First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada     

Quantile regression for interval censored data

As direct generalization of the quantile regression for complete observed data, an estimation method for quantile regression models with interval censored data is proposed, and the property of consistency is obtained. The property of asymptotic normality is also established with a bias converging to zero, and to reduce the bias, two bias correction methods are proposed. Methods proposed in this paper do not require the censoring vectors to be identically distributed, and can be applied to models with various covariates. Simulation results show that the proposed methods work well.     

Comparison of bootstrap estimation intervals to forecast arithmetic mean and median air passenger demand

The aim of this paper is to compare passenger (pax) demand between airports based on the arithmetic mean (MPD) and the median pax demand (MePD). A three phases approach is applied. First phase, we use bootstrap procedures to estimate the distribution of the arithmetic MPD and the MePD for each block of routes distance; second phase, we use percentile, standard, bias corrected, and bias corrected accelerated methods to calculate bootstrap confidence bands for the MPD and the MePD; and third phase, we implement Monte Carlo (MC) experiments to analyse the finite sample performance of the applied bootstrap. Our results conclude that it is more meaningful to use the estimation of MePD rather than the estimation of MPD in the air transport industry. By carrying out MC experiments, we demonstrate that the bootstrap methods produce coverages close to the nominal for the MPD and the MePD.     

Evaluating bias correction in weighted proportional hazards regression

Often in observational studies of time to an event, the study population is a biased (i.e., unrepresentative) sample of the target population. In the presence of biased samples, it is common to weight subjects by the inverse of their respective selection probabilities. Pan and Schaubel (Can J Stat 36:111–127, 2008) recently proposed inference procedures for an inverse selection probability weighted (ISPW) Cox model, applicable when selection probabilities are not treated as fixed but estimated empirically. The proposed weighting procedure requires auxiliary data to estimate the weights and is computationally more intense than unweighted estimation. The ignorability of sample selection process in terms of parameter estimators and predictions is often of interest, from several perspectives: e.g., to determine if weighting makes a significant difference to the analysis at hand, which would in turn address whether the collection of auxiliary data is required in future studies; to evaluate previous studies which did not correct for selection bias. In this article, we propose methods to quantify the degree of bias corrected by the weighting procedure in the partial likelihood and Breslow-Aalen estimators. Asymptotic properties of the proposed test statistics are derived. The finite-sample significance level and power are evaluated through simulation. The proposed methods are then applied to data from a national organ failure registry to evaluate the bias in a post-kidney transplant survival model.     

Coherent mortality forecasting by the weighted multilevel functional principal component approach

In human mortality modelling, if a population consists of several subpopulations it can be desirable to model their mortality rates simultaneously while taking into account the heterogeneity among them. The mortality forecasting methods tend to result in divergent forecasts for subpopulations when independence is assumed. However, under closely related social, economic and biological backgrounds, mortality patterns of these subpopulations are expected to be non-divergent in the future. In this article, we propose a new method for coherent modelling and forecasting of mortality rates for multiple subpopulations, in the sense of nondivergent life expectancy among subpopulations. The mortality rates of subpopulations are treated as multilevel functional data and a weighted multilevel functional principal component (wMFPCA) approach is proposed to model and forecast them. The proposed model is applied to sex-specific data for nine developed countries, and the results show that, in terms of overall forecasting accuracy, the model outperforms the independent model and the Product-Ratio model as well as the unweighted multilevel functional principal component approach.     

Controlling the false-discovery rate by procedures adapted to the length bias of RNA-Seq

In RNA-Seq experiments, the number of mapped reads for a given gene is proportional to its expression level and length. Because longer genes contribute more sequencible fragments than do shorter ones, it is expected that even if two genes have the same expression level, the longer gene will have a greater number of total reads. This characteristic creates a length bias such that the proportion of significant genes increases with the gene length. However, genes with a long length are not more biologically meaningful than genes with a short length. Therefore, the length bias should be properly corrected to determine the accurate list of significant genes in RNA-Seq. For this purpose, we proposed two multiple-testing procedures based on a weighted-FDR and a separate-FDR approach. These two methods use prior information on differential gene length while keeping the false-discovery rate (FDR) controlled at $\alpha$. In the weighted-FDR controlling procedure, we incorporated prior weights for the length of each gene. These weights increase the power when the gene’s length is short and decrease the power when its length is long. In the separate-FDR controlling procedure, we sequentially ordered all genes according to their lengths and then split these genes into two subgroups of short and long genes. The adaptive Benjamini–Hochberg procedure was then performed separately for each subgroup. The proposed procedures were compared with existing methods and evaluated in two numerical examples and one simulation study. We concluded that the weighted $p$-value procedure properly reduced the length bias of RNA-Seq.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.     

Biases in bias elicitation

Abstract

We consider the biases that can arise in bias elicitation when expert assessors make random errors. After presenting a general framework of the phenomenon, we illustrate it for two examples: the case of omitting variables bias and that of the bias arising in adjusting relative risks. Results show that, even when assessors’ elicitations of bias have desirable properties, the nonlinear nature of biases can lead to elicitations of bias that are, themselves, biased. We show the corrections which can be made to remove this bias and discuss the implications for the applied literature which employs these methods.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.     

Variable selection for functional density trees

In this paper, the exhaustive search principle used in functional trees for classifying densities is shown to select variables with more split points. A new variable selection scheme is proposed to correct this bias. The Pearson chi-squared tests for associated two-way contingency tables are used to select the variables. Through simulation, we show that the new method can control bias and is more powerful in selecting split variable.     

Influence analysis in multivariate linear general models

In this paper we obtain several influence measures for the multivariate linear general model through the approach proposed by Muñoz-Pichardo et al. (1995), which is based on the concept of conditional bias. An interesting charasteristic of this approach is that it does not require any distributional hypothesis. Appling the obtained results to the multivariate regression model, we obtain some measures proposed by other authors. Nevertheless, on the results obtained in this paper, we emphasize two aspects. First, they provide a theoretical foundation for measures proposed by other authors for the mul¬tivariate regression model. Second, they can be applied to any linear model that can be formulated as a particular case of the multivariate linear general model. In particular, we carry out an application to the multivariate analysis of covariance.     

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.     

Price bias and common practice in option pricing

Generally, the semiclosed-form option pricing formula for complex financial models depends on unobservable factors such as stochastic volatility and jump intensity. A popular practice is to use an estimate of these latent factors to compute the option price. However, in many situations this plug-and-play approximation does not yield the appropriate price. This article examines this bias and quantifies its impacts. We decompose the bias into terms that are related to the bias on the unobservable factors and to the precision of their point estimators. The approximated price is found to be highly biased when only the history of the stock price is used to recover the latent states. This bias is corrected when option prices are added to the sample used to recover the states' best estimate. We also show numerically that such a bias is propagated on calibrated parameters, leading to erroneous values. The Canadian Journal of Statistics 48: 8–35; 2020 © 2019 Statistical Society of Canada     

Kernel density classification and boosting: an L2 analysis

Kernel density estimation is a commonly used approach to classification. However, most of the theoretical results for kernel methods apply to estimation per se and not necessarily to classification. In this paper we show that when estimating the difference between two densities, the optimal smoothing parameters are increasing functions of the sample size of the complementary group, and we provide a small simluation study which examines the relative performance of kernel density methods when the final goal is classification.A relative newcomer to the classification portfolio is boosting, and this paper proposes an algorithm for boosting kernel density classifiers. We note that boosting is closely linked to a previously proposed method of bias reduction in kernel density estimation and indicate how it will enjoy similar properties for classification. We show that boosting kernel classifiers reduces the bias whilst only slightly increasing the variance, with an overall reduction in error. Numerical examples and simulations are used to illustrate the findings, and we also suggest further areas of research.     

Limiting bias-reduced Amoroso kernel density estimators for non-negative data

The Amoroso kernel density estimator (Igarashi and Kakizawa 2017 Igarashi, G., and Y. Kakizawa. 2017. Amoroso kernel density estimation for nonnegative data and its bias reduction. Department of Policy and Planning Sciences Discussion Paper Series No. 1345, University of Tsukuba. [Google Scholar]) for non-negative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n^{? 4/5}), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n^{? 8/9}, if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.     

Small Sample Bias in the Gamma Frailty Model for Univariate Survival

The gamma frailty model is a natural extension of the Cox proportional hazards model in survival analysis. Because the frailties are unobserved, an E-M approach is often used for estimation. Such an approach is shown to lead to finite sample underestimation of the frailty variance, with the corresponding regression parameters also being underestimated as a result. For the univariate case, we investigate the source of the bias with simulation studies and a complete enumeration. The rank-based E-M approach, we note, only identifies frailty through the order in which failures occur; additional frailty which is evident in the survival times is ignored, and as a result the frailty variance is underestimated. An adaption of the standard E-M approach is suggested, whereby the non-parametric Breslow estimate is replaced by a local likelihood formulation for the baseline hazard which allows the survival times themselves to enter the model. Simulations demonstrate that this approach substantially reduces the bias, even at small sample sizes. The method developed is applied to survival data from the North West Regional Leukaemia Register.     

High-order data sharpening with dependent errors for regression bias reduction

Abstract

In this paper, we show that Y can be introduced into data sharpening to produce non-parametric regression estimators that enjoy high orders of bias reduction. Compared with those in existing literature, the proposed data-sharpening estimator has advantages including simplicity of the estimators, good performance of expectation and variance, and mild assumptions. We generalize this estimator to dependent errors. Finally, we conduct a limited simulation to illustrate that the proposed estimator performs better than existing ones.     

A lack-of-fit test for parametric zero-inflated Poisson models

Count data often contain many zeros. In parametric regression analysis of zero-inflated count data, the effect of a covariate of interest is typically modelled via a linear predictor. This approach imposes a restrictive, and potentially questionable, functional form on the relation between the independent and dependent variables. To address the noted restrictions, a flexible parametric procedure is employed to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. The semiparametric zero-inflated Poisson regression model is fitted by maximizing the likelihood function through an expectation–maximization algorithm. The smooth estimate of the functional form of the covariate effect can enhance modelling flexibility. Within this modelling framework, a log-likelihood ratio test is used to assess the adequacy of the covariate function. Simulation results show that the proposed test has excellent power in detecting the lack of fit of a linear predictor. A real-life data set is used to illustrate the practicality of the methodology.     

Control charts based on quasi-likelihood estimation for monitoring profiles

In some applications, the quality of the process or product is characterized and summarized by a functional relationship between a response variable and one or more explanatory variables. Profile monitoring is a technique for checking the stability of the relationship over time. Existing linear profile monitoring methods usually assumed the error distribution to be normal. However, this assumption may not always be true in practice. To address this situation, we propose a method for profile monitoring under the framework of generalized linear models when the relationship between the mean and variance of the response variable is known. Two multivariate exponentially weighted moving average control schemes are proposed based on the estimated profile parameters obtained using a quasi-likelihood approach. The performance of the proposed methods is evaluated by simulation studies. Furthermore, the proposed method is applied to a real data set, and the R code for profile monitoring is made available to users.     

Bias-correction for Weibull common shape estimation

A general method for correcting the bias of the maximum likelihood estimator (MLE) of the common shape parameter of Weibull populations, allowing a general right censorship, is proposed in this paper. Extensive simulation results show that the new method is very effective in correcting the bias of the MLE, regardless of censoring mechanism, sample size, censoring proportion and number of populations involved. The method can be extended to more complicated Weibull models.