for a complex multivariate Gaussian data model, which is then incorporated into the Cramer-Rao bound and Barankin bound respectively to obtain MSE approximations. The simulations show an improved threshold region error prediction compared to the same bounds without bias correction.

Bias and variance reduction techniques for bootstrap information criteria

We discuss the problem of constructing information criteria by applying the bootstrap methods. Various bias and variance reduction methods are presented for improving the bootstrap bias correction term in computing the bootstrap information criterion. The properties of these methods are investigated both in theoretical and numerical aspects, for which we use a statistical functional approach. It is shown that the bootstrap method automatically achieves the second-order bias correction if the bias of the first-order bias correction term is properly removed. We also show that the variance associated with bootstrapping can be considerably reduced for various model estimation procedures without any analytical argument. Monte Carlo experiments are conducted to investigate the performance of the bootstrap bias and variance reduction techniques.

Orthogonal Locally Ancillary Estimating Functions with Application to Stratified Clustered Data

We propose an orthogonal locally ancillary estimating function that provides first-order bias correction of inferences. It requires the specification of merely the first two moments of the observations when applying to analysis of stratified clustered (continuous or binary) data with the parameters of interest in both the first and second joint moments of dependent data. Simulation results confirm that the estimators obtained using the proposed method are substantially improved over those using regular profile estimating functions.

HIGHER ORDER ASYMPTOTIC THEORY WHEN A PARAMETER IS ON A BOUNDARY WITH AN APPLICATION TO GARCH MODELS

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

Orthogonal Locally Ancillary Estimating Functions for Matched Pair Studies and Errors in Covariates

SummaryWe propose an estimating function method for two related applications, matched pair studies and studies with errors in covariates under a functional model, where a mismeasured unknown scalar covariate is treated as a fixed nuisance parameter. Our method addresses the severe inferential problem that is posed by an abundance of nuisance parameters in these two applications. We propose orthogonal locally ancillary estimating functions for these two applications that depend on merely the mean model and partial modelling of the variances of the observations (and observed mismeasured covariate, if applicable), and we achieve first-order bias correction of inferences under a ‘small dispersion and large sample size’ asymptotic. Simulation results confirm that the estimator proposed is largely improved over that using a regular profile estimating function. We apply the approach proposed to a length of hospital stay study with a mismeasured covariate.

Bias Reduction through First-order Mean Correction, Bootstrapping and Recursive Mean Adjustment

Standard methods of estimation for autoregressive models are known to be biased in finite samples, which has implications for estimation, hypothesis testing, confidence interval construction and forecasting. Three methods of bias reduction are considered here: first-order bias correction, FOBC, where the total bias is approximated by the O(T−1) bias; bootstrapping; and recursive mean adjustment, RMA. In addition, we show how first-order bias correction is related to linear bias correction. The practically important case where the AR model includes an unknown linear trend is considered in detail. The fidelity of nominal to actual coverage of confidence intervals is also assessed. A simulation study covers the AR(1) model and a number of extensions based on the empirical AR(p) models fitted by Nelson & Plosser (1982). Overall, which method dominates depends on the criterion adopted: bootstrapping tends to be the best at reducing bias, recursive mean adjustment is best at reducing mean squared error, whilst FOBC does particularly well in maintaining the fidelity of confidence intervals.

Conditional quantile estimation by local logistic regression†

In this article, we propose a new nonparametric estimator of the conditional quantile function. It is based on locally fitting a logistic model. We compare the new proposal with some existing methods. Those include the double-kernel technique of Yu and Jones (1998), the adjusted version of the Nadaraya–Watson estimator suggested by Hall et al. (1999) and the approach by Koenker and Bassett (1978) based on the ‘check function’ loss. The comparison is done by asymptotic mean squared error and a simulation study. The four estimators have the same asymptotic variance, but their first-order biases are different. We also propose a new automatic smoothing parameter selection method in quantile estimation. We analyze the finite sample properties of the quantile estimators using the proposed bandwidth selectors. We find that the new method outperforms the others in most cases of the numerical experiments.

Confidence intervals for multinomial logistic regression in sparse data

Logistic regression is one of the most widely used regression models in practice, but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples. Modification of the logistic regression score function to remove first-order bias is equivalent to penalizing the likelihood by the Jeffreys prior, and yields penalized maximum likelihood estimates (PLEs) that always exist, even in samples in which maximum likelihood estimates (MLEs) are infinite. PLEs are an attractive alternative in small-to-moderate-sized samples, and are preferred to exact conditional MLEs when there are continuous covariates. We present methods to construct confidence intervals (CI) in the penalized multinomial logistic regression model, and compare CI coverage and length for the PLE-based methods to that of conventional MLE-based methods in trinomial logistic regressions with both binary and continuous covariates. Based on simulation studies in sparse data sets, we recommend profile CIs over asymptotic Wald-type intervals for the PLEs in all cases. Furthermore, when finite sample bias and data separation are likely to occur, we prefer PLE profile CIs over MLE methods.

Inference for Change-Point and Post-Change Mean with Possible Change in Variance

For a sequence of independent normal random variables, we consider the estimation of the change-point and the post-change mean after a change in the mean is detected by a CUSUM procedure, subject to a possible change in variance. Conditional on the event that a change is detected and it occurred far away from the starting point and the threshold is large, the (absolute) bias of the maximum likelihood estimator for the change-point (obtained at the reference value) is found. The first-order biases for the post-change mean and variance estimators are also obtained by using Wald's Likelihood Ratio Identity and the renewal theorem. In the local case when the reference value and the post-change mean are both small, accurate approximations are derived. A confidence interval for post-change mean based on a corrected normal pivot is then discussed.

Inference for post-change mean by a CUSUM procedure

In this paper, the inference problem for the post-change mean is considered after a change is detected by a CUSUM process in a sequence of independent normal variables. The change-point is estimated as the maximum likelihood estimate at the reference value and the post-change mean is estimated as the sample mean after the change-point estimate. By assuming the change-point is large and the monitoring limit approaches infinity, the first-order bias of the post-change mean estimate and a corrected asymptotic normal pivot are derived conditioning on that a change is detected. Local approximations for small reference value and post-change mean are obtained for numerical evaluation.