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We study estimation of the correlation ρ in the multivariate linear model with uniform correlation structure. Unbiased estimator of ρ based on moment estimator proposed by Žežula (2006) is derived. Biased and unbiased estimators are compared by bias and mean square error under normal and t-distribution.

Background In infectious disease transmission dynamics, the high heterogeneity in individual infectiousness indicates that few index cases generate large numbers of secondary cases, which is commonly known as superspreading events. The heterogeneity in transmission can be measured by describing the distribution of the number of secondary cases as a negative binomial (NB) distribution with dispersion parameter, k. However, such inference framework usually neglects the under-ascertainment of sporadic cases, which are those without known epidemiological link and considered as independent clusters of size one, and this may potentially bias the estimates. Methods In this study, we adopt a zero-truncated likelihood-based framework to estimate k. We evaluate the estimation performance by using stochastic simulations, and compare it with the baseline non-truncated version. We exemplify the analytical framework with three contact tracing datasets of COVID-19. Results We demonstrate that the estimation bias exists when the under-ascertainment of index cases with 0 secondary case occurs, and the zero-truncated inference overcomes this problem and yields a less biased estimator of k. We find that the k of COVID-19 is inferred at 0.32 (95%CI: 0.15, 0.64), which appears slightly smaller than many previous estimates. We provide the simulation codes applying the inference framework in this study. Conclusions The zero-truncated framework is recommended for less biased transmission heterogeneity estimates. These findings highlight the importance of individual-specific case management strategies to mitigate COVID-19 pandemic by lowering the transmission risks of potential super-spreaders with priority.

This study develops a nearly unbiased estimator of the ratio of the contemporary effective mother size to the census size in a population, as a proxy of the ratio of contemporary effective size (or effective breeding size) to census size (Ne/N or Nb/N). The proposed estimator is based on both known mother–offspring (MO) and maternal‐sibling (MS) relationships observed within the same cohort, in which sampled individuals in the cohort probably share MO relationships with sampled mothers. The rationale is that the frequency of MO and MS pairs contains information regarding the contemporary effective mother size and the (mature) census size, respectively. Therefore, the estimator can be obtained only from genetic data. We also evaluate the performance of the estimator by running an individual‐based model. The results of this study provide the following: (a) parameter range for satisfying the unbiasedness, and (b) guidance for sample sizes to ensure the required accuracy and precision, especially when the order of the ratio is available. Furthermore, the results demonstrate the usefulness of a sibship assignment method for genetic monitoring, providing insights for interpreting environmental and/or anthropological factors fluctuating Ne/N (or Nb/N), especially in the context of conservation biology and wildlife management., We developed a nearly unbiased estimator of the ratio of the contemporary effective mother size to the census size in a population, as a proxy of Ne/N. The proposed estimator is based on a known mother–offspring (MO) relationship and on a known maternal‐sibling (MS) relationship found within the same cohort, in which sampled individuals in the cohort potentially share MO relationships with sampled mothers.

In personalized medicine, it is often desired to determine if all patients or only a subset of them benefit from a treatment. We consider estimation in two-stage adaptive designs that in stage 1 recruit patients from the full population. In stage 2, patient recruitment is restricted to the part of the population, which, based on stage 1 data, benefits from the experimental treatment. Existing estimators, which adjust for using stage 1 data for selecting the part of the population from which stage 2 patients are recruited, as well as for the confirmatory analysis after stage 2, do not consider time to event patient outcomes. In this work, for time to event data, we have derived a new asymptotically unbiased estimator for the log hazard ratio and a new interval estimator with good coverage probabilities and probabilities that the upper bounds are below the true values. The estimators are appropriate for several selection rules that are based on a single or multiple biomarkers, which can be categorical or continuous.

In ecological and environmental sampling the quantification of units is either difficult or overly demanding in terms of the time, money, workload, it requires. For this reason efficient and cost-effective sampling methods need to be devised for data collecting. The most commonly used method for this purpose is the Ranked Set Sampling (RSS). In this paper, a sampling scheme called Improved Paired Ranked Set Sampling (IPRSS) is proposed to estimate the population mean. The performance of the proposed IPRSS is evaluated under perfect and imperfect rankings. A simulation study based on selected hypothetical distributions and a real-life data set showed that IPRSS is more precise than RSS, Paired RSS (PRSS) or Extreme RSS (ERSS).

In this paper, we use the combined-unified hybrid censoring samples to obtain the maximum likelihood estimates of the unknown parameters, survival, and hazard functions of Pareto distribution. Next, we discuss some efficiency criteria of the maximum likelihood estimators, including, the unbiasedness, consistency, and sufficiency. Additionally, we use MCMC to obtain the Bayesian estimates of the unknown parameters. In addition, we calculate the intervals estimation of the unknown parameters. Finally, we analyze a set of real data in view of the theoretical findings of the paper.

The assessment of the probability of a rare event with a naive Monte Carlo method is computationally intensive, so faster estimation or variance reduction methods are needed. We focus on one of these methods which is the interacting particle system (IPS) method. The method is not intrusive in the sense that the random Markov system under consideration is simulated with its original distribution, but selection steps are introduced that favor trajectories (particles) with high potential values. An unbiased estimator with reduced variance can then be proposed. The method requires to specify a set of potential functions. The choice of these functions is crucial because it determines the magnitude of the variance reduction. So far, little information was available on how to choose the potential functions. This paper provides the expressions of the optimal potential functions minimizing the asymptotic variance of the estimator of the IPS method and it proposes recommendations for the practical design of the potential functions.

This paper aims at semi-parametrically estimating the input process to a Lévy-driven queue by sampling the workload process at Poisson times. We construct a method-of-moments based estimator for the Lévy process' characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimating a constant (related to both the input distribution and the sampling rate); this constant also features in the asymptotic analysis. For subordinator Lévy input, a partial MLE is constructed for the intermediate step and we show that it satisfies the consistency and asymptotic normality conditions. For general spectrally-positive Lévy input a biased estimator is proposed that only uses workload observations above some threshold; the bias can be made arbitrarily small by appropriately choosing the threshold.

This review investigates the market performance of salmon forward contracts. It studies whether the forward price is an unbiased estimator of the spot price and whether the forward market generates price discovery information. The focus is on the Fish Pool market for the period 2006-2017 and relates to forward contracts with maturities up to 60 months. The main finding is that there is strong cointegration up to a period of seven months. After this window, there is marginally significant cointegration up to a period of twelve months and the cointegration relationship disappears for contracts with maturities longer than twelve months, pointing to the inefficiency of these forward markets. The results from error-correction models and Granger causality tests suggest that the salmon forward market does not fulfil the expected price discovery role and that the spot market drives the forward market. These findings suggest the salmon forward market is still immature and cast doubt on the viability of longer-term salmon forward contracts. Publisher PDF

The fixed-effects estimator is biased in the presence of dynamic misspecification and omitted within variation correlated with one of the regressors. We argue and demonstrate that fixed-effects estimates can amplify the bias from dynamic misspecification and that with omitted time-invariant variables and dynamic misspecifications, the fixed-effects estimator can be more biased than the ‘naïve’ OLS model. We also demonstrate that the Hausman test does not reliably identify the least biased estimator when time-invariant and time-varying omitted variables or dynamic misspecifications exist. Accordingly, empirical researchers are ill-advised to rely on the Hausman test for model selection or use the fixed-effects model as default unless they can convincingly justify the assumption of correctly specified dynamics. Our findings caution applied researchers to not overlook the potential drawbacks of relying on the fixed-effects estimator as a default. The results presented here also call upon methodologists to study the properties of estimators in the presence of multiple model misspecifications. Our results suggest that scholars ought to devote much more attention to modeling dynamics appropriately instead of relying on a default solution before they control for potentially omitted variables with constant effects using a fixed-effects specification.

Penalized Least Squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise without requiring to invert the covariance matrix. Then, in order to avoid expansive grid search, it is necessary to design algorithmic scheme minimizing the quadratic risk with respect to regularization parameters. This work extends the Stein's Unbiased GrAdient estimator of the Risk of Deledalle et al. to the case of correlated Gaussian noise, deriving a general automatic tuning of regularization parameters. First, the theoretical asymptotic unbiasedness of the gradient estimator is demonstrated in the case of general correlated Gaussian noise. Then, the proposed parameter selection strategy is particularized to fractal texture segmentation, where problem formulation naturally entails inter-scale and spatially correlated noise. Numerical assessment is provided, as well as discussion of the practical issues., Submitted to SIAM Imaging Science

A new method to estimate longevity risk based on the kernel estimation of the extreme quantiles of truncated age-at-death distributions is proposed. Its theoretical properties are presented and a simulation study is reported. The flexible yet accurate estimation of extreme quantiles of age-at-death conditional on having survived a certain age is fundamental for evaluating the risk of lifetime insurance. Our proposal combines a parametric distributions with nonparametric sample information, leading to obtain an asymptotic unbiased estimator of extreme quantiles for alternative distributions with different right tail shape, i.e., heavy tail or exponential tail. A method for estimating the longevity risk of a continuous temporary annuity is also shown. We illustrate our proposal with an application to the official age-at-death statistics of the population in Spain.

The choice of negative examples is important in noise contrastive estimation. Recent works find that hard negatives -- highest-scoring incorrect examples under the model -- are effective in practice, but they are used without a formal justification. We develop analytical tools to understand the role of hard negatives. Specifically, we view the contrastive loss as a biased estimator of the gradient of the cross-entropy loss, and show both theoretically and empirically that setting the negative distribution to be the model distribution results in bias reduction. We also derive a general form of the score function that unifies various architectures used in text retrieval. By combining hard negatives with appropriate score functions, we obtain strong results on the challenging task of zero-shot entity linking., NAACL 2021

International audience; This article tackles the topic of performance analysis for Spectrum Sensing based on Compressive Sampling (CS). More precisely, the lower bound on the variance of any unbiased estimator, the Cramér-Rao Bound (CRB), is investigated in the context of spectral parametric estimation. Compressed samples are obtained from a multiband architecture like the Modulated Wideband Converter, the Quadrature-Analog-to-Information Converter or the Periodic Non Uniform Sampler. An expression of the Fisher information matrix, which allows to compute the CRB, is established for a compressive multiband architecture assuming a disjoint spectral subband model. The relationships between Fisher matrices in a generic framework are exposed: first between compressive multiband and subsampling architectures, then between subsampling and Nyquist sampling architectures. Based on these new considerations, the issue of interferer detection, a canonical case and also a huge thorn in the side of wideband radiofrequency receivers is tackled. A benchmark between Nyquist, Subsampling and Compressive Multiband Sampling approaches is provided for frequency and amplitude estimation of dual-tone signals. This analysis illustrates the way in which interferences between parameters occur in estimation with Compressive Sampling. It is then shown how properties of the sensing matrix for popular compressive architectures enable to control the precision of spectral parametric estimation in specific subbands. This control opportunity opens the door to adaptive methods.

Biodistance analysis can elucidate various aspects of past population structure. The most commonly adopted measure of divergence when estimating biodistances is the mean measure of divergence (MMD). The MMD is an unbiased estimator of population divergence but this property is lost when the dataset includes variables with very high or low frequency. In the present paper, we examine new measures of divergence based on untransformed binary data and the logit and probit transformations. It is shown that a measure of divergence based on untransformed data is a better unbiased estimator of population divergence. The conventional MMD is a satisfactory distance measure for binary data; however, it may produce biased estimations of population divergence when there are many traits with frequencies lower than 0.1 or/and greater than 0.9. Finally, the measures of divergence based on the probit and logit transformations are usually biased estimators.

One of the main focus in federated learning (FL) is the communication efficiency since a large number of participating edge devices send their updates to the edge server at each round of the model training. Existing works reconstruct each model update from edge devices and implicitly assume that the local model updates are independent over edge devices. In FL, however, the model update is an indirect multi-terminal source coding problem, also called as the CEO problem where each edge device cannot observe directly the gradient that is to be reconstructed at the decoder, but is rather provided only with a noisy version. The existing works do not leverage the redundancy in the information transmitted by different edges. This paper studies the rate region for the indirect multiterminal source coding problem in FL. The goal is to obtain the minimum achievable rate at a particular upper bound of gradient variance. We obtain the rate region for the quadratic vector Gaussian CEO problem under unbiased estimator and derive an explicit formula of the sum-rate-distortion function in the special case where gradient are identical over edge device and dimension. Finally, we analyse communication efficiency of convex Minibatched SGD and non-convex Minibatched SGD based on the sum-rate-distortion function, respectively., 6 pages, 2 figures

This paper develops the analytical form of the degrees of freedom in functional principal components analysis. Under the framework of unbiased risk estimation, we derive an unbiased estimator with a clear analytical formula for the degrees of freedom in the one-way penalized functional principal components analysis paradigm. Specifically, a new analytical formula incorporating binary smoothing parameters is also derived based on the singular value decomposition and half-smoothed method regarding the two-way penalized functional principal components analysis framework. The performance of our procedures is demonstrated by simulation studies.

New ways to combine data from multiple environmental area frame surveys of a finite population are being introduced. Environmental surveys often sample finite populations through area frames. However, to combine multiple surveys without risking bias, design components (inclusion probabilities, etc.) are needed at unit level of the finite population. We show how to derive the design components and exemplify this for three commonly used area frame sampling designs. We show how to produce an unbiased estimator using data from multiple surveys, and how to reduce the risk of introducing significant bias in linear combinations of estimators from multiple surveys. If separate estimators and variance estimators are used in linear combinations, there’s a risk of introducing negative bias. By using pooled variance estimators, the bias of a linear combination estimator can be reduced. National environmental surveys often provide good estimators at national level, while being too sparse to provide sufficiently good estimators for some domains. With the proposed methods, one can plan extra sampling efforts for such domains, without discarding readily available information from the aggregate/national survey. Through simulation, we show that the proposed methods are either unbiased, or yield low variance with small bias, compared to traditionally used methods.

Large-scale serological testing in the population is essential to determine the true extent of the current SARS-CoV-2 pandemic. Serological tests measure antibody responses against pathogens and use predefined cutoff levels that dichotomize the quantitative test measures into sero-positives and negatives and use this as a proxy for past infection. With the imperfect assays that are currently available to test for past SARS-CoV-2 infection, the fraction of seropositive individuals in serosurveys is a biased estimator of the cumulative incidence and is usually corrected to account for the sensitivity and specificity. Here we use an inference method—referred to as mixture-model approach—for the estimation of the cumulative incidence that does not require to define cutoffs by integrating the quantitative test measures directly into the statistical inference procedure. We confirm that the mixture model outperforms the methods based on cutoffs, leading to less bias and error in estimates of the cumulative incidence. We illustrate how the mixture model can be used to optimize the design of serosurveys with imperfect serological tests. We also provide guidance on the number of control and case sera that are required to quantify the test’s ambiguity sufficiently to enable the reliable estimation of the cumulative incidence. Lastly, we show how this approach can be used to estimate the cumulative incidence of classes of infections with an unknown distribution of quantitative test measures. This is a very promising application of the mixture-model approach that could identify the elusive fraction of asymptomatic SARS-CoV-2 infections. An R-package implementing the inference methods used in this paper is provided. Our study advocates using serological tests without cutoffs, especially if they are used to determine parameters characterizing populations rather than individuals. This approach circumvents some of the shortcomings of cutoff-based methods at exactly the low cumulative incidence levels and test accuracies that we are currently facing in SARS-CoV-2 serosurveys., PLoS Computational Biology, 17 (2), ISSN:1553-734X, ISSN:1553-7358

Learning to quantify (a.k.a.\ quantification) is a task concerned with training unbiased estimators of class prevalence via supervised learning. This task originated with the observation that "Classify and Count" (CC), the trivial method of obtaining class prevalence estimates, is often a biased estimator, and thus delivers suboptimal quantification accuracy; following this observation, several methods for learning to quantify have been proposed that have been shown to outperform CC. In this work we contend that previous works have failed to use properly optimised versions of CC. We thus reassess the real merits of CC (and its variants), and argue that, while still inferior to some cutting-edge methods, they deliver near-state-of-the-art accuracy once (a) hyperparameter optimisation is performed, and (b) this optimisation is performed by using a true quantification loss instead of a standard classification-based loss. Experiments on three publicly available binary sentiment classification datasets support these conclusions., This is the final version of the paper, identical to the one that is going to appear on the Proceedings of the 43rd European Conference on Information Retrieval (ECIR 2021)

In this chapter, the principles of statistical parameter estimation theory for a quantitative analysis of atomic-resolution electron microscopy images are introduced. Within this framework, electron microscopy images are described by a parametric statistical model. Here, parametric models are introduced for different types of electron microscopy images: reconstructed exit waves, annular dark-field (ADF) scanning transmission electron microscopy (STEM) images, and simultaneously acquired ADF and annular bright-field (ABF) STEM images. Furthermore, the Cramer-Rao lower bound (CRLB) is introduced, i.e. a theoretical lower bound on the variance of any unbiased estimator. This CRLB is used to quantify the precision of the structure parameters of interest, such as the atomic column positions and the integrated atomic column intensities.

Multilevel Models are widely used in organizational research, educational research, epidemiology, psychology, biology and medical fields. In this paper, we recommend the situations where Bootstrap procedures through Minimum Norm Quadratic Unbiased Estimator (MINQUE) can be extremely handy than that of Restricted Maximum Likelihood (REML) in multilevel level linear regression models. In our simulation study the bootstrap by means of MINQUE is superior to REML in conditions where normality does not hold. Moreover, the real data application also supports our findings in terms of accuracy of estimates and their standard errors.

The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter estimation and model evaluation. However, the log-likelihood of complex models in fields such as computational biology and neuroscience is often intractable to compute analytically or numerically. In those cases, researchers can often only estimate the log-likelihood by comparing observed data with synthetic observations generated by model simulations. Standard techniques to approximate the likelihood via simulation either use summary statistics of the data or are at risk of producing substantial biases in the estimate. Here, we explore another method, inverse binomial sampling (IBS), which can estimate the log-likelihood of an entire data set efficiently and without bias. For each observation, IBS draws samples from the simulator model until one matches the observation. The log-likelihood estimate is then a function of the number of samples drawn. The variance of this estimator is uniformly bounded, achieves the minimum variance for an unbiased estimator, and we can compute calibrated estimates of the variance. We provide theoretical arguments in favor of IBS and an empirical assessment of the method for maximum-likelihood estimation with simulation-based models. As case studies, we take three model-fitting problems of increasing complexity from computational and cognitive neuroscience. In all problems, IBS generally produces lower error in the estimated parameters and maximum log-likelihood values than alternative sampling methods with the same average number of samples. Our results demonstrate the potential of IBS as a practical, robust, and easy to implement method for log-likelihood evaluation when exact techniques are not available., Author summary Researchers often validate scientific hypotheses by comparing data with the predictions of a mathematical or computational model. This comparison can be quantified by the ‘log-likelihood’, a number that captures how well the model explains the data. However, for complex models common in neuroscience and computational biology, obtaining exact formulas for the log-likelihood can be difficult. Instead, the log-likelihood is usually approximated by simulating synthetic observations from the model (‘sampling’), and seeing how often the simulated data match the actual observations. To reach correct scientific conclusions, it is crucial that the log-likelihood estimates produced by such sampling methods are accurate (unbiased). Here, we introduce inverse binomial sampling (IBS), a method which differs from traditional approaches in that the number of samples drawn from the model is not fixed, but adaptively adjusted in a simple way. For each data point, IBS samples from the model until it matches the observation. We show that IBS is unbiased and has other desirable statistical properties, both theoretically and via empirical validation on three case studies from computational and cognitive neuroscience. Across all examples, IBS outperforms fixed sampling methods, demonstrating the utility of IBS as a practical, robust, and easy to implement method for log-likelihood evaluation.

The unrelated design has been shown to improve the efficiency of a randomized response method and reduces respondents’ suspicion. In the light of this, the paper proposes a new Unrelated Randomized Response Model constructed by incorporating an unrelated question into the alternative unbiased estimator in the dichotomous randomized response model proposed by Ewemooje in 2019. An unbiased estimate and variance of the model are thus obtained. The variance of the proposed model decreases as the proportion of the sensitive attribute π_A and the unrelated attribute π_U increases, in contrast to the earlier Ewemooje model, whose variance increases as the proportion of the sensitive attribute increases. The relative efficiency of the proposed model over the earlier Ewemooje model decreases as π_U increases when 0.1≤π_A≤ 0.3 and increases as π_U increases when 0.35≤π_A≤ 0.45. Application of the proposed model also revealed its efficiency over the direct method in estimating the prevalence of examination malpractices among university students;the direct method gave an estimate of 19.0%, compared to the proposed method’s estimate of 23.0%. Hence, the proposed model is more efficient than the direct method and the earlier Ewemooje model as the proportion of people belonging to the sensitive attribute increases.

We abstract the concept of a randomized controlled trial (RCT) as a triple (beta,b,s), where beta is the primary efficacy parameter, b the estimate and s the standard error (s>0). The parameter beta is either a difference of means, a log odds ratio or a log hazard ratio. If we assume that b is unbiased and normally distributed, then we can estimate the full joint distribution of (beta,b,s) from a sample of pairs (b_i,s_i). We have collected 23,747 such pairs from the Cochrane database to do so. Here, we report the estimated distribution of the signal-to-noise ratio beta/s and the achieved power. We estimate the median achieved power to be 0.13. We also consider the exaggeration ratio which is the factor by which the magnitude of beta is overestimated. We find that if the estimate is just significant at the 5% level, we would expect it to overestimate the true effect by a factor of 1.7. This exaggeration is sometimes referred to as the winner's curse and it is undoubtedly to a considerable extent responsible for disappointing replication results. For this reason, we believe it is important to shrink the unbiased estimator, and we propose a method for doing so., 13 pages, 5 figures

A fuzzy least squares estimator in the multiple with fuzzy-input&ndash, fuzzy-output linear regression model is considered. The paper provides a formula for the L2 estimator of the fuzzy regression model. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. Using the proposed operations, the formula for the variance, provided and this paper, proves that the estimators have several important optimal properties and asymptotic properties: they are Best Linear Unbiased Estimator (BLUE), asymptotic normality and strong consistency. The confidence regions of the coefficient parameters and the asymptotic relative efficiency (ARE) are also discussed. In addition, several examples are provided including a Monte Carlo simulation study showing the validity of the proposed theorems.

It is well known that multicollinearity, which occurs among the explanatory variables, has adverse effects on the maximum likelihood estimator in the inverse Gaussian regression model. Biased estimators are proposed to cope with the multicollinearity problem in the inverse Gaussian regression model. The main interest of this article is to introduce a new biased estimator. Also, we compare newly proposed estimator with the other estimators given in the literature. We conduct a Monte Carlo simulation and provide a real data example to illustrate the performance of the proposed estimator over the maximum likelihood and Ridge estimators. As a result of the simulation study and real data example, the newly proposed estimator is superior to the other estimators used in this study.

I formulate three basic biomedical/statistical assumptions that should ideally guide well-designed population prevalence studies of the present or past disease including COVID-19. On the basis of these assumptions alone, I compute several probability distributions required for statistical analysis of testing data collected from a sample of individuals drawn from a heterogeneous population. I also construct a consistent asymptotically unbiased estimator of the population prevalence of the disease or infection from the collected data and derive a simple upper bound for its variance. All the results are rigorously proved and valid for any test for COVID-19 or other disease provided that the sum of the test&rsquo, s sensitivity and specificity is larger than 1. A few recommendations for the design of COVID-19 prevalence studies informed by the results of this work are formulated. The methodology developed in this article may prove applicable to diseases and conditions other than COVID-19 as well as in some non-epidemiological settings.

Background Doubly robust estimation produces an unbiased estimator for the average treatment effect unless both propensity score (PS) and outcome models are incorrectly specified. Studies have shown that the doubly robust estimator is subject to more bias than the standard weighting estimator when both PS and outcome models are incorrectly specified. Method We evaluated whether various machine learning methods can be used for estimating conditional means of the potential outcomes to enhance the robustness of the doubly robust estimator to various degrees of model misspecification in terms of reducing bias and standard error. We considered four types of methods to predict the outcomes: least squares, tree-based methods, generalized additive models and shrinkage methods. We also considered an ensemble method called the Super Learner (SL), which is a linear combination of multiple learners. We conducted simulations considering different scenarios by the complexity of PS and outcome-generating models and some ranges of treatment prevalence. Results The shrinkage methods performed well with robust doubly robust estimates in term of bias and mean squared error across the scenarios when the models became rich by including all 2-way interactions of the covariates. The SL performed similarly to the best method in each scenario. Conclusions Our findings indicate that machine learning methods such as the SL or the shrinkage methods using interaction models should be used for more accurate doubly robust estimators.

Estimating the cardinality (number of distinct elements) of a large multiset is a classic problem in streaming and sketching, dating back to Flajolet and Martin's classic Probabilistic Counting (PCSA) algorithm from 1983. In this paper we study the intrinsic tradeoff between the space complexity of the sketch and its estimation error in the random oracle model. We define a new measure of efficiency for cardinality estimators called the Fisher-Shannon (Fish) number $\mathcal{H}/\mathcal{I}$. It captures the tension between the limiting Shannon entropy ($\mathcal{H}$) of the sketch and its normalized Fisher information ($\mathcal{I}$), which characterizes the variance of a statistically efficient, asymptotically unbiased estimator. Our results are as follows. We prove that all base-$q$ variants of Flajolet and Martin's PCSA sketch have Fish-number $H_0/I_0 \approx 1.98016$ and that every base-$q$ variant of (Hyper)LogLog has Fish-number worse than $H_0/I_0$, but that they tend to $H_0/I_0$ in the limit as $q\rightarrow \infty$. Here $H_0,I_0$ are precisely defined constants. We describe a sketch called Fishmonger that is based on a smoothed, entropy-compressed variant of PCSA with a different estimator function. It is proved that with high probability, Fishmonger processes a multiset of $[U]$ such that at all times, its space is $O(\log^2\log U) + (1+o(1))(H_0/I_0)b \approx 1.98b$ bits and its standard error is $1/\sqrt{b}$. We give circumstantial evidence that $H_0/I_0$ is the optimum Fish-number of mergeable sketches for Cardinality Estimation. We define a class of linearizable sketches and prove that no member of this class can beat $H_0/I_0$. The popular mergeable sketches are, in fact, also linearizable., 50 pages, 7 figures, submitted to STOC 2021

The win ratio method has received much attention in methodological research, ad hoc analyses, and designs of prospective studies. As the primary analysis, it supported the approval of tafamidis for the treatment of cardiomyopathy to reduce cardiovascular mortality and cardiovascular-related hospitalization. However, its dependence on censoring is a potential shortcoming. In this article, we propose the inverse-probability-of-censoring weighting (IPCW) adjusted win ratio statistic (i.e., the IPCW-adjusted win ratio statistic) to overcome censoring issues. We consider independent censoring, common censoring across endpoints, and right censoring. We develop an asymptotic variance estimator for the logarithm of the IPCW-adjusted win ratio statistic and evaluate it via simulation. Our simulation studies show that, as the amount of censoring increases, the unadjusted win proportions may decrease greatly. Consequently, the bias of the unadjusted win ratio estimate may increase greatly, producing either an overestimate or an underestimate. We demonstrate theoretically and through simulation that the IPCW-adjusted win ratio statistic gives an unbiased estimate of treatment effect.

This paper provides the first sample complexity lower bounds for the estimation of simple diffusion models which seek to explain the diffusion of an epidemic in a network. The Susceptible-Infected-Recovered (SIR) model is a classic example, proposed nearly a century ago [2]. The SIR model remains a cornerstone for the forecasting of epidemics. The so-called Bass model [1] remains a basic building block in forecasting consumer adoption of new products and services. The durability of these models arises from the fact that they have shown an excellent fit to data, in numerous studies spanning both the epidemiology and marketing literatures. Somewhat paradoxically, using these same models as reliable forecasting tools presents a challenge. While we are ultimately motivated by the problem of forecasting a diffusion model, this paper asks a more basic question that is surprisingly unanswered: What are the limits to learning a diffusion model? We answer this question by characterizing sample complexity lower bounds for a class of stochastic diffusion models that encompass both the Bass model and the SIR model. We show that the time to collect a number of observations that exceeds these lower bounds is too large to allow for accurate forecasts early in the process. In the context of the Bass model our results imply that when adoption is driven by imitation, one cannot hope to predict the eventual number of adopting customers until one is at least two-thirds of the way to the time at which the rate of new adopters is at its peak. In a similar vein, our results imply that in the case of an SIR model, one cannot hope to predict the eventual number of infections until one is approximately two-thirds of the way to the time at which the infection rate has peaked. Our analysis is conceptually simple and relies on the Cramer-Rao bound. The core technical difficulty in our analysis rests in characterizing the Fisher information in the observations available due to the fact that they have a non-trivial correlation structure. Maximum likelihood estimation of diffusion models on product adoption datasets (for products on Amazon.com), and epidemic data (from the ongoing COVID-19 epidemic) illustrate precisely the behavior predicted by our theory. As a byproduct of our analysis, we see that the difficulty in learning a diffusion model stems solely from uncertainty in a single unknown 'effective population size' parameter. In particular, other parameters, including those related to the 'rate of imitation' (in the Bass model) or the 'reproduction number' (in the SIR model) are easy to learn. This suggests that estimators that rely on an (informative) bias in this population size parameter can in fact overcome the limitations presented by our analysis. Although not a primary contribution of the present work, we describe a heuristic procedure used to construct such a biased estimator that yielded one of the first US county-level forecasters available for COVID-19. The full paper is available at https://arxiv.org/abs/2006.06373.

The problem of inferring pair-wise and higher-order interactions in complex systems involving large numbers of interacting variables, from observational data, is fundamental to many fields. Known to the statistical physics community as the inverse problem, it has become accessible in recent years due to real and simulated 'big' data being generated. Current approaches to the inverse problem rely on parametric assumptions, physical approximations, e.g. mean-field theory, and ignoring higher-order interactions which may lead to biased or incorrect estimates. We bypass these shortcomings using a cross-disciplinary approach and demonstrate that none of these assumptions and approximations are necessary: We introduce a universal, model-independent, and fundamentally unbiased estimator of all-order symmetric interactions, via the non-parametric framework of Targeted Learning, a subfield of mathematical statistics. Due to its universality, our definition is readily applicable to any system at equilibrium with binary and categorical variables, be it magnetic spins, nodes in a neural network, or protein networks in biology. Our approach is targeted, not requiring fitting unnecessary parameters. Instead, it expends all data on estimating interactions, hence substantially increasing accuracy. We demonstrate the generality of our technique both analytically and numerically on (i) the 2-dimensional Ising model, (ii) an Ising-like model with 4-point interactions, (iii) the Restricted Boltzmann Machine, and (iv) simulated individual-level human DNA variants and representative traits. The latter demonstrates the applicability of this approach to discover epistatic interactions causal of disease in population biomedicine., 25 pages, 25 figures. Comments welcome

Background The Shannon diversity index has been widely used in population genetics studies. Recently, it was proposed as a unifying measure of diversity at different levels—from genes and populations to whole species and ecosystems. The index, however, was proven to be negatively biased at small sample sizes. Modifications to the original Shannon’s formula have been proposed to obtain an unbiased estimator. Methods In this study, the performance of four different estimators of Shannon index—the original Shannon’s formula and those of Zahl, Chao and Shen and Chao et al.—was tested on simulated microsatellite data. Both the simulation and analysis of the results were performed in the R language environment. A new R function was created for the calculation of all four indices from the genind data format. Results Sample size dependence was detected in all the estimators analysed; however, the deviation from parametric values was substantially smaller in the derived measures than in the original Shannon’s formula. Error rate was negatively associated with population heterozygosity. Comparisons among loci showed that fast-mutating loci were less affected by the error, except for the original Shannon’s estimator which, in the smallest sample, was more strongly affected by loci with a higher number of alleles. The Zahl and Chao et al. estimators performed notably better than the original Shannon’s formula. Conclusion The results of this study show that the original Shannon index should no longer be used as a measure of genetic diversity and should be replaced by Zahl’s unbiased estimator.

This paper is devoted to the state and input estimation of a linear time varying system in the presence of an unknown input (UI) in both state and measurement equations, and affected by Gaussian noises. The classical rank condition used in this kind of approach is relaxed in order to be able to be used in a wider range of systems. A state observer, that is an unbiased estimator with minimum error variance, is proposed. Then a UI observer is constructed, in order to be a best linear unbiased estimator, it follows a unique construction whether the direct feedthrough matrix is null or not. In a sense the proposed approach, generalizes and unifies the existing ones. Besides, a stability result is given for linear time invariant systems, which is a novelty for unbiased minimum variance observers relaxing the classical rank condition.

We thank the anonymous referees for their useful suggestions., The raise regression has been proposed as an alternative to ordinary least squares estimation when a model presents collinearity. In order to analyze whether the problem has been mitigated, it is necessary to develop measures to detect collinearity after the application of the raise regression. This paper extends the concept of the variance inflation factor to be applied in a raise regression. The relevance of this extension is that it can be applied to determine the raising factor which allows an optimal application of this technique. The mean square error is also calculated since the raise regression provides a biased estimator. The results are illustrated by two empirical examples where the application of the raise estimator is compared to the application of the ridge and Lasso estimators that are commonly applied to estimate models with multicollinearity as an alternative to ordinary least squares.

© 2000-2011 IEEE. Traffic data estimation plays an important role because traffic data often suffers from missing data problems, caused by a variety of reasons, i.e., temporary deployment of sensors, sensor malfunction and communication failure. Existing research on missing data estimation has mostly focused on using data-driven or model-driven models to estimate the missing data, and there is a lack of study on the achievable estimation accuracy and the conditions to achieve accurate missing data estimation. In this paper, we investigate the fundamental limits of missing traffic data estimation accuracy in urban networks using the spatial-temporal random effects model. We derive the squared flow error bound (SFEB) for the cases of the Fisher matrix being a singular and non-singular matrix, respectively. We show that the sufficient and necessary condition of the existence of an unbiased estimator is that the number of missing points is less than or equal to the rank of the Fisher matrix. For the case that no unbiased estimator can be found, we derive an inequality for the SPEB and show that the SFEB is readily determined by the covariance matrix of the unknown (missing) parameter vector, flow correlation between the unknown and the available data, and the sensor locations. Furthermore, we develop an optimal spatial-temporal Kriging estimator which is efficient in both cases where the causal relationship among available data points exists or does not exist. Our theoretical findings can be used to develop a sensor location optimization strategy to minimize the SFEB.

Suppose that $\pi_{1},\pi_{2},\ldots ,\pi_{k}$ be $k(\geq2)$ independent exponential populations having unknown location parameters $\mu_{1},\mu_{2},\ldots,\mu_{k}$ and known scale parameters $\sigma_{1},\ldots,\sigma_{k}$. Let $\mu_{[k]}=\max \{\mu_{1},\ldots,\mu_{k}\}$. For selecting the population associated with $\mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $\mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $\delta_{N,1},\delta_{N,2}$ and $\delta_{N,3}$ of $\mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $\mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $\mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $\mu_{S}$ is derived. Using this result, estimator better than the natural estimator $\delta_{N,1}$ is obtained. We also shown that the estimator $\delta_{N,1}$ is dominated by the natural estimator $\delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $\mu_{S}$.

A great deal of experimental effort is currently being devoted to the precise measurements of the cosmic microwave background (CMB) sky in temperature and polarisation. Satellites, balloon-borne, and ground-based experiments scrutinize the CMB sky at multiple scales, and therefore enable to investigate not only the evolution of the early Universe, but also its late-time physics with unprecedented accuracy. The pipeline leading from time ordered data as collected by the instrument to the final product is highly structured. Moreover, it has also to provide accurate estimates of statistical and systematic uncertainties connected to the specific experiment. In this paper, we review likelihood approaches targeted to the analysis of the CMB signal at different scales, and to the estimation of key cosmological parameters. We consider methods that analyze the data in the spatial (i.e., pixel-based) or harmonic domain. We highlight the most relevant aspects of each approach and compare their performance., 71 pages, 5 figures. Matching version to be published in Frontiers in Physics, section Cosmology, as a contribution to the collection "Status and Prospects of Cosmic Microwave Background Analysis"

A weakly-supervised learning framework named as complementary-label learning has been proposed recently, where each sample is equipped with a single complementary label that denotes one of the classes the sample does not belong to. However, the existing complementary-label learning methods cannot learn from the easily accessible unlabeled samples and samples with multiple complementary labels, which are more informative. In this paper, to remove these limitations, we propose the novel multi-complementary and unlabeled learning framework that allows unbiased estimation of classification risk from samples with any number of complementary labels and unlabeled samples, for arbitrary loss functions and models. We first give an unbiased estimator of the classification risk from samples with multiple complementary labels, and then further improve the estimator by incorporating unlabeled samples into the risk formulation. The estimation error bounds show that the proposed methods are in the optimal parametric convergence rate. Finally, the experiments on both linear and deep models show the effectiveness of our methods., 22 pages, 5 figure

The Cramer&ndash, Rao&ndash, Frechet inequality is reviewed and extended to track fitting. A diffused opinion attributes to this inequality the limitation of the resolution of the track fits with the number N of observations. It will be shown that this opinion is incorrect, the weighted least squares method is not subjected to that N-limitation and the resolution can be improved beyond those limits. In previous publications, simulations with realistic models and simple Gaussian models produced interesting results: linear growths of the peaks of the distributions of the fitted parameters with the number N of observations, much faster than the N of the standard least-squares. These results could be considered a violation of a well-known 1 / N -rule for the variance of an unbiased estimator, frequently reported as the Cramer&ndash, Frechet bound. To clarify this point beyond any doubt, a direct proof of the consistency of those results with this inequality would be essential. Unfortunately, such proof is lacking. Hence, the Cramer&ndash, Frechet developments are applied to prove the efficiency (optimality) of the simple Gaussian model and the consistency of the linear growth. The inequality remains valid even for irregular models supporting the similar improvement of resolution for the realistic models.

In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and confidence intervals can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly-informative observations, and a realistic Lotka-Volterra model with an intractable transition density., This document is a self-contained and direct description of the smoothing method introduced in Coupling of Particle Filters (arXiv:1606.01156). Code is available at github.com/pierrejacob/CoupledCPF. Compared to the previous version, a bug was fixed in the code, and the numerical results were updated

The general unbalanced mixed linear model with two variance components is considered. Through resampling it is demonstrated how the fixed effects can be estimated explicitly. It is shown that the obtained nonlinear estimator is unbiased and its variance is also derived. A condition is given when the proposed estimator is recommended instead of the ordinary least squares estimator. Funding Agencies|Swedish Foundation for Humanities and Social sciences [P14-0641:1]

We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two -- implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits., Published version

This study develops a nearly unbiased estimator of the ratio of the contemporary effective mother size to the census size in a population, as a proxy of the ratio of contemporary effective size to census size (Ne/N). The proposed estimator is based on both known mother–offspring (MO) and maternal-sibling (MS) relationships observed within the same cohort, in which sampled individuals in the cohort probably share MO relationships with sampled mothers. The rationale is that the frequency of MO and MS pairs contains information regarding the contemporary effective mother size and the (mature) census size, respectively. Therefore, the estimator can be obtained only from genetic data. Moreover, We also evaluate the performance of the estimator by running an individual-based model. The results of this study provide the following: i) parameter range for satisfying the unbiasedness, and ii) guidance for sample sizes to ensure the required accuracy and precision, especially when the order of the ratio is available. Furthermore, the results demonstrate the usefulness of a sibship assignment method for genetic monitoring, providing insights for interpreting environmental and/or anthropological factors fluctuatingNe/N, especially in the context of conservation biology and wildlife management.

It is known that the variance of the maximum likelihood estimator (MLE) inflates when the explanatory variables are correlated. This situation is called the multicollinearity problem. As a result, the estimations of the model may not be trustful. Therefore, this paper introduces a new restricted estimator (RLTE) that may be applied to get rid of the multicollinearity when the parameters lie in some linear subspace in logistic regression. The mean squared errors (MSE) and the matrix mean squared errors (MMSE) of the estimators considered in this paper are given. A Monte Carlo experiment is designed to evaluate the performances of the proposed estimator, the restricted MLE (RMLE), MLE and Liu-type estimator (LTE). The criterion of performance is chosen to be MSE. Moreover, a real data example is presented. According to the results, proposed estimator has better performance than MLE, RMLE and LTE.

In this study we have proposed a multivariate regression-cum-exponential type estimator for estimating a vector of population variance. In the present study, unknown population variance vector estimation has been discussed using multi-auxiliary variables in two-phase sampling and different cases have also been derived. A comparison between existing and the proposed multivariate, bivariate and univariate estimators has been prepared with the help of a real data for estimating population variance. A simulation study for multivariate estimator using multi-auxiliary variables has also been carried out to demonstrate the performance of the estimators. Keywords: Multivariate estimator, Multi-auxiliary variables, Two-phase sampling, Regression estimator, Variance-covariance matrix, Simulation

International audience; is paper introduces a new algorithm to approximate smoothed additive functionals of partially observed diffusion processes. This method relies on a new sequential Monte Carlo method which allows to compute such approximations online, i.e., as the observations are received, and with a computational complexity growing linearly with the number of Monte Carlo samples. The original algorithm cannot be used in the case of partially observed stochastic differential equations since the transition density of the latent data is usually unknown. We prove that it may be extended to partially observed continuous processes by replacing this unknown quantity by an unbiased estimator obtained for instance using general Poisson estimators. This estimator is proved to be consistent and its performance are illustrated using data from two models.

We investigate the probabilities of large deviations of the maximum likelihood estimator and Bayes estimator of the drift parameter for a mixed fractional Ornstein-Uhlenbeck type process.