Constructing Confidence Intervals Research Articles

Retraction: A New Approach to Constructing Confidence Intervals for Population Means Based on Small Samples.

Retraction: A New Approach to Constructing Confidence Intervals for Population Means Based on Small Samples.

Estimation of process performance index for the two-parameter exponential distribution with measurement error

Measurement errors are inevitable in practice, but they are not considered in the existing process performance index. Therefore, we propose an estimation method of process performance index for the two-parameter exponential distribution with measurement errors to fill this gap. In this paper, the relationship between the unobservable actual value and measurement value is considered as full error model, and the maximum likelihood estimation method is considered to obtain the unknown parameters. In addition, we also use the Bootstrap method to construct confidence intervals of process performance index. The performance of the proposed estimation is investigated in terms of bias, mean square error (MSE) and average interval length. Simulation results show that the proposed estimator outperforms other estimators. Finally, an example of the mileage data of the military personnel carrier is given to illustrate the implementation of the proposed estimation method.

Double-Estimation-Friendly Inference for High-Dimensional Misspecified Models

All models may be wrong—but that is not necessarily a problem for inference. Consider the standard t-test for the significance of a variable X for predicting response Y while controlling for p other covariates Z in a random design linear model. This yields correct asymptotic type I error control for the null hypothesis that X is conditionally independent of Y given Z under an arbitrary regression model of Y on (X,Z), provided that a linear regression model for X on Z holds. An analogous robustness to misspecification, which we term the “double-estimation-friendly” (DEF) property, also holds for Wald tests in generalised linear models, with some small modifications. In this expository paper, we explore this phenomenon, and propose methodology for high-dimensional regression settings that respects the DEF property. We advocate specifying (sparse) generalised linear regression models for both Y and the covariate of interest X; our framework gives valid inference for the conditional independence null if either of these hold. In the special case where both specifications are linear, our proposal amounts to a small modification of the popular debiased Lasso test. We also investigate constructing confidence intervals for the regression coefficient of X via inverting our tests; these have coverage guarantees even in partially linear models where the contribution of Z to Y can be arbitrary. Numerical experiments demonstrate the effectiveness of the methodology.

The use of fractional order statistics for estimating nonparametric confidence intervals for quantiles of the fatigue damage computed in service random loadings

This paper deals with the statistical variability of fatigue damage in random loadings as caused by the inherent randomness of the loading. It examines the use of a nonparametric method based on fractional order statistics as a tool for constructing confidence intervals for the quantile of fatigue damage, regardless of the knowledge of the damage probability distribution. The approach here discussed relies on a statistical method existing in the literature. After a short review of this method, with also an emphasis on possible advantages and limitations, the paper presents a numerical benchmark, followed by a practical application to time-histories measured on a mountain-bike in off-road tracks. The paper concludes with some critical remarks and an outlook.

Uncertainty quantification in the Bradley–Terry–Luce model

Abstract The Bradley–Terry–Luce (BTL) model is a benchmark model for pairwise comparisons between individuals. Despite recent progress on the first-order asymptotics of several popular procedures, the understanding of uncertainty quantification in the BTL model remains largely incomplete, especially when the underlying comparison graph is sparse. In this paper, we fill this gap by focusing on two estimators that have received much recent attention: the maximum likelihood estimator (MLE) and the spectral estimator. Using a unified proof strategy, we derive sharp and uniform non-asymptotic expansions for both estimators in the sparsest possible regime (up to some poly-logarithmic factors) of the underlying comparison graph. These expansions allow us to obtain: (i) finite-dimensional central limit theorems for both estimators; (ii) construction of confidence intervals for individual ranks; (iii) optimal constant of $\ell _2$ estimation, which is achieved by the MLE but not by the spectral estimator. Our proof is based on a self-consistent equation of the second-order remainder vector and a novel leave-two-out analysis.

On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models

Most threshold models to-date contain a single threshold variable. However, in many empirical applications, models with multiple threshold variables may be needed and are the focus of this article. For the sake of readability, we start with the Two-Threshold-Variable Autoregressive (2-TAR) model and study its Least Squares Estimation (LSE). Among others, we show that the respective estimated thresholds are asymptotically independent. We propose a new method, namely the weighted Nadaraya-Watson method, to construct confidence intervals for the threshold parameters, that turns out to be, as far as we know, the only method to-date that enjoys good probability coverage, regardless of whether the threshold variables are endogenous or exogenous. Finally, we describe in some detail how our results can be extended to the K-Threshold-Variable Autoregressive (K-TAR) model, K > 2. We assess the finite-sample performance of the LSE by simulation and present two real examples to illustrate the efficacy of our modeling.

Statistical Inference for Aggregation of Malmquist Productivity Indices

A Comprehensive Set of Asymptotic Properties for a Meaningful Aggregation of Malmquist Indices The Malmquist productivity index (MPI) has become one of the most widely used tools for analyzing dynamic performance of decision-making units. Whereas accounting for economic weights of individual units in aggregations of indices is emphasized in the literature, statistical theory for constructing confidence intervals and performing hypothesis tests based on weighted aggregation of the MPI are still unavailable. In “Statistical Inference for Aggregation of Malmquist Productivity Indices,” Pham, Simar, and Zelenyuk use a novel approach (based on the uniform delta method) to develop new asymptotic theory (including new central limit theorems) for aggregate MPIs as the basis for the statistical inference and test. They also verify the finite-sample performance of their approach via extensive Monte Carlo experiments and provide an illustration using real-world data.

A resampling approach for confidence intervals in linear time-series models after model selection

Linear models are commonly used in econometrics to describe structural relations and draw inferences about underlying parameters. Although the theoretical and practical value of these models has been justified under ideal circumstances, empirical studies have shown that they did not well fit the observed data in reality, and their selection is therefore an important problem. In the big-data era, one approach is to consider a large set of covariates and apply model selection. However, many model selection methods can provide consistent coefficient estimators and confidence intervals, but perform poorly in practice due to an insufficient sample size. Especially for financial data, which have relatively small sample sizes and are usually correlated, classical methods to construct confidence intervals after model selection are usually inaccurate. We consider the problem of constructing confidence intervals for selected coefficients in linear time-series models. To overcome the difficulties of post-selection confidence intervals, we introduce consistent estimators of the selected parameters and work on a resampling approach for confidence interval construction. Theory, simulations, and empirical studies demonstrate the advantages of the proposed method.

Adjusting for publication bias in meta-analysis via inverse probability weighting using clinical trial registries.

Publication bias is a major concern in conducting systematic reviews and meta-analyses. Various sensitivity analysis or bias-correction methods have been developed based on selection models, and they have some advantages over the widely used trim-and-fill bias-correction method. However, likelihood methods based on selection models may have difficulty in obtaining precise estimates and reasonable confidence intervals, or require a rather complicated sensitivity analysis process. Herein, we develop a simple publication bias adjustment method by utilizing the information on conducted but still unpublished trials from clinical trial registries. We introduce an estimating equationfor parameter estimation in the selection function by regarding the publication bias issue as a missing data problem under the missing not at random assumption. With the estimated selection function, we introduce the inverse probability weighting (IPW) method to estimate the overall mean across studies. Furthermore, the IPW versions of heterogeneity measures such as the between-study variance and the I2 measure are proposed. We propose methods to construct confidence intervals based on asymptotic normal approximation as well as on parametric bootstrap. Through numerical experiments, we observed that the estimators successfully eliminated bias, and the confidence intervals had empirical coverage probabilities close to the nominal level. On the other hand, the confidence interval based on asymptotic normal approximation is much wider in some scenarios than the bootstrap confidence interval. Therefore, the latter is recommended for practicaluse.

О получении стохастических прогнозов в детерминированной модели банковской системы России

In this article we propose the method of obtaining forecasts in stochastic terms for deterministic models. The proposed method is computationally simpler than the one used in dyna­mic stochastic general equilibrium (DSGE) models. The method is based on the estimation of parameters in the deterministic paradigm and the estimation of the vector of sample means and the covariance matrix for the increments of exogeneous variables on the in-sample period. For every realization of exogeneous variables, the trajectories of endogenous variables is calculated. The methods of mathematical statistics such as moments calculation, construction of confidence intervals, testing various hypotheses can be applied to them. The approach is illustrated on the model of the Russian banking system, which successfully reproduces the wide set of its indicators. Several interesting properties of the obtained stochastic forecasts were found, including the violation of their normality and the nontrivial dynamics of confidence intervals. Several scenarios of key rate and exchange rate resembling their actual dynamics in the beginning of the 2022. Several conclusions on the influence of key rate and exchange rate on the basic indicators of the banking systems are made. In particular, several effects are found which could not be discovered in the purely deterministic modelling paradigm.

ПРИМЕНЕНИЕ ИНСТРУМЕНТОВ PYTHON В КУРСЕ СТАТИСТИКИ

The methodological aspects of teaching mathematical statistics in a technical university using laboratory work are considered. Examples of tasks that can be solved using the SciPy and NumPy libraries of the Python language are given. The advantages of this programming language over other computer mathematical systems are substantiated. The problems of generating data from a given distribution, constructing confidence intervals, testing hypotheses, correlation analysis, and some others are considered.

Confidence intervals for the difference between coefficients of variation of zero-inflated gamma distributions

<abstract><p>The problem of constructing confidence intervals (CIs) for the difference between coefficients of variation of two zero-inflated gamma distributions was considered. As gamma distribution does not have closed form maximum likelihood estimators, the parameters of gamma distribution have to be estimated numerically. To this end, we proposed here four different generalized confidence intervals (GCIs) based on fiducial inference, Box-Cox transformation, parametric bootstrap and the method of variance of estimates recovery (MOVER). Performances of the four GCIs were evaluated and compared via extensive simulation. The simulation results showed that all four methods returned satisfactory results according to coverage probabilities, even for the setting of small sample sizes.</p></abstract>

Inference for joint quantile and expected shortfall regression

AbstractQuantiles and expected shortfalls are commonly used risk measures in financial risk management. The two measurements are correlated while having distinguished features. In this project, our primary goal is to develop a stable and practical inference method for the conditional expected shortfall. We consider the joint modelling of conditional quantile and expected shortfall to facilitate the statistical inference procedure. While the regression coefficients can be estimated jointly by minimizing a class of strictly consistent joint loss functions, the computation is challenging, especially when the dimension of parameters is large since the loss functions are neither differentiable nor convex. We propose a two‐step estimation procedure to reduce the computational effort by first estimating the quantile regression parameters with standard quantile regression. We show that the two‐step estimator has the same asymptotic properties as the joint estimator, but the former is numerically more efficient. We develop a score‐type inference method for hypothesis testing and confidence interval construction. Compared to the Wald‐type method, the score method is robust against heterogeneity and is superior in finite samples, especially for cases with many confounding factors. The advantages of our proposed method over existing approaches are demonstrated by simulations and empirical studies based on income and college education data.

УНИВЕРСАЛЬНЫЙ СТАТИСТИЧЕСКИЙ АНАЛИЗ РАСПРЕДЕЛЕНИЙ, ЗАВИСЯЩИХ ОТ ПАРАМЕТРА

The development of the digital economy, the creation of intelligent and neural systems put forward the problem of finding adequate mathematical tools and improving the technological content of probabilistic and statistical methods that serve as the basis for their development. The aim of the study is to improve methods and create new universal algorithms for constructing statistical estimates together with their properties for special classes of distributions. The unification of methods significantly optimizes costs and saves time for obtaining the properties of estimates. The main attention is paid to the presentation of universal algorithms for statistical estimation of parameters of certain classes of distributions and the properties of these estimates, as well as algorithms for constructing confidence intervals and parametric criteria for hypothesis testing. The normal distribution is usually used to estimate the spread of the forecast. But it is not characteristic of all indicators. Therefore, the analysis of distributions of other types is very relevant. The scientific value of the article consists in the creation of universal algorithms for constructing statistical estimates together with their properties for special classes of distributions. The developed tools allow combining the necessary professional competencies of the course students with the basic theoretical knowledge from the basic course of probability theory and mathematical statistics with effective statistical estimation algorithms for practical application in a certain economic field. The acquired knowledge makes it possible to perform various professional tasks based on probabilistic and statistical estimates in various formulations of initial tasks, competently explain the results and conclusions obtained; work with experimental representations using the latest digital and computer technologies. The application of the developed concept leads to the practical implementation of the tasks of finding the optimal solution for various problem statements related to the problems of creating neural and intelligent systems with active elements and having significant initial uncertainty.

Weak Signal Identification and Inference in Penalized Likelihood Models for Categorical Responses

Penalized likelihood models are widely used to simultaneously select variables and estimate model parameters. However, the existence of weak signals can lead to inaccurate variable selection, biased parameter estimation, and invalid inference. Thus, identifying weak signals accurately and making valid inferences are crucial in penalized likelihood models. We develop a unified approach to identify weak signals and make inferences in penalized likelihood models, including the special case when the responses are categorical. To identify weak signals, we use the estimated selection probability of each covariate as a measure of the signal strength and formulate a signal identification criterion. To construct confidence intervals, we propose a two-step inference procedure. Extensive simulation studies show that the proposed procedure outperforms several existing methods. We illustrate the proposed method by applying it to the Practice Fusion diabetes data set.

Functional spherical autocorrelation: A robust estimate of the autocorrelation of a functional time series

We propose a new autocorrelation measure for functional time series that we term “spherical autocorrelation”. It is based on measuring the average angle between lagged pairs of series after having been projected onto the unit sphere. This new measure enjoys several complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of “sign” or “direction” of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and are used to construct confidence intervals and portmanteau white noise tests. These confidence intervals and tests are shown to be effective in simulation experiments, and demonstrated in applications to model selection for daily electricity price curves, and measuring the volatility in densely observed asset price data.

Online confidence interval estimation for federated heterogeneous optimization

From a statistical viewpoint, it is essential to perform statistical inference in federated learning to understand the underlying data distribution. Due to the heterogeneity in the number of local iterations and in local datasets, traditional statistical inference methods are not competent in federated learning. This paper studies how to construct confidence intervals for federated heterogeneous optimization problems. We introduce the rescaled federated averaging estimate and prove the consistency of the estimate. Focusing on confidence interval estimation, we establish the asymptotic normality of the parameter estimate produced by our algorithm and show that the asymptotic covariance is inversely proportional to the client participation rate. We propose an online confidence interval estimation method called separated plug-in via rescaled federated averaging. This method can construct valid confidence intervals online when the number of local iterations is different across clients. Since there are variations in clients and local datasets, the heterogeneity in the number of local iterations is common. Consequently, confidence interval estimation for federated heterogeneous optimization problems is important.

An improved score-type confidence interval for stratified risk differences involving rare events.

A stratified analysis of the differences in proportions has been widely employed in epidemiological research, social sciences, and drug development. It provides a useful framework for combining data across strata to produce a common effect. However, for rare events with incidence rates close to zero, popular confidence intervals for risk differences in a stratified analysis may not have appropriate coverage probabilities that approach the nominal confidence levels and the algorithms may fail to produce a valid confidence interval because of zero events in both the arms of a stratum. The main objective of this study is to evaluate the performance of certain methods commonly employed to construct confidence intervals for stratified risk differences when the response probabilities are close to a boundary value of zero or one. Additionally, we propose an improved stratified Miettinen-Nurminen confidence interval that exhibits a superior performance over standard methods while avoiding computational difficulties involving rare events. The proposed method can also be employed when the response probabilities are close to one.

Global Temperature and Carbon Dioxide Nexus: Evidence from a Maximum Entropy Approach

The connection between Earth’s global temperature and carbon dioxide (CO2) emissions is one of the highest challenges in climate change science since there is some controversy about the real impact of CO2 emissions on the increase of global temperature. This work contributes to the existing literature by analyzing the relationship between CO2 emissions and the Earth’s global temperature for 61 years, providing a recent review of the emerging literature as well. Through a statistical approach based on maximum entropy, this study supports the results of other techniques that identify a positive impact of CO2 in the increase of the Earth’s global temperature. Given the well-known difficulties in the measurement of global temperature and CO2 emissions with high precision, this statistical approach is particularly appealing around climate change science, as it allows the replication of the original time series with the subsequent construction of confidence intervals for the model parameters. To prevent future risks, besides the present urgent decrease of greenhouse gas emissions, it is necessary to stop using the planet and nature as if resources were infinite.

A sampling strategy for habitat selection, mapping, and abundance estimation of deer by pellet counts

AbstractWe propose a design‐based strategy to exploit pellet group counts performed within plots of a prefixed size using the clearance count technique with the purposes of analyzing habitat selection, mapping the pellet group presence throughout the study area, and estimating the abundance of deer populations. As is customary in design‐based inference, the strategy is free of model assumptions, and the precision and statistical consistency of the proposed estimators are determined by the probabilistic sampling scheme adopted to locate plots. The unique necessary assumptions are the absence of migratory movements during the survey period, the accurate recording of the number of pellet groups deposited within sample plots between the 2 visits, and a precise approximation of the daily defecation rate of the population. In addition to these assumptions, which can be attained by a suitable design of the survey, the statistical soundness of the strategy rests on the use of tessellation stratified sampling, a stratified sampling scheme that ensures an even distribution of plots throughout the study area. The scheme also allows for the estimation of the standard errors and the construction of confidence intervals without involving any other assumptions. We applied this strategy in summer 2019 in a protected area of a Mediterranean coastal region to estimate the density of a fallow deer (Dama dama) population. We estimated the corresponding standard error considering the uncertainty entailed by the estimation of the daily defecation rate, with the purpose of performing reliable monitoring. The proposed strategy provided precise estimates of deer abundance and is readily implementable in the field, standardized, and easily repeatable over time, thus allowing reliable monitoring and comparisons across time and space, which are fundamental attributes for management of deer populations.