Convergence Rate Of Estimator Research Articles

Minimax Rates of ℓp-Losses for High-Dimensional Linear Errors-in-Variables Models over ℓq-Balls.

In this paper, the high-dimensional linear regression model is considered, where the covariates are measured with additive noise. Different from most of the other methods, which are based on the assumption that the true covariates are fully obtained, results in this paper only require that the corrupted covariate matrix is observed. Then, by the application of information theory, the minimax rates of convergence for estimation are investigated in terms of the -losses under the general sparsity assumption on the underlying regression parameter and some regularity conditions on the observed covariate matrix. The established lower and upper bounds on minimax risks agree up to constant factors when , which together provide the information-theoretic limits of estimating a sparse vector in the high-dimensional linear errors-in-variables model. An estimator for the underlying parameter is also proposed and shown to be minimax optimal in the -loss.

Comparison of four bio-inspired algorithms to optimize KNEA for predicting monthly reference evapotranspiration in different climate zones of China

Accurate estimation of reference crop evapotranspiration (ET0) is of great significance to crop water use and agricultural water resources management. This study evaluated the performance of four bio-inspired algorithm optimized kernel-based nonlinear extension of Arps decline (KNEA) models, namely KNEA with grasshopper optimization algorithm (GOA-KNEA), KNEA with grey wolf optimizer algorithm (GWO-KNEA), KNEA with particle swarm optimization algorithm (PSO-KNEA), and KNEA with salp swarm algorithm (SSA-KNEA), on estimating monthly ET0 across China. Monthly meteorological data [including maximum air temperature (Tmax), minimum air temperature (Tmin), extra-terrestrial solar radiation (Ra), relative humidity (RH), global solar radiation (Rs), wind speed (U)] during 1966–2015 from 51 weather stations across the seven different climate zones of China were used for model training and testing. Four different combinations of meteorological data were applied as model input, and results from the FAO-56 Penman-Monteith formula were used as a control. Results showed that the GWO-KNEA model overall performed better than the other three coupling models, of which the GWO-KNEA2 model (i.e., the model with input combination 2) was the best (on average R2 = 0.9814, RMSE = 0.2143 mm d−1). The convergence rate and the population size of the GWO-KNEA model were also superior to the other three models. Among input combinations, models with combination 2 had the best overall performance, while models with combination 3 were the worst on average. In terms of the importance of each meteorological parameter contributing to model accuracy, Rs was greater than Ra, RH, or U. Among different climate zones, station specific models in the semi-arid steppe of Inner Mongolia showed the best estimating performance in general, while models in the Qinghai-Tibetan Plateau overall performed relatively poorly. The GOA-KNEA and SSA-KNEA models with the combination 4 showed large increases (29.7% and 28.7%, respectively), indicating a problem of overfitting. Conversely, the GWO-KNEA model was the most stable model with the smallest increase in RMSE during the testing phase (4.4–15.7%) among all models. In summary, taking the model's estimation accuracy, stability, convergence rate, and climate environment into account, our results suggest that GWO-KNEA would be a suitable model for estimating ET0 in different climate zones of China and the most practical input combination would be combination 2.

Perbandingan Estimator Histogram dan Estimator Kernel

Salah satu hal penting dalam analisis statistik adalah prosedur estimasi suatu fungsi padat peluang yang biasa disebut fungsi densitas. Ada dua metode pendekatan yang biasanya digunakan, yaitu pendekatan parameter yang terkait dengan asumsi distribusi tertentu dan metode estimasi densitas secara non parametrik. Metode non parametrik yang sering kita jumpai adalah metode histogram.
 Beberapa kelemahan metode histogram menjadi acuan untuk dikembangkannya metode yang lain yaitu metode kernel, dimana estimator densitas kernel ini mempunyai parameter yang perlu diestimasi yaitu bandwidth h. Yang menjadi rumusan masalah dalam kajian pustaka ini adalah bagaimana memilih bandwidth dari estimator densitas kernel pada suatu fungsi densitas f di R dan membandingkannya dengan estimator histogram.
 Kesimpulan yang dapat diambil antara lain: (1) Estimasi densitas histogram adalah:
 dengan , (2) Estimator densitas kernel adalah :
 , (3) Estimator kernel mengatasi kelemahan- kelemahan dari estimator histogram, (4) Tingkat konvergensi dari estimator kernel lebih baik dari pada histogram, (5) Pemilihan bandwidth dengan metode unbiased cross validation (least cross validation) secara asymtotik menghasilkan bandwidth yang optimum

One step ahead adaptive control technique featured by fuzzy-based least square estimator for wind systems

The present paper proposes the application to a wind system of the One Step Ahead control scheme featured by a Fuzzy-based Least Square Estimator. The considered wind system power generation supplies an electrical load disconnected from the power supply grid. It is composed of a three bladed horizontal-axis wind turbine which drives a synchronous generator by means of gearbox: the mathematical model for both the horizontal-axis wind-turbine and the synchronous generator will be briefly outlined. The adaptive nature of the One-Step-Ahead control algorithm relies on providing consistent estimation of the controlled system. This is achieved by means of a Least Square Algorithm that is able to provide a good estimation of a linear discrete time model of the controlled system, nevertheless its convergence rate reduces very quickly. For this reason, Least Square Algorithm needs a resetting strategy, which allows the achievement of a compromise between estimation accuracy and convergence rate. This not only represents a very problem-dependent issue but also introduces weaknesses in term of control tracking errors, which in turns needs an extra control contribution (integral correction). The proposed Least Square Algorithm enhancement overcomes these issues managing differently the estimation accuracy and the convergence rate. In the Results section, the achievements of the application of the One-Step-Ahead algorithm to the wind system will prove the reliability of the suggested enhanced control technique.

Homogeneity Test of Many-to-One Risk Differences for Correlated Binary Data under Optimal Algorithms

In clinical studies, it is important to investigate the effectiveness of different therapeutic designs, especially, multiple treatment groups to one control group. The paper mainly studies homogeneity test of many-to-one risk differences from correlated binary data under optimal algorithms. Under Donner’s model, several algorithms are compared in order to obtain global and constrained MLEs in terms of accuracy and efficiency. Further, likelihood ratio, score, and Wald-type statistics are proposed to test whether many-to-one risk differences are equal based on optimal algorithms. Monte Carlo simulations show the performance of these algorithms through the total averaged estimation error, SD, MSE, and convergence rate. Score statistic is more robust and has satisfactory power. Two real examples are given to illustrate our proposed methods.

Convergence rate of estimators of clustered panel models with misclassification

We study kmeans clustering estimation of panel data models with a latent group structure and N units and T time periods under long panel asymptotics. We show that the group-specific coefficients can be estimated at the parametric NT-rate even if error variances diverge as T→∞ and consequently some units are asymptotically misclassified. This limit case approximates empirically relevant settings and is not covered by existing asymptotic results.

Time-varying threshold cointegration with an application to the Fisher hypothesis

Abstract This paper extends the threshold cointegration model developed by Gonzalo, J., and J. Y. Pitarakis. 2006. “Threshold Effects in Cointegrating Relationships.” Oxford Bulletin of Economics & Statistics 68: 813–33 and Chen, H. 2015. “Robust Estimation and Inference for Threshold Models with Integrated Regressors.” Econometric Theory 31 (4): 778–810 to allow for a time-varying threshold, which is a function of candidate variables that affect the separation of regimes. We derive the asymptotic distribution of the proposed least-square estimator of the threshold, and study the convergence rate of the threshold estimator. We also suggest test statistics for threshold effect and threshold constancy. Monte Carlo simulations point out that the convergence rate of the threshold estimator is consistent with the asymptotic theory, and the proposed tests have good size and power properties. The empirical usefulness of the proposed model is illustrated by an application to the US data to investigate the Fisher hypothesis.

ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

Robust estimation of superhedging prices

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with $d$ traded assets. We introduce a plug-in estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this, we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We then extend our study, in part, to estimation of risk measures, to the case of markets with traded options, to a multi-period setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of super-hedging strategies.

Sequential Shrinkage Estimate for COX Regression Models with Uncertain Number of Effective Variables

In the applications of COX regression models, we always encounter data sets that contain too many variables that only a few of them contribute to the model. Therefore, it will waste much more samples to estimate the “noneffective” variables in the inference. In this paper, we use a sequential procedure for constructing the fixed size confidence set for the “effective” parameters to the model based on an adaptive shrinkage estimate such that the “effective” coefficients can be efficiently identified with the minimum sample size. Fixed design is considered for numerical simulation. The strong consistency, asymptotic distributions and convergence rates of estimates under the fixed design are obtained. In addition, the sequential procedure is shown to be asymptotically optimal in the sense of Chow and Robbins (1965).

Uniform Convergence Rates for Nonparametric Estimators of a Density Function and its Derivatives when the Density has a Known Pole

Uniform Convergence Rates for Nonparametric Estimators of a Density Function and its Derivatives when the Density has a Known Pole

Parameter estimation for SPDEs based on discrete observations in time and space

Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in both coordinates, we prove central limit theorems for realized quadratic variations based on temporal and spatial increments as well as on double increments in time and space. Resulting method of moments estimators for the diffusivity and the volatility parameter inherit the asymptotic normality and can be constructed robustly with respect to the sampling frequencies in time and space. Upper and lower bounds reveal that in general the optimal convergence rate for joint estimation of the parameters is slower than the usual parametric rate. The theoretical results are illustrated in a numerical example.

Convergence rates for Bayesian estimation and testing in monotone regression

Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the estimation of a monotone regression function and testing for monotonicity. We construct a prior distribution using piecewise constant functions. For estimation, a prior imposing monotonicity of the heights of these steps is sensible, but the resulting posterior is harder to analyze theoretically. We consider a “projection-posterior” approach, where a conjugate normal prior is used, but the monotonicity constraint is imposed on posterior samples by a projection map onto the space of monotone functions. We show that the resulting posterior contracts at the optimal rate n−1∕3 under the L1-metric and at a nearly optimal rate under the empirical Lp-metrics for 0<p≤2. The projection-posterior approach is also computationally more convenient. We also construct a Bayesian test for the hypothesis of monotonicity using the posterior probability of a shrinking neighborhood of the set of monotone functions. We show that the resulting test has a universal consistency property and obtain the separation rate which ensures that the resulting power function approaches one.

Complete consistency and convergence rate of the nearest neighbor estimator of the density function based on WOD samples

Complete consistency and convergence rate of the nearest neighbor estimator of the density function based on WOD samples

Antithetical stratified sampling estimator for filtering signals with discontinuities

A novel approach to signal filtering using digital alias-free signal processing (DASP) is presented in this paper. We propose an unbiased, fast-converging estimator of the output of a finite impulse response (FIR) continuous-time filter. The estimator processes 2N signal samples collected with the use of random antithetical stratified (AnSt) sampling technique. To assess the estimator convergence rate as a function of N, we consider various forms of smoothness of the input signal, filter impulse response and windowing function. The cases are piecewise-continuous second-order derivative (SOD), piecewise-continuous first-order derivative (FOD) and piecewise-continuous zero-order derivative (ZOD). In each case we assume that the respective derivative has a finite number of bounded discontinuities. We prove that the proposed estimator converges to the true filter output at the rate of N−5 in the first case. But for the other two the rate drops to N−4 and N−2 respectively.

Invariant density adaptive estimation for ergodic jump–diffusion processes over anisotropic classes

We consider the solution X=(Xt)t≥0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of observations XT=(Xt)0≤t≤T is available.In the case without jumps, Dalalyan and Reiss (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic Hölder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation.We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d≥2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method (Goldenshluger and Lepski, 2011) which leads us to an adaptive non-parametric kernel estimator of the stationary density μ of the jump–diffusion X.

Multiple block sizes and overlapping blocks for multivariate time series extremes

Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice, however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead of disjoint blocks leads to a uniform improvement in the asymptotic variance of the multivariate empirical distribution function of rescaled block maxima and any smooth functionals thereof (such as the empirical copula), without any sacrifice in the asymptotic bias. We further derive functional central limit theorems for multivariate empirical distribution functions and empirical copulas that are uniform in the block size parameter, which seems to be the first result of this kind for estimators based on block maxima in general. The theory allows for various aggregation schemes over multiple block sizes, leading to substantial improvements over the single block length case and opens the door to further methodology developments. In particular, we consider bias correction procedures that can improve the convergence rates of extreme-value estimators and shed some new light on estimation of the second-order parameter when the main purpose is bias correction.

Block bootstrap optimality and empirical block selection for sample quantiles with dependent data

Summary We establish a general theory of optimality for block bootstrap distribution estimation for sample quantiles under mild strong mixing conditions. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the sub- sampling bootstrap and the moving block bootstrap, in which the number of blocks is between 1 and the ratio of sample size to block length. The hybrid block bootstrap is shown to give theoretical benefits, and startling improvements in accuracy in distribution estimation in important practical settings. The conclusion that bootstrap samples should be of smaller size than the original sample has significant implications for computational efficiency and scalability of bootstrap methodologies with dependent data. Our main theorem determines the optimal number of blocks and block length to achieve the best possible convergence rate for the block bootstrap distribution estimator for sample quantiles. We propose an intuitive method for empirical selection of the optimal number and length of blocks, and demonstrate its value in a nontrivial example.

Adaptive elastic-net selection in a quantile model with diverging number of variable groups

In real applications of the linear model, the explanatory variables are very often naturally grouped, the most common example being the multivariate variance analysis. In the present paper, a quantile model with group structure is considered, where the number of groups can diverge with the sample size. We introduce and study the adaptive elastic-net group quantile estimator, for improving the parameter estimation accuracy. This method allows automatic selection, with a probability converging to 1, of the non-zero coefficient vectors and further, the asymptotic normality of the non-zero parameter estimators. We also give the convergence rate of the adaptive elastic-net group quantile estimator, rate which depends on the number of the groups. In order to put the estimation method into practice, an algorithm based on the subgradient method is proposed and implemented. The performed Monte Carlo simulations show that the adaptive elastic-net group quantile estimations are more accurate than other existing group estimations in the literature. Moreover, the numerical study confirms the theoretical results and the usefulness of the proposed estimation method.

Uniform convergence rate of the nonparametric maximum likelihood estimator for current status data with competing risks

ABSTRACT We study the uniform convergence rate of the nonparametric maximum likelihood estimator (MLE) for the sub-distribution functions of failure times in the current status data with competing risks model. It is known that the MLE have -norm convergence rate in the absolutely continuous case, but no agreement has emerged on a uniform convergence rate. We specify conditions under which is the uniform convergence rate for the MLE of the sub-distribution functions on finite intervals. The obtained result refines known uniform convergence rate in the particular case of current status data. The main result is applied in order to get the uniform convergence rate of the MLE for the survival function of failure time in the current status right-censored data model.