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Abstract.The negative multinomial (NM) distribution is of interest in various application studies. Based on closed-form n-consistent estimators, new and fast closed-form efficient estimators are proposed for the NM distribution. The theorem applied to derive the new estimators guarantees two important properties of the new closed-form efficient estimators: asymptotic efficiency and normality. The new closed-form efficient estimators are denoted as MLE-CEs, because the asymptotic distribution is the same as that of the maximum likelihood estimators (MLEs). Simulation studies suggest that the MLE-CE performs similarly to its MLE. The estimated accuracies of the MLE and MLE-CE are generally better than the method of moments estimator (MME) for relatively large 풑 values. The MLE-CE is 10–30 times faster than the MLE, especially for large sample sizes, which is good for the big data era. Considering the estimated accuracy and computing time, the MLE-CE is recommended for large 풑 values, whereas the MME is recommended for other conditions. [ABSTRACT FROM AUTHOR]

For some distributions, there is no available closed-form maximum likelihood estimator (MLE). This could be a problem when estimating the parameters of state-space models or of real-time processing models, because of time consuming iterations required to obtain an MLE. A simple method of obtaining a closed-form and efficient estimators for the univariate Weibull distribution is presented. Furthermore, an alternative efficient estimator for bivariate Weibull distribution parameters other than the MLE is also suggested. For the bivariate Weibull distribution, one estimator is not in closed-form, but the others are all in closed-form. In general, if we have a closed-form n -consistent estimator for each parameter, then we have a closed-form solution set which is efficient and has asymptotic normality. This is quite striking since we can obtain such a closed-form and efficient estimator using a very simple strategy. Roughly speaking, our suggested estimator is approximately 11 times as fast as the MLE for univariate case and 85 times for bivariate case. A simulation study is done for univariate and bivariate Weibull distributions to find some small sample properties. A real data illustration is also provided. [ABSTRACT FROM AUTHOR]

The negative binomial (NB) distribution is of interest in various application studies. New closed-form efficient estimators are proposed for the two NB parameters, based on closed-form n -consistent estimators. The asymptotic efficiency and normality of the new closed-form efficient estimators are guaranteed by the theorem applied to derive the new estimators. Since the new closed-form efficient estimators have the same asymptotic distribution as the maximum likelihood estimators (MLEs), these are denoted as MLE-CEs. Simulation studies suggest that the MLE-CE of dispersion parameter r performs better than its MLE and the method of moments estimator (MME) for some parameter ranges. The MLE-CE of the probability parameter p exhibits the best performance for relatively large p values, where the positive-definite expected Fisher information matrix exists. MLE performs better than MME in this parameter space. The MLE-CE is over 200 times faster than the MLE, especially for large sample sizes, which is good for the big data era. Considering the estimated accuracy and computing time, MLE-CE is recommended for small r values and large p values, whereas MME is recommended for other conditions. [ABSTRACT FROM AUTHOR]

This study aimed to develop new closed-form and efficient estimators for the parameters of the bivariate Weibull distribution. New estimators can be produced using closed-form n -consistent estimators for all parameters, except for the association parameter for which the estimator is not in closed form. This is carried out by utilizing a theorem that produces asymptotically efficient estimators. To achieve this, n -consistent estimators are introduced. Fisher observed and expected information matrices are derived and used to develop new estimators. A simulation study and real data application are included to validate the new estimators. Given that the new estimators are as asymptotically efficient as maximum likelihood estimators and are in closed form, except for the association parameter, they can be used effectively in state-space modelling or real-time processing models. This is because of the shorter computing time associated with them than with maximum likelihood estimators. [ABSTRACT FROM AUTHOR]

Computational data analysis is an essential component of modern statistics. The most important challenge in classical statistics was an estimation of the location parameter. In addition, the selection of estimators among potential estimators is another problem in the computational data analysis. Therefore, the main objective of this study is to develop a location estimator that has been modified compared to all common location estimators. In order to ensure the objective of the new estimator, combinations weighted mean (Alex-mean) was developed and computed with R version 3.6.3. Further, the properties of a good location estimator diagnosed with all location estimators compared to Alex-mean with a COVID-19 new case data set. In addition, the new estimator was compared with mean, median and trimmed-mean based on relative efficiency and bootstrap simulation techniques. The new estimator is better than the mean, median and trimmed-mean of relative efficiency estimated using bootstrap and simulation techniques. The sensitivity graph shows that Alex-mean is insensitive to the outlier and is computed on the basis of all observations. Thus, the researcher suggested that Alex-mean is the good location estimator than all common measures of central tendency since it is undisturbed by extreme values. [ABSTRACT FROM AUTHOR]

We present a framework for using existing external data to identify and estimate the relative efficiency of a covariate-adjusted estimator compared to an unadjusted estimator in a future randomized trial. Under conditions, these relative efficiencies approximate the ratio of sample sizes needed to achieve a desired power. We develop semiparametrically efficient estimators of the relative efficiencies for several treatment effect estimands of interest with either fully or partially observed outcomes, allowing for the application of flexible statistical learning tools to estimate the nuisance functions. We propose an analytic Wald-type confidence interval and a double bootstrap scheme for statistical inference. We demonstrate the performance of the proposed methods through simulation studies and apply these methods to estimate the efficiency gain of covariate adjustment in Covid-19 therapeutic trials. [ABSTRACT FROM AUTHOR]

Since the commonly available time series on micro units are typically quite short, this paper considers a different estimation of linear panel data models where the unobserved individual effects are permitted to have time-varying effects on the response variable. We allow flexible possible correlations between included regressors and unobserved individual effects, and the model can accommodate both time varying and time invariant covariates. The quasi-maximum likelihood method is then proposed to obtain the estimates, which are easily executed by a simple iterative method. Two types of approaches to estimate the covariance matrix are introduced. The large sample properties are established when n → ∞ and T is fixed. The estimates are efficient when both the individual effects and random errors follow normal distributions. Simulation studies show that our estimates perform well even when the correlations between the regressors and unobserved individual effects are misspecified. The proposed method is further illustrated by applications to a real data. [ABSTRACT FROM AUTHOR]

In many settings, multiple data collections and analyses on the same topic are summarised separately through statistical estimators of parameters and variances, and yet there are scientific reasons for sharing some statistical parameters across these different studies. This paper summarises what is known from large-sample theory about when estimators of a common structural parameter from several independent samples can be combined functionally, or more specifically linearly, to obtain an asymptotically efficient estimator from the combined sample. The main idea is that such combination can be done when the separate-sample nuisance parameters, if any exist, vary freely and independently of one another. The issues are illustrated using data from a multi-centre lung cancer clinical trial. Examples are presented to show that separate estimators cannot always be combined in this way, and that the functionally combined separate estimators may have low or 0 efficiency compared to the unified analysis that could be performed by pooling the datasets.

Summary: Error variance estimation plays a key role in the analysis of homogeneous non‐parametric regression models. For a random design model, most methods in the literature for error variance estimation assume the independence between the predictor variable X and the error ε. In this work, we derive the optimal semi‐parametric efficiency bound for the error variance σ2=var(ϵ) without such an independence assumption. A residual‐based efficient estimator for σ2 is proposed and its asymptotic normality is established. An extensive simulation study is conducted, which shows that our proposed estimator works very favourably against competitors. A simple real‐data example is also presented. Derived the optimal semi‐parametric efficiency bound and efficient estimator for the error variance without the independence assumption [ABSTRACT FROM AUTHOR]

The perinatal mortality rate is very high throughout the world. A fetal monitor may be used remotely, and this would tackle the problem of continuous monitoring of high-risk pregnancies. There is evidence that current technology is of low reliability, and, therefore, of low precision to identify fetal health. In medical technological implementation, a safe, efficient, and reliable operation must be guaranteed, and the main problem is that remote fetal monitor gathers just a few samples, so the hypothesis of classical theory is not met. We are proposing an approach that improves the data's lack of reliability that accompanies the use of a remote fetal monitor. The method refers to how, by using the existing technologies and the initial experimental data, it is possible to apply probabilistic models that are truly representative of each application. This leads to the characterization of properties of the statistics used to generate a representative probabilistic model without the need to consider the hard suppositions. Results show that, for every case study, it was possible to improve estimations of measurement uncertainty. The proposed method is a useful tool to increase the reliability of medical technology, especially for pieces of equipment where a health care professional is not available. [ABSTRACT FROM AUTHOR]

We consider the estimation of the error distribution function of partial linear single-index models. The estimation methods for the error distribution function based on the classical empirical distribution function as well as empirical likelihood method are discussed, the latter method allows for incorporation of additional information on the error distribution function into estimation. We show weak convergence of the corresponding empirical processes to Gaussian processes and compare both approaches with the asymptotic theory and by means of simulation studies. [ABSTRACT FROM AUTHOR]

Abstract Assume a (semi)parametrically efficient estimator is given of the Euclidean parameter in a (semi)parametric model. A submodel is obtained by constraining this model in that a continuously differentiable function of the Euclidean parameter vanishes. We present an explicit method to construct (semi)parametrically efficient estimators of the Euclidean parameter in such equality constrained submodels and prove their efficiency. Our construction is based solely on the original efficient estimator and the constraining function. [ABSTRACT FROM AUTHOR]

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a "delete‐one‐component" estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

In traditional Reinforcement Learning (RL), agents learn to optimize actions in a dynamic context based on recursive estimation of expected values. We show that this form of machine learning fails when rewards (returns) are affected by tail risk, i.e., leptokurtosis. Here, we adapt a recent extension of RL, called distributional RL (disRL), and introduce estimation efficiency, while properly adjusting for differential impact of outliers on the two terms of the RL prediction error in the updating equations. We show that the resulting “efficient distributional RL” (e-disRL) learns much faster, and is robust once it settles on a policy. Our paper also provides a brief, nontechnical overview of machine learning, focusing on RL.

The perinatal mortality rate is very high throughout the world. A fetal monitor may be used remotely, and this would tackle the problem of continuous monitoring of high-risk pregnancies. There is evidence that current technology is of low reliability, and, therefore, of low precision to identify fetal health. In medical technological implementation, a safe, efficient, and reliable operation must be guaranteed, and the main problem is that remote fetal monitor gathers just a few samples, so the hypothesis of classical theory is not met. We are proposing an approach that improves the data’s lack of reliability that accompanies the use of a remote fetal monitor. The method refers to how, by using the existing technologies and the initial experimental data, it is possible to apply probabilistic models that are truly representative of each application. This leads to the characterization of properties of the statistics used to generate a representative probabilistic model without the need to consider the hard suppositions. Results show that, for every case study, it was possible to improve estimations of measurement uncertainty. The proposed method is a useful tool to increase the reliability of medical technology, especially for pieces of equipment where a health care professional is not available.

Suppose we observe n independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285-1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this L1-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than n-1/4 w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient. [ABSTRACT FROM AUTHOR]

Case-cohort study designs are widely used to reduce the cost of large cohort studies. When several diseases are of interest, we can use the same subcohort. In this paper, we will study the casecohort design of marginal additive hazards model for multiple outcomes by a more efficient version. Instead of analyzing each disease separately, ignoring the additional exposure measurements collected on subjects with other diseases, we propose a new weighted estimating equation approach to improve the efficiency by utilizing as much information collected as possible. The consistency and asymptotic normality of the resulting estimator are established. Simulation studies are conducted to examine the finite sample performance of the proposed estimator, which confirm the efficiency gains. [ABSTRACT FROM AUTHOR]

Clustered failure times often arise from studies with stratified sampling designs where it is desired to reduce both cost and sampling error. Semiparametric accelerated failure time (AFT) models have not been used as frequently as Cox relative risk models in such settings due to lack of efficient and reliable computing routines for inferences. The challenge roots in the nonsmoothness of the rank-based estimating functions, and for clustered data, the asymptotic properties of the estimator from the weighted version have not been available. The recently proposed induced smoothing approach, which provides fast and accurate rank-based inferences for AFT models, is generalized to incorporate weights to accommodate stratified sampling designs. The estimator from the induced smoothing weighted estimating equations are shown to be consistent and have the same asymptotic distribution as that from the nonsmooth version, which has not been developed before. The variance of the estimator is estimated by computationally efficient sandwich estimators aided by a multiplier bootstrap. The proposed method is assessed in extensive simulation studies where the estimators appear to provide valid and efficient inferences. A stratified case-cohort design with clustered times to tooth extraction in a dental study illustrates the usefulness of the method. [ABSTRACT FROM AUTHOR]

In this paper we present a definition of an efficient estimator for an unknown parameter and an example. The optimal estimator is called the minimum variance unbiased estimator. This type of estimator will be our choice for the optimum or best estimator. The entire article consists my own point of view concerning the estimation theory in general and the efficient estimators in particular. [ABSTRACT FROM AUTHOR]

Highlights: [•] The maximum likelihood estimate (MLE) of T 2 from spin-echo data is presented. [•] The MLE is computationally fast and agnostic to the echo shape. [•] Compared to existing methods, the MLE provides 3-fold or more time savings. [•] The method is illustrated by application to simulated and experimental EPR data. [ABSTRACT FROM AUTHOR]

This article considers nonparametric regression models with multivariate covariates and with responses missing at random. We estimate the regression function with a local polynomial smoother. The residual-based empirical distribution function that only uses complete cases, i.e. residuals that can actually be constructed from the data, is shown to be efficient in the sense of Hájek and Le Cam. In the proofs we derive, more generally, the efficient influence function for estimating an arbitrary linear functional of the error distribution; this covers the distribution function as a special case. We also show that the complete case residual-based empirical distribution function admits a functional central limit theorem. This is done by applying the transfer principle for complete case statistics developed by Koul et al. [(2012), ‘The Transfer Principle: a Tool for Complete Case Analysis’,Annals of Statistics, 40, 3031–3049], which makes it possible to adapt known results for fully observed data to the missing data case. The article concludes with a small simulation study investigating the performance of the complete case residual-based empirical distribution function. [ABSTRACT FROM AUTHOR]

The estimation of a source location using directly the range or range difference measurements is difficult and requires numerical solution, which is caused by the highly non-linear relationship between the measurements and the unknown. We can obtain a computationally efficient and non-iterative algebraic solution by squaring the measurements first before solving for the unknown. However, a recent study has shown that such a solution is suboptimum in reaching the CRLB performance and the localization accuracy could be significantly worse in some localization geometries. This paper demonstrates that when range weighting factors are introduced to the squared measurements, the resulting solution will be able to reach the CRLB accuracy. Both the squared range and squared range difference cases are considered, and the mean-square error (MSE) and the bias of the resulting solutions are derived. The asymptotic efficiency of the proposed cost functions are proven theoretically and validated by simulations. The effects of range weighting factors on the localization performance under different sensor number, noise correlation, and localization geometry are examined. Introducing range weightings to the squared range measurements increases the bias but it is negligible in the MSE. Having range weightings in the squared range difference measurements improves both the MSE and bias. [ABSTRACT FROM PUBLISHER]

In randomized clinical trials, we are often concerned with comparing two-sample survival data. Although the log-rank test is usually suitable for this purpose, it may result in substantial power loss when the two groups have nonproportional hazards. In a more general class of survival models of Yang and Prentice (Biometrika 92:1-17, ), which includes the log-rank test as a special case, we improve model efficiency by incorporating auxiliary covariates that are correlated with the survival times. In a model-free form, we augment the estimating equation with auxiliary covariates, and establish the efficiency improvement using the semiparametric theories in Zhang et al. (Biometrics 64:707-715, ) and Lu and Tsiatis (Biometrics, 95:674-679, ). Under minimal assumptions, our approach produces an unbiased, asymptotically normal estimator with additional efficiency gain. Simulation studies and an application to a leukemia study show the satisfactory performance of the proposed method. [ABSTRACT FROM AUTHOR]

Despite of the increased interest in independent component analysis (ICA) during the past two decades, a simple closed form expression of the Cramér-Rao bound (CRB) for the demixing matrix estimation has not been established in the open literature. In the present paper we fill this gap by deriving a simple closed-form expression for the CRB of the demixing matrix directly from its definition. A simulation study comparing ICA estimators with the CRB is given. [ABSTRACT FROM AUTHOR]

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model $Y_{t}=\int_{0}^{t}\sqrt{V_{s}}\,\mathrm {d}W_{s}$ , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling $(Y_{i\delta_{n}})_{i=0,\dots,n}$ with δ n →0 and n δ n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies identification of systems in which only quantized output observations are available. An identification algorithm for system gains is introduced that employs empirical measures from multiple sensor thresholds and optimizes their convex combinations. Strong convergence of the algorithm is first derived. The algorithm is then extended to a scenario of system identification with communication constraints, in which the sensor output is transmitted through a noisy communication channel and observed after transmission. The main results of this paper demonstrate that these algorithms achieve the Cramér–Rao lower bounds asymptotically, and hence are asymptotically efficient algorithms. Furthermore, under some mild regularity conditions, these optimal algorithms achieve error bounds that approach optimal error bounds of linear sensors when the number of thresholds becomes large. These results are further extended to finite impulse response and rational transfer function models when the inputs are designed to be periodic and full rank. [Copyright &y& Elsevier]

Motivated by two practical problems, we propose a new procedure for estimating a semivarying‐coefficient model. Asymptotic properties are established which show that the bias of the parameter estimator is of order h3 when a symmetric kernel is used, where h is the bandwidth, and the variance is of order n−1 and efficient in the semiparametric sense. Undersmoothing is unnecessary for the root‐n consistency of the estimators. Therefore, commonly used bandwidth selection methods can be employed. A model selection method is also developed. Simulations demonstrate how the proposed method works. Some insights are obtained into the two motivating problems by using the proposed models. [ABSTRACT FROM AUTHOR]

We consider semiparametric models of semi-Markov processes with arbitrary state space. Assuming that the process is geometrically ergodic, we characterize efficient estimators, in the sense of Hájek and Le Cam, for arbitrary real-valued smooth functionals of the distribution of the embedded Markov renewal process. We construct efficient estimators of the parameter and of linear functionals of the distribution. In particular we treat the two cases in which we have a parametric model for the transition distribution of the embedded Markov chain and an arbitrary conditional distribution of the inter-jump times, and vice versa. [ABSTRACT FROM AUTHOR]

This paper addresses estimation of linear functionals of the error distribution in nonparametric regression models. It derives an i.i.d. representation for the empirical estimator based on residuals, using undersmoothed estimators for the regression curve. Asymptotic efficiency of the estimator is proved. Estimation of the error variance is discussed in detail. In this case, undersmoothing is not necessary. [Copyright &y& Elsevier]