During an epidemic outbreak of a new disease, the probability of dying once infected is considered an important though difficult task to be computed. Since it is very hard to know the true number of infected people, the focus is placed on estimating the case fatality rate, which is defined as the probability of dying once tested and confirmed as infected. The estimation of this rate at the beginning of an epidemic remains challenging for several reasons, including the time gap between diagnosis and death, and the rapid growth in the number of confirmed cases. In this work, an unbiased estimator of the case fatality rate of a virus is presented. The consistency of the estimator is demonstrated, and its asymptotic distribution is derived, enabling the corresponding confidence intervals (C.I.) to be established. The proposed method is based on the distribution F of the time between confirmation and death of individuals who die because of the virus. The estimator's performance is analyzed in both simulation scenarios and the real-world context of Argentina in 2020 for the COVID-19 pandemic, consistently achieving excellent results when compared to an existing proposal as well as to the conventional \naive" estimator that was employed to report the case fatality rates during the last COVID-19 pandemic. In the simulated scenarios, the empirical coverage of our C.I. is studied, both using the F employed to generate the data and an estimated F, and it is observed that the desired level of confidence is reached quickly when using real F and in a reasonable period of time when estimating F.
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
Based on observations of points uniformly distributed over a convex set in $\R^d$, a new estimator for the volume of the convex set is proposed. The estimator is minimax optimal and also efficient non-asymptotically: it is nearly unbiased with minimal variance among all unbiased oracle-type estimators. Our approach is based on a Poisson point process model and as an ingredient, we prove that the convex hull is a sufficient and complete statistic. No hypotheses on the boundary of the convex set are imposed. In a numerical study, we show that the estimator outperforms earlier estimators for the …
An important challenge in statistical analysis concerns the control of the finite sample bias of estimators. For example, the maximum likelihood estimator has a bias that can result in a significant inferential loss. This problem is typically magnified in high-dimensional settings where the number of variables $p$ is allowed to diverge with the sample size $n$. However, it is generally difficult to establish whether an estimator is unbiased and therefore its asymptotic order is a common approach used (in low-dimensional settings) to quantify the magnitude of the bias. As an alternative, we int…
Based on the methods provided in Caeiro and Gomes (2002) and Fraga Alves (2001), a new class of location invariant Hill-type estimators is derived in this paper. Its asymptotic distributional representation and asymptotic normality are presented, and the optimal choice of sample fraction by Mean Squared Error is also discussed for some special cases. Finally comparison studies are provided for some familiar models by Monte Carlo simulations.
We employ a parameter-free distribution estimation framework where estimators are random distributions and utilize the Kullback-Leibler (KL) divergence as a loss function. Wu and Vos [J. Statist. Plann. Inference 142 (2012) 1525-1536] show that when an estimator obtained from an i.i.d. sample is viewed as a random distribution, the KL risk of the estimator decomposes in a fashion parallel to the mean squared error decomposition when the estimator is a real-valued random variable. In this paper, we explore how conditional versions of distribution expectation ($E^{\dagger}$) can be defined so th…
For analyzing unit-level multivariate data in small area estimation, we consider the multivariate nested error regression model (MNER) and provide the empirical best linear unbiased predictor (EBLUP) of a small area characteristic based on second-order unbiased and consistent estimators of the `within' and `between' multivariate components of variance. The second-order approximation of the mean squared error (MSE) matrix of the EBLUP and its unbiased estimator are derived in closed forms. The confidence interval with second-order accuracy is also provided analytically.
The linear regression models are widely used statistical techniques in numerous practical applications. The standard regression model requires several assumptions about the regres- sors and the error term. The regression parameters are estimated using the least-squares method. In this paper, we consider the regression model with arbitrary regressors and with- out the error term. An explicit expression for the regression parameters vector is obtained. The unbiasedness approach is used to estimate the regression parameters and its various properties are investigated. It is shown that the resulti…
This manuscript develops a general purpose inner-product norm for the Kendall \(τ\) and Spearman's \(ρ\), which operates as an unbiased MLE even in the presence of ties. We derive and prove the strict sub-Gaussianity of the Kemeny norm-space, thereby disproving conclusions developed by both \textcite{kendall1948} and \textcite{diaconis1977} as to the nature of the appropriate, finite sample, probability distribution and test statistics. A non-parametric MLE framework for all bivariate pairs is developed, thereby resolving an hypothesis of \textcite{olkin1994} concerning an exponential multivar…
Many simulation problems require the estimation of a ratio of two expectations. In recent years Monte Carlo estimators have been proposed that can estimate such ratios without bias. We investigate the theoretical properties of such estimators for the estimation of $β= 1/\mathbb{E}\, Z$, where $Z \geq 0$. The estimator, $\widehat β(w)$, is of the form $w/f_w(N) \prod_{i=1}^N (1 - w\, Z_i)$, where $w < 2β$ and $N$ is any random variable with probability mass function $f_w$ on the positive integers. For a fixed $w$, the optimal choice for $f_w$ is well understood, but less so the choice of $w$. W…
This paper introduces a general framework for estimating variance components in the linear mixed models via general unbiased estimating equations, which include some well-used estimators such as the restricted maximum likelihood estimator. We derive the asymptotic covariance matrices and second-order biases under general estimating equations without assuming the normality of the underlying distributions and identify a class of second-order unbiased estimators of variance components. It is also shown that the asymptotic covariance matrices and second-order biases do not depend on whether the re…
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Thus, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yi…
The Garman-Klass unbiased estimator of the variance per unit time of a zero-drift Brownian Motion B, based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1=(CLOSE-OPEN, OPEN-MIN, MAX-OPEN). This estimator, with efficiency 7.4 with respect to the classical estimator (CLOSE-OPEN)^2, is widely believed to be of minimal variance. The current report disproves this belief by exhibiting an unbiased estimator with slightly but strictly higher efficiency 7.7322. The essence …
In this note, we revisit the $g$ and $h$ control charts that are commonly used for monitoring the number of conforming cases between the two consecutive appearances of nonconformities. It is known that the process parameter of these charts is usually unknown and estimated by using the maximum likelihood estimator and the minimum variance unbiased estimator. However, the minimum variance unbiased estimator in the control charts has been inappropriately used in the quality engineering literature. This observation motivates us to provide the correct minimum variance unbiased estimator and investi…
An empirical best linear unbiased prediction (EBLUP) estimator is utilized for efficient inference in small-area estimation. To measure its uncertainty, we need to estimate its mean squared error (MSE) since the true MSE cannot generally be derived in a closed form. The "naive MSE estimator", one of the estimators available for small-area inference, is unlikely to be chosen, since it does not achieve the desired asymptotic property, namely second-order unbiasedness, although it maintains strict positivity and tractability. Therefore, users tend to choose the second-order unbiased MSE estimator…
In this paper the problem of best linear unbiased estimation is investigated for continuous-time regression models. We prove several general statements concerning the explicit form of the best linear unbiased estimator (BLUE), in particular when the error process is a smooth process with one or several derivatives of the response process available for construction of the estimators. We derive the explicit form of the BLUE for many specific models including the cases of continuous autoregressive errors of order two and integrated error processes (such as integrated Brownian motion). The results…
We study the existence of algorithms generating almost surely nonnegative unbiased estimators. We show that given a nonconstant real-valued function $f$ and a sequence of unbiased estimators of $λ\in\mathbb{R}$, there is no algorithm yielding almost surely nonnegative unbiased estimators of $f(λ)\in\mathbb{R}^+$. The study is motivated by pseudo-marginal Monte Carlo algorithms that rely on such nonnegative unbiased estimators. These methods allow "exact inference" in intractable models, in the sense that integrals with respect to a target distribution can be estimated without any systematic er…
The objective of this paper is to propose an unbiased ratio-type estimator for finite population mean when the variables are negatively correlated. Hartley and Ross[2] and Singh and Singh [6] estimators are identified as particular cases of the proposed unbiased estimator. The variance expression of the proposed estimator to the first degree of approximation has been obtained. An empirical study is carried out to demonstrate the performance of the proposed estimator over, Robson [5] estimator and Singh and Singh [6] estimator.
Causal inference methods have been applied in various fields where researchers want to estimate treatment effects. In traditional causal inference settings, one assumes that the outcome of a unit does not depend on treatments of other units. However, as causal inference methods are extended to more applications, there is a greater need for estimators of general causal effects. We use an exposure mapping framework [Aronow and Samii, 2017] to map the relationship between the treatment allocation and the potential outcomes. Under the exposure model, we propose linear unbiased estimators (LUEs) fo…
When evaluating and comparing models using leave-one-out cross-validation (LOO-CV), the uncertainty of the estimate is typically assessed using the variance of the sampling distribution. Considering the uncertainty is important, as the variability of the estimate can be high in some cases. An important result by Bengio and Grandvalet (2004) states that no general unbiased variance estimator can be constructed, that would apply for any utility or loss measure and any model. We show that it is possible to construct an unbiased estimator considering a specific predictive performance measure and m…
Debiased estimation has long been an area of research in the group testing literature. This has led to the development of several estimators with the goal of bias minimization and, recently, an unbiased estimator based on sequential binomial sampling. Previous research, however, has focused heavily on the simple case where no misclassification is assumed and only one trait is to be tested. In this paper, we consider the problem of unbiased estimation in these broader areas, giving constructions of such estimators for several cases. We show that, outside of the standard case addressed previousl…
In this paper we have proposed an almost unbiased estimator using known value of some population parameter(s) with known population proportion of an auxiliary variable. A class of estimators is defined which includes [1], [2] and [3] estimators. Under simple random sampling without replacement (SRSWOR) scheme the expressions for bias and mean square error (MSE) are derived. Numerical illustrations are given in support of the present study. Key words: Auxiliary information, proportion, bias, mean square error, unbiased estimator.
Shrinkage estimation usually reduces variance at the cost of bias. But when we care only about some parameters of a model, I show that we can reduce variance without incurring bias if we have additional information about the distribution of covariates. In a linear regression model with homoscedastic Normal noise, I consider shrinkage estimation of the nuisance parameters associated with control variables. For at least three control variables and exogenous treatment, I establish that the standard least-squares estimator is dominated with respect to squared-error loss in the treatment effect eve…
Constructing unbiased estimators from Markov chain Monte Carlo (MCMC) outputs is a difficult problem that has recently received a lot of attention in the statistics and machine learning communities. However, the current unbiased MCMC framework only works when the quantity of interest is an expectation, which excludes many practical applications. In this paper, we propose a general method for constructing unbiased estimators for functions of expectations and extend it to construct unbiased estimators for nested expectations. Our approach combines and generalizes the unbiased MCMC and Multilevel…
When an unbiased estimator of the likelihood is used within a Metropolis--Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of averages computed under this chain. Many Monte Carlo samples will typically result in Metropolis--Hastings averages with lower asymptotic variances than the corresponding Metropolis--Hastings averages using fewer samples. However, the computing time required to construct the likelihood estimator increases with the number of Monte Carlo samples. Under the assumption that the d…
Posterior distributions often feature intractable normalizing constants, called marginal likelihoods or evidence, that are useful for model comparison via Bayes factors. This has motivated a number of methods for estimating ratios of normalizing constants in statistics. In computational physics the logarithm of these ratios correspond to free energy differences. Combining unbiased Markov chain Monte Carlo estimators with path sampling, also called thermodynamic integration, we propose new unbiased estimators of the logarithm of ratios of normalizing constants. As a by-product, we propose unbia…
The expected value of partial perfect information (EVPPI) denotes the value of eliminating uncertainty on a subset of unknown parameters involved in a decision model. The EVPPI can be regarded as a decision-theoretic sensitivity index, and has been widely used for identifying relatively important unknown parameters. It follows from Jensen's inequality, however, that the standard nested Monte Carlo computation of the EVPPI results in biased estimates. In this paper we introduce two unbiased Monte Carlo estimators for the EVPPI based on multilevel Monte Carlo method, introduced by Heinrich (1998…
In this article we consider the development of an unbiased estimator for the ensemble Kalman--Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology which can be viewed as a continuous-time analogue of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work [Rhee \& Glynn 2010, [31]] which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization, and through the number of samples at each level. Our estimator will be specific to …
We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. These methods provide consistent but biased estimators when run for a finite time. We propose a simple way of coupling two MCMC chains built using Particle Independent Metropolis-Hastings (PIMH) to produce unbiased smoothing estimators. Unbiased estimators are appealing in the context of parallel computing, and facilitate the construction of co…
Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；City University of Hong Kong（香港城市大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
In this article, we consider computing expectations w.r.t. probability measures which are subject to discretization error. Examples include partially observed diffusion processes or inverse problems, where one may have to discretize time and/or space, in order to practically work with the probability of interest. Given access only to these discretizations, we consider the construction of unbiased Monte Carlo estimators of expectations w.r.t. such target probability distributions. It is shown how to obtain such estimators using a novel adaptation of randomization schemes and Markov simulation m…
In this article we consider the development of unbiased estimators of the Hessian, of the log-likelihood function with respect to parameters, for partially observed diffusion processes. These processes arise in numerous applications, where such diffusions require derivative information, either through the Jacobian or Hessian matrix. As time-discretizations of diffusions induce a bias, we provide an unbiased estimator of the Hessian. This is based on using Girsanov's Theorem and randomization schemes developed through Mcleish [2011] and Rhee & Glynn [2015]. We demonstrate our developed estimato…
We consider the problem of calibration and the GREG method as suggested and studied in Deville and Sarndal (1992). We show that a GREG type estimator is typically not minimal variance unbiased estimator even asymptotically. We suggest a similar estimator which is unbiased but is asymptotically with a minimal variance.
