New Class Of Estimators Research Articles

Combining Instrumental Variable Estimators for a Panel Data Model with Factors

We address the estimation of factor-augmented panel data models using observed measurements to proxy for unobserved factors or loadings and explore the use of internal instruments to address the resulting endogeneity. The main challenge consists in that economic theory rarely provides insights into which measurements to choose as proxies when several are available. To overcome this problem, we propose a new class of estimators that are linear combinations of instrumental variable estimators and establish large sample results. We also show that an optimal weighting scheme exists, leading to efficiency gains relative to an instrumental variable estimator. Simulations show that the proposed approach performs better than existing methods. We illustrate the new method using data on test scores across US school districts.

Improved Variance Estimation From Trimmed Samples

ABSTRACTSeveral methods for the robust estimation of the variance of a normal random variable are based on trimming. For example, the popular estimator is based on trimming the pairs of samples with the largest absolute distance. Another example is least trimmed squares regression, where all unusually large residuals are removed before the estimation of the variance. Here, in this work, we propose two new classes of estimators that use an optimal linear combination of the trimmed samples to achieve a lower mean squared error (MSE). The first class guarantees the smallest variance among all unbiased estimators that are based on linear combinations, while the second class of estimators guarantees the smallest MSE at the cost of some small bias. Through simulations, we demonstrate that our proposed estimators can have considerably smaller MSE than other robust estimators both in the presence and absence of outliers.

Improved Liu-ridge-type estimates for the beta regression model

The main objective of this paper is to introduce a new class of estimators for the ill-conditioned beta regression model. The new class, named the Liu-ridge-type (LRT), incorporates two shrinkage parameters that allow it to include the maximum-likelihood estimator (MLE), the ridge estimator (RE), and the Liu estimator (LE) as special cases. The LRT estimator has been shown to outperform the LE, the RE, and the MLE in terms of the mean squared error (MSE) matrix under certain conditions. Simulated and real applications illustrate the potential benefits of the new LRT estimator.

On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

$U$-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. $U$-statistics generalise the empirical mean of a random variable $X$ to sums over every $m$-tuple of distinct observations of $X$. Stute [Conditional U -statistics, Ann. Probab., 1991] introduced a class of estimators called conditional $U$-statistics. In the present work, we provide a new class of estimators of conditional $U$-statistics. More precisely, we investigate the conditional $U$-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over $\varphi \in \mathscr{F}$ for a suitably restricted class $\mathscr{F}$, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric Inverse Probability of Censoring Weighted estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional $U$-statistics processes. We discuss the wild bootstrap of the conditional $U$-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.

Efficient class of estimators for finite population mean using auxiliary attribute in stratified random sampling

The aim of this paper is to develop more effective methods for estimating population means in sample surveys using auxiliary attributes. To achieve this goal, we introduce a modified version of the estimators proposed by Koyuncu (2013b) and Shahzad et al. (2019), as well as a new class of estimators. We derive expressions for the bias and mean squared error of these new estimators up to the first degree of approximation. Our results show that the suggested classes of estimators perform better than other existing methods, with the lowest mean squared error under optimal conditions. We also conduct an empirical investigation to support our findings.

Improved Estimators For The Population Mean Under Non-Response

We propose a novel family of estimators for the population mean under non-response and obtain the MSE equation of the suggested estimator for each situation in theory. These theoretical conditions are applied to three popular data sets in literature and we see that the suggested estimators are more efficient than the traditional estimators, such as ratio, regression estimators, in Case 1; whereas, in Case 2, the suggested estimators are also more efficient than the Unal-Kadilar exponential estimators that are more efficient than the traditional estimators for the same data sets.

A New Class of Generalized Probability-Weighted Moment Estimators for the Pareto Distribution

Estimation based on probability-weighted moments is a well-established method and an excellent alternative to the classic method of moments or the maximum likelihood method, especially for small sample sizes. In this research, we developed a new class of estimators for the parameters of the Pareto type I distribution. A generalization of the probability-weighted moments approach is the foundation for this new class of estimators. It has the advantage of being valid in the entire parameter space of the Pareto distribution. We established the asymptotic normality of the new estimators and applied them to simulated and real datasets in order to illustrate their finite sample behavior. The results of comparisons with the most used estimation methods were also analyzed.

Martingale product estimators for sensitivity analysis in computational statistical physics

Abstract We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name ‘martingale product estimators’. We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so that its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green–Kubo (GK) estimator, motivated, in part, by its variance growing linearly in time. We compare on illustrative numerical tests the new estimators with results obtained by the GK method.

On the study of efficiency of exponential‐type estimator of population mean using robust regression methods

AbstractThis study aimed at enhancing the efficiency of Zaman estimators using exponential transformation technique. A new class of estimators was obtained using the concept of Bahl and Tuteja. The bias and mean squared error (MSE) of the new class of suggested estimators was derived up to second degree approximation. The empirical study through simulations was conducted using Normal, exponential, gamma, chi‐square and beta distributions under robust regression methods (Huber‐M, Huber‐MM, LTS (least trimmed squares) and LMS (least median of squares)) and the results revealed that proposed estimators were more efficient.

A new estimator of a jump discontinuity in regression

We propose a new class of estimators for a jump discontinuity on nonparametric regression. While there is a vast literature in econometrics that addresses this issue (e.g., Hahn et al., 2001; Porter, 2003; Imbens and Lemieux, 2008; Cattaneo and Escanciano, 2017), the main approach in these studies is to use local polynomial (linear) estimators on both sides of the discontinuity to produce an estimator for the jump that has desirable boundary properties. Our approach extends the regression from both sides of the discontinuity using a theorem of Hestenes (1941). The extended regressions are then estimated and used to construct an estimator for the jump discontinuity that solves the boundary problems normally associated with classical Nadaraya–Watson estimators. We provide asymptotic characterizations for the jump estimators, including bias and variance orders, and asymptotic distributions after suitable centering and normalization. Monte Carlo simulations show that our jump estimators can outperform those based on local polynomial (linear) regression.

On Generalized Schürmann Entropy Estimators.

We present a new class of estimators of Shannon entropy for severely undersampled discrete distributions. It is based on a generalization of an estimator proposed by T. Schürmann, which itself is a generalization of an estimator proposed by myself.For a special set of parameters, they are completely free of bias and have a finite variance, something which is widely believed to be impossible. We present also detailed numerical tests, where we compare them with other recent estimators and with exact results, and point out a clash with Bayesian estimators for mutual information.

High-dimensional linear regression via implicit regularization

SummaryMany statistical estimators for high-dimensional linear regression are $M$-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that, under suitable restricted isometry conditions, overparameterization leads to implicit regularization: if we directly apply gradient descent to the residual sum of squares with sufficiently small initial values then, under some proper early stopping rule, the iterates converge to a nearly sparse rate-optimal solution that improves over explicitly regularized approaches. In particular, the resulting estimator does not suffer from extra bias due to explicit penalties, and can achieve the parametric root-$n$ rate when the signal-to-noise ratio is sufficiently high. We also perform simulations to compare our methods with high-dimensional linear regression with explicit regularization. Our results illustrate the advantages of using implicit regularization via gradient descent after overparameterization in sparse vector estimation.

Cosmology without window functions. II. Cubic estimators for the galaxy bispectrum

When analyzing the galaxy bispectrum measured from spectroscopic surveys, it is imperative to account for the effects of non-uniform survey geometry. Conventionally, this is done by convolving the theory model with the the window function; however, the computational expense of this prohibits full exploration of the bispectrum likelihood. In this work, we provide a new class of estimators for the unwindowed bispectrum; a quantity that can be straightforwardly compared to theory. This builds upon the work of Philcox (2021) for the power spectrum, and comprises two parts (both obtained from an Edgeworth expansion): a cubic estimator applied to the data, and a Fisher matrix, which deconvolves the bispectrum components. In the limit of weak non-Gaussianity, the estimator is minimum-variance; furthermore, we give an alternate form based on FKP weights that is close-to-optimal and easy to compute. As a demonstration, we measure the binned bispectrum monopole of a suite of simulations both using conventional estimators and our unwindowed equivalents. Computation times are comparable, except that the unwindowed approach requires a Fisher matrix, computable in an additional $\mathcal{O}(100)$ CPU-hours. Our estimator may be straightforwardly extended to measure redshift-space distortions and the components of the bispectrum in arbitrary separable bases. The techniques of this work will allow the bispectrum to straightforwardly be included in the cosmological analysis of current and upcoming survey data.

A new GEE method to account for heteroscedasticity using asymmetric least-square regressions

ABSTRACT Generalized estimating equations are widely used to analyze longitudinal data; however, they are not appropriate for heteroscedastic data, because they only estimate regressor effects on the mean response – and therefore do not account for data heterogeneity. Here, we combine the with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations . The model estimates regressor effects on the expectiles of the response distribution, which provides a detailed view of regressor effects on the entire response distribution. In addition to capturing data heteroscedasticity, the GEEE extends the various working correlation structures to account for within-subject dependence. We derive the asymptotic properties of the estimators and propose a robust estimator of its covariance matrix for inference (see our R package, github.com/AmBarry/expectgee). Our simulations show that the GEEE estimator is non-biased and efficient, and our real data analysis shows it captures heteroscedasticity.

Factor models with local factors — Determining the number of relevant factors

We extend the theory on factor models by incorporating “local” factors into the model. Local factors affect only an unknown subset of the observed variables. This implies a continuum of eigenvalues of the covariance matrix, as is commonly observed in applications. We derive which factors are pervasive enough to be economically important and which factors are pervasive enough to be estimable using the common principal component estimator. We then introduce a new class of estimators to determine the number of those relevant factors. Unlike existing estimators, our estimators use not only the eigenvalues of the covariance matrix, but also its eigenvectors. We find that incorporating partial sums of the eigenvectors into our estimators leads to significant gains in performance in simulations.

Asymptotic distributions for weighted power sums of extreme values

Let $X_{1,n}\le\cdots\le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail with tail index $\gamma$. Given known constants $d_{i,n}$, $1\le i\le n$, consider the weighted power sums $\sum^{k_n}_{i=1}d_{n+1-i,n}\log^pX_{n+1-i,n}$, where $p>0$ and the $k_n$ are positive integers such that $k_n\to\infty$ and $k_n/n\to0$ as $n\to\infty$. Under some constraints on the weights $d_{i,n}$, we prove asymptotic normality for the power sums over the whole heavy-tail model. We apply the obtained result to construct a new class of estimators for the parameter $\gamma$.

A New Class of Estimators Based on a General Relative Loss Function

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

Use of some conventional and non-conventional parameters for improving the efficiency of ratio-type estimators

In the extant calligraphy, we ponder the prominent estimation of population mean of peppermint yield by dint of a new class of estimators. We suggest class of estimators employing known non-traditional parameters of auxiliary variable along with some traditional as well. We study the sampling properties of the proposed class of estimators up to approximation of order one using Taylor’s series expansion. We compare the suggested class of estimators with the competing class of estimators using auxiliary variable. The conditions under which proposed class is more efficient then the competing estimators are obtained. These conditions of efficiency are verified using primary data collected from Siddhaur Block of Barabanki District of Uttar Pradesh State in INDIA. The mean squared errors of suggested and competing classes are calculated to compare them. We used a numerical example in environment engineering to support these theoretical results, thus enriching the sampling theory by the development of new estimators.

On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation

Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional U-statistics of In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. Our theorems allow data-driven local bandwidths for these statistics. The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in copula regression analysis. The performance of these procedures is evaluated through simulation in the context of the conditional Kendall's tau.

New class of estimators for enhanced estimation of average production of peppermint yield utilising known auxiliary variable

New class of estimators for enhanced estimation of average production of peppermint yield utilising known auxiliary variable