We investigate the joint asymptotic behavior of so-called blocks estimator of the extremal index, that determines the mean length of clusters of extremes, based on the exceedances over different thresholds. Due to the large bias of these estimators, the resulting estimates are usually very sensitive to the choice of the threshold and thus difficult to interpret. We propose and examine a bias correction that asymptotically removes the leading bias term while the rate of convergence of the random error is preserved.
The extremal index parameter theta characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance and environmental studies. In this study, a novel estimator for theta based on the asymptotic scaling of block-maxima and resampling is introduced. It is shown to be consistent and asymptotically normal for a large class of m-dependent time series. Further, a procedure for the automatic selection of its tuning parameter is developed and different types of confidence interva…
The occurrence of successive extreme observations can have an impact on society. In extreme value theory there are parameters to evaluate the effect of clustering of high values, such as the extremal index. The estimation of the extremal index is a recurrent theme in the literature and there are several methodologies for this purpose. The majority of existing methods depend on two parameters whose choice affects the performance of the estimators. Here we consider a new estimator depending only on one of the parameters, thus contributing to a decrease in the degree of uncertainty. A simulation …
The extremal index $θ$, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate $θ$ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are c…
The $α$-Brownian bridge, or scaled Brownian bridge, is a generalization of the Brownian bridge with a scaling parameter that determines how strong the force that pulls the process back to 0 is. The bias of the maximum likelihood estimator of the parameter $α$ is derived and a bias-correction that improves the estimator substantially is proposed. The properties of the bias-corrected estimator and four Bayesian estimators based on non-informative priors are evaluated in a simulation study.
In the causal inference literature an estimator belonging to a class of semi-parametric estimators is called robust if it has desirable properties under the assumption that at least one of the working models is correctly specified. In this paper we propose a crude analytical approach to study the large sample bias of semi-parameteric estimators of the average causal effect when all working models are misspecified. We apply our approach to three prototypical estimators, two inverse probability weighting (IPW) estimators, using a misspecified propensity score model, and a doubly robust (DR) esti…
Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate.
Understanding dependence structure among extreme values plays an important role in risk assessment in environmental studies. In this work we propose the $χ$ network and the annual extremal network for exploring the extremal dependence structure of environmental processes. A $χ$ network is constructed by connecting pairs whose estimated upper tail dependence coefficient, $\hat χ$, exceeds a prescribed threshold. We develop an initial $χ$ network estimator and we use a spatial block bootstrap to assess both the bias and variance of our estimator. We then develop a method to correct the bias of t…
The pseudo-Lindley distribution which was introduced in Zeghdoudi and Nedjar (2016) is studied with regards to its upper tail. In that regard, and when the underlying distribution function follows the Pseudo-Lindley law, we investigate the behavior of its values, the asymptotic normality of the Hill estimator and the double-indexed generalized Hill statistic process (Ngom and Lo), the asymptotic normality of the records values and the moment problem.
We use bias-reduced estimators of high quantiles, of heavy-tailed distributions, to introduce a new estimator of the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked, in a simulation study, by four of the most popular goodness-of-fit tests for different sample sizes. Moreover, we compare, in terms of bias and mean squared error, our estimator with Peng's estimator (Peng, 2001) and we evaluate the accuracy of some resulting confidence intervals.
This paper establishes a global bias-correction divide-and-conquer (GBC-DC) rule for biased estimation under the case of memory constraint. In order to introduce the new estimation, a closed representation of the local estimators obtained by the data in each batch is adopted, aiming to formulate a pro forma linear regression between the local estimators and the true parameter of interest. Least square method is then used within this framework to composite a global estimator of the parameter. Thus, the main advantage over the classical DC method is that the new GBC-DC method can absorb the info…
In this paper we discuss improved estimators for the regression and the dispersion parameters in an extended class of dispersion models (Jørgensen, 1996). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the second-order bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The pra…
An important challenge in statistical analysis concerns the control of the finite sample bias of estimators. This problem is magnified in high-dimensional settings where the number of variables $p$ diverges with the sample size $n$, as well as for nonlinear models and/or models with discrete data. For these complex settings, we propose to use a general simulation-based approach and show that the resulting estimator has a bias of order $\mathcal{O}(0)$, hence providing an asymptotically optimal bias reduction. It is based on an initial estimator that can be slightly asymptotically biased, makin…
The issues of bias-correction and robustness are crucial in the strategy of divide-and-conquer (DC), especially for asymmetric nonparametric models with massive data. It is known that quantile-based methods can achieve the robustness, but the quantile estimation for nonparametric regression has non-ignorable bias when the error distribution is asymmetric. This paper explores a global bias-corrected DC by quantile-matched composite for nonparametric regressions with general error distributions. The proposed strategies can achieve the bias-correction and robustness, simultaneously. Unlike common…
Estimating the limiting shape of bivariate scaled sample clouds: with additional benefits of self-consistent inference for existing extremal dependence properties
In this paper, we propose a reduced-bias estimator of the EVI for Pareto-type tails (heavy-tailed) distributions. This is derived using the weighted least squares method. It is shown that the estimator is unbiased, consistent and asymptotically normal under the second-order conditions on the underlying distribution of the data. The finite sample properties of the proposed estimator are studied through a simulation study. The results show that it is competitive to the existing estimators of the extreme value index in terms of bias and Mean Square Error. In addition, it yields estimates of $γ>0$…
In this paper, we study the strong consistency of a bias reduced kernel density estimator and derive a strongly con- sistent Kullback-Leibler divergence (KLD) estimator. As application, we formulate a goodness-of-fit test and an asymptotically standard normal test for model selection. The Monte Carlo simulation show the effectiveness of the proposed estimation methods and statistical tests.
An important challenge in statistical analysis lies in controlling the bias of estimators due to the ever-increasing data size and model complexity. Approximate numerical methods and data features like censoring and misclassification often result in analytical and/or computational challenges when implementing standard estimators. As a consequence, consistent estimators may be difficult to obtain, especially in complex and/or high dimensional settings. In this paper, we study the properties of a general simulation-based estimation framework that allows to construct bias corrected consistent est…
The distributed Hill estimator is a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the choice of the number of the exceedance ratios used in each machine. Even when choosing the number at a low level, a high asymptotic bias may arise. We overcome this potential drawback by designing a bias correction procedure for the distributed Hill estimator, which adheres to the setup of distributed inference. The asymptotically unbiased distributed estimato…
The stable tail dependence function provides a full characterization of the extremal dependence structures. Unfortunately, the estimation of the stable tail dependence function often suffers from significant bias, whose scale relates to the Peaks-Over-Threshold (POT) second-order parameter. For this second-order parameter, this paper introduces a penalized estimator that discourages it from being too close to zero. This paper then establishes this estimator's asymptotic consistency, uses it to correct the bias in the estimation of the stable tail dependence function, and illustrates its desira…
This work proposes a new minimum distance estimator (MDE) for the parameters of short and long memory models. This bias corrected minimum distance estimator (BCMDE) considers a correction in the usual MDE to account for the bias of the sample autocorrelation function when the mean is unknown. We prove the weak consistency of the BCMDE for the general fractional autoregressive moving average (ARFIMA(p, d, q)) model and derive its asymptotic distribution for some particular cases. Simulation studies show that the BCMDE presents a good performance compared to other procedures frequently used in t…
This paper investigates the use of bootstrap-based bias correction of semi-parametric estimators of the long memory parameter in fractionally integrated processes. The re-sampling method involves the application of the sieve bootstrap to data pre-filtered by a preliminary semi-parametric estimate of the long memory parameter. Theoretical justification for using the bootstrap techniques to bias adjust log-periodogram and semi-parametric local Whittle estimators of the memory parameter is provided. Simulation evidence comparing the performance of the bootstrap bias correction with analytical bia…
We use the jackknife to bias correct the log-periodogram regression(LPR) estimator of the fractional parameter in a stationary fractionally integrated model. The weights for the jackknife estimator are chosen in such a way that bias reduction is achieved without the usual increase in asymptotic variance, with the estimator viewed as `optimal' in this sense. The theoretical results are valid under both the non-overlapping and moving-block sub-sampling schemes that can be used in the jackknife technique, and do not require the assumption of Gaussianity for the data generating process. A Monte Ca…
The median absolute deviation is a widely used robust measure of statistical dispersion. Using a scale constant, we can use it as an asymptotically consistent estimator for the standard deviation under normality. For finite samples, the scale constant should be corrected in order to obtain an unbiased estimator. The bias-correction factor depends on the sample size and the median estimator. When we use the traditional sample median, the factor values are well known, but this approach does not provide optimal statistical efficiency. In this paper, we present the bias-correction factors for the …
This paper presents a novel estimation approach for cumulative link models, based on median bias reduction as developed in Kenne Pagui et al. (2017). The median bias reduced estimator is obtained as solution of an estimating equation based on an adjustment of the score. It allows to obtain higher-order median centering of maximum likelihood estimates without requiring their finiteness. Moreover, the estimator is equivariant under componentwise monotone reparameterizations and the method is effective in preventing boundary estimates. We evaluate the properties of the median bias reduced estimat…
In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. (2008). The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set are provided in order to prove th…
In this paper, we consider the beta prime regression model recently proposed by \cite{bour18}, which is tailored to situations where the response is continuous and restricted to the positive real line with skewed and long tails and the regression structure involves regressors and unknown parameters. We consider two different strategies of bias correction of the maximum-likelihood estimators for the parameters that index the model. In particular, we discuss bias-corrected estimators for the mean and the dispersion parameters of the model. Furthermore, as an alternative to the two analytically b…
The manuscript discusses how to incorporate random effects for quantile regression models for clustered data with focus on settings with many but small clusters. The paper has three contributions: (i) documenting that existing methods may lead to severely biased estimators for fixed effects parameters; (ii) proposing a new two-step estimation methodology where predictions of the random effects are first computed {by a pseudo likelihood approach (the LQMM method)} and then used as offsets in standard quantile regression; (iii) proposing a novel bootstrap sampling procedure in order to reduce bi…
For studying or reducing the bias of functionals of the Kaplan-Meier survival estimator, the jackknifing approach of Stute and Wang (1994) is natural. We have studied the behavior of the jackknife estimate of bias under different configurations of the censoring level, sample size, and the censoring and survival time distributions. The empirical research reveals some new findings about robust calculation of the bias, particularly for higher censoring levels. We have extended their jackknifing approach to cover the case where the largest observation is censored, using the imputation methods for …
The main approach to inference for multivariate extremes consists in approximating the joint upper tail of the observations by a parametric family arising in the limit for extreme events. The latter may be expressed in terms of componentwise maxima, high threshold exceedances or point processes, yielding different but related asymptotic characterizations and estimators. The present paper clarifies the connections between the main likelihood estimators, and assesses their practical performance. We investigate their ability to estimate the extremal dependence structure and to predict future extr…
This paper investigates bootstrap-based bias correction of semiparametric estimators of the long memory parameter, $d$, in fractionally integrated processes. The re-sampling method involves the application of the sieve bootstrap to data pre-filtered by a preliminary semiparametric estimate of the long memory parameter. Theoretical justification for using the bootstrap technique to bias adjust log periodogram and semiparametric local Whittle estimators of the memory parameter is provided in the case where the true value of $d$ lies in the range $0\leq d<0.5$. That the bootstrap method provides …
The strategy of divide-and-combine (DC) has been widely used in the area of big data. Bias-correction is crucial in the DC procedure for validly aggregating the locally biased estimators, especial for the case when the number of batches of data is large. This paper establishes a race-DC through a residual-adjustment composition estimate (race). The race-DC applies to various types of biased estimators, which include but are not limited to Lasso estimator, Ridge estimator and principal component estimator in linear regression, and least squares estimator in nonlinear regression. The resulting g…
The crosswise model is an increasingly popular survey technique to elicit candid answers from respondents on sensitive questions. Recent studies, however, point out that in the presence of inattentive respondents, the conventional estimator of the prevalence of a sensitive attribute is biased toward 0.5. To remedy this problem, we propose a simple design-based bias correction using an anchor question that has a sensitive item with known prevalence. We demonstrate that we can easily estimate and correct for the bias arising from inattentive respondents without measuring individual-level attenti…
In an influential critique of empirical practice, Freedman (2008) showed that the linear regression estimator was biased for the analysis of randomized controlled trials under the randomization model. Under Freedman's assumptions, we derive exact closed-form bias corrections for the linear regression estimator with and without treatment-by-covariate interactions. We show that the limiting distribution of the bias corrected estimator is identical to the uncorrected estimator, implying that the asymptotic gains from adjustment can be attained without introducing any risk of bias. Taken together …
Standard Monte Carlo computation is widely known to exhibit a canonical square-root convergence speed in terms of sample size. Two recent techniques, one based on control variate and one on importance sampling, both derived from an integration of reproducing kernels and Stein's identity, have been proposed to reduce the error in Monte Carlo computation to supercanonical convergence. This paper presents a more general framework to encompass both techniques that is especially beneficial when the sample generator is biased and noise-corrupted. We show our general estimator, which we call the doub…
