Constructing Confidence Intervals Research Articles

Confidence Intervals for Mean and Difference between Means of Delta-Lognormal Distributions Based on Left-Censored Data

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution is used in medical and environmental sciences. This study considers the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare two daily rainfall average areas in Thailand. Three different approaches for constructing confidence intervals for the mean of the delta-lognormal distribution containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, and the parametric bootstrap approach, are developed. Moreover, four different approaches for constructing confidence intervals for the difference between means of delta-lognormal distributions containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, the parametric bootstrap approach, and the method of variance estimates recovery approach, are considered. The performance of the proposed confidence intervals is evaluated by Monte Carlo simulation. The simulation studies indicate that the Bayesian approach can be considered as an alternative to construct a credible interval for the mean of the delta-lognormal distribution containing left-censored data. Additionally, the generalized confidence interval and Bayesian approaches can be recommended as alternatives to estimate the confidence interval for the difference between means of delta-lognormal distributions containing left-censored data. All approaches are illustrated using the daily rainfall data from Chiang Mai and Lampang provinces in Thailand.

Confidence intervals for the cross product ratio under the special case of direct-inverse sampling scheme and its applications

This article focuses on the estimation of the cross-product ratio under so-called special case of the direct-inverse sampling scheme, where the number of successes in the direct sampling scheme is used in the second sampling scheme of the inverse binomial scheme. Asymptotic confidence intervals are constructed. Our goal is to investigate the cases when the normal approximations for estimators of the cross-product ratio are reliable for the construction of confidence intervals. We use the closeness of the confidence coefficient to the nominal confidence level as our main evaluation criterion, and use the Monte-Carlo method to investigate the key probability characteristics of intervals. We present estimations of the coverage probability and interval width in tables. In the last section, Cytochrome Psychotropic Genotyping Under Investigation for Decision Support case study is discussed where the standard and genetically guided therapy is compared and estimates for the cross-product ratio are presented and interpreted when the participants are enrolled according to the special case of the direct-inverse sampling scheme.

Neighborhood-based cross fitting approach to treatment effects with high-dimensional data

High-dimensional data are increasingly popular in various physical, biological and social disciplines. A common existing approach of repeatedly splitting data was suggested to address the overfitting problem in high-dimensional statistics, however it is computationally expensive in high dimensions. A computationally efficient data splitting method is proposed and referred to as Neighborhood-Based Cross Fitting (NBCF) double machine learning in causal inference for structural causal models with high-dimensional data. The proposed method deals well with the problem of post-selection bias in causal inference in the presence of high-dimensional confounding. It provides an equivalent basis in unbiased estimation as repeated data splitting, which is suggested to expand the complexity scope of function class by empirical process methods. Numerical simulation studies were conducted to demonstrate that the proposed neighborhood-based approach is not only more computationally efficient than the existing sample splitting methods, but also better in bias reduction compared with other existing methods. Under certain conditions, simulation results further showed that the proposed estimators are asymptotically unbiased and normally distributed, which allows construction of valid confidence intervals. The practical application of NBCF is illustrated with a real dataset.

Seq2R: An R Package to Detect Change Points in DNA Sequences

Identifying the mutational processes that shape the nucleotide composition of the mitochondrial genome (mtDNA) is fundamental to better understand how these genomes evolve. Several methods have been proposed to analyze DNA sequence nucleotide composition and skewness, but most of them lack any measurement of statistical support or were not developed taking into account the specificities of mitochondrial genomes. A new methodology is presented, which is specifically developed for mtDNA to detect compositional changes or asymmetries (AT and CG skews) based on nonparametric regression models and their derivatives. The proposed method also includes the construction of confidence intervals, which are built using bootstrap techniques. This paper introduces an R package, known as seq2R, that implements the proposed methodology. Moreover, an illustration of the use of seq2R is provided using real data, specifically two publicly available complete mtDNAs: the human (Homo sapiens) sequence and a nematode (Radopholus similis) mitogenome sequence.

A frequentist approach to dynamic borrowing.

There has been growing interest in leveraging external control data to augment a randomized control group data in clinical trials and enable more informative decision making. In recent years, the quality and availability of real-world data have improved steadily as external controls. However, information borrowing by directly pooling such external controls with randomized controls may lead to biased estimates of the treatment effect. Dynamic borrowing methods under the Bayesian framework have been proposed to better control the false positive error. However, the numerical computation and, especially, parameter tuning, of those Bayesian dynamic borrowing methods remain a challenge in practice. In this paper, we present a frequentist interpretation of a Bayesian commensurate prior borrowing approach and describe intrinsic challenges associated with this method from the perspective of optimization. Motivated by this observation, we propose a new dynamic borrowing approach using adaptive lasso. The treatment effect estimate derived from this method follows a known asymptotic distribution, which can be used to construct confidence intervals and conduct hypothesis tests. The finite sample performance of the method is evaluated through extensive Monte Carlo simulations under different settings. We observed highly competitive performance of adaptive lasso compared to Bayesian approaches. Methods for selecting tuning parameters are also thoroughly discussed based on results from numerical studies and an illustrationexample.

Statistical Inference of Two Gamma Distributions under the Joint Type-II Censoring Scheme

The joint Type-II censoring scheme is a useful model when carrying out comparative lifecycle tests of units from various production lines. This article takes into account the estimation problem of the joint Type-II censoring data coming from two Gamma distributions with the same shape parameter but various scale parameters. The maximum likelihood estimators of the parameters from Gamma populations and asymptotic confidence intervals based on the observed Fisher information matrix are obtained. Bootstrap methods are also applied to construct confidence intervals. The Metropolis–Hastings algorithm is considered to draw Markov Chain Monte Carlo samples when computing Bayesian estimates as well as establishing the corresponding credible intervals. Monte Carlo simulations are adopted to compare the performance of the estimates. Finally, two real engineering datasets are analyzed.

A nonparametric approach for setting safety stock levels

In practice, lead time demand (LTD) can be skewed, multi‐modal or highly variable, and these factors compromise the validity of typical approaches used for setting safety stock levels. Motivated by encountering this problem at our industry partner, we develop an approach for setting safety stock levels using the bootstrap, a widely used statistical procedure. Existing bootstrap approaches for inventory management either operate directly on observed LTD or assume deterministic lead times, permitting direct application of the bootstrap approach for univariate quantile estimation. As LTD is a convolution of multiple random demands over a random lead time, a multivariate bootstrap approach is required. As we demonstrate, when lead times are stochastic, our multivariate approach provides improved safety stock estimates. We develop a multivariate central limit theorem for the bootstrap mean and bootstrap quantile—components of the safety stock calculation—highlighting why the generalization of these bootstrap methods is critical for inventory management. These results provide a theoretical underpinning for the bootstrap estimator of safety stock and permit the construction of confidence intervals for safety stock estimates, allowing decision makers to understand the reliability with which the desired service level will be achieved. Building on our theoretical results, and supported by numerical experiments, we provide insights on the behavior of the bootstrap for various LTD distributions, which our results demonstrate are critical when employing the bootstrap method. Implementation of our approach with our industry partner resulted in an inventory investment reduction of $1.17 million combined with an overall increase in service level. Our approach is general and can be implemented without modification in other settings.

Bayesian confidence intervals for the ratio of the means of zero-inflated gamma distributions with application to rainfall data

Herein, we propose eight Bayesian methods (credible and highest posterior density intervals using the Jeffreys rule, independence Jeffreys, uniform, or normal-gamma priors) and one based on the fiducial quantity for constructing confidence intervals for the ratio of the means of zero-inflated gamma distributions. Simulation studies were conducted to evaluate the efficacies of the confidence intervals and compare their performances in terms of coverage probabilities and expected lengths, in which the Bayesian highest posterior density interval using the Jeffreys rule prior performed the best. We also applied the methods to rainfall datasets from northern Thailand to demonstrate their efficacies with real data.

Tail Spectral Density Estimation and Its Uncertainty Quantification: Another Look at Tail Dependent Time Series Analysis

We consider the estimation and uncertainty quantification of the tail spectral density, which provide a foundation for tail spectral analysis of tail dependent time series. The tail spectral density has a particular focus on serial dependence in the tail, and can reveal dependence information that is otherwise not discoverable by the traditional spectral analysis. Understanding the convergence rate of tail spectral density estimators and finding rigorous ways to quantify their statistical uncertainty, however, still stand as a somewhat open problem. The current article aims to fill this gap by providing a novel asymptotic theory on quadratic forms of tail statistics in the double asymptotic setting, based on which we develop the consistency and the long desired central limit theorem for tail spectral density estimators. The results are then used to devise a clean and effective method for constructing confidence intervals to gauge the statistical uncertainty of tail spectral density estimators, and it can be turned into a visualization tool to aid practitioners in examining the tail dependence for their data of interest. Numerical examples including data applications are presented to illustrate the developed results.

Original article Forecasting trends in the development of time series by estimating the Hodges-Lehman median

Introduction. The lack of stationarity in the time series’ development, small samples, the outliers presence, jumps do not allow to find estimates of model parameters that have good properties of statistical estimates and, as a result, to find reliable forecasts, both in the development trend of the process and in its numerical expression. The means to solve these problems is the use of ordinal or robust statistics. Scientific monographs and special chapters in books on mathematical statistics contain a deep and extensive theory on the study of the properties of ordinal statistics, which are the justification for their application in forecasting methods. The aim of the work is to develop and verify method for obtaining one-step forecasts of trends in the development of time series based on stable statistical estimates.Materials and methods. The article presents the results of the development of a method for obtaining one-step forecasts of trends in the development of time series based on the construction of confidence intervals of a selective stable HodgesLehman estimate based on Walsh averages. In particular, the Hodges-Lehman median is used to solve the problem of small samples obtained during the procedure of shifting the time series window. The proposed method is considered in detail in the article: the basic definitions, the theoretical justification of the method, calculation formulas, a detailed description of the algorithm, formulas for calculating the quality metric of forecasts are given.The results of the study. The method was implemented in computational experiment using the example of forecasting the Urals crude oil’s spot price. The article presents the computational experiment’s results. The parameters of the proposed method can be configured to obtain reliable one-step forecasts.Discussion and conclusions. The method proposed in the article has shown its effectiveness on experimental data and can be used as an independent method for constructing one-step forecasts of trends in the development of time series. Further development of the method involves the improvement of computational procedures, verification of the method in case of jumps in the dynamics of the time series.

Prediction Models for the Plant Coverage Percentage of a Vertical Green Wall System: Regression Models and Artificial Neural Network Models

(1) Background: The expansion that most cities have been showing for more than half a century has also brought with it an increase in the density of buildings, most of the time at the expense of green areas. This has led to negative effects, such as overpopulation of cities, rising urban temperatures, pollution of water, air, soil, and others, affecting daily urban life. As a result, specialists from different fields form multidisciplinary teams are looking for solutions to counteract these effects. The subject of visible facades has registered an increased interest among researchers in recent years because they can represent a viable solution that can contribute to increasing the degree of urban comfort. However, for such a system to be effective, it is necessary that the plants used grow and develop harmoniously and ensure the best possible coverage of the facade. The aim of this research is to find an adequate mathematical model that can predict, with a high degree of accuracy, the percentage of plant coverage of a green wall system, which is positioned in the city of Iasi, northeastern Romania. (2) Methods: The models used for this purpose were a multiple linear regression model (MLR) and a model based on a feed-forward artificial neural network (ANN). Four independent variables (soil temperature, soil moisture, week of the year, and cardinal wall orientation) and the interaction between two variables (soil temperature and week of the year) were used for the multiple linear regression model. Artificial neural networks were also trained to estimate the percentage of plant coverage in the analyzed system, and the network with the best mean squared error performance was chosen in doing predictions. For both MLR and ANN models, we constructed confidence intervals for the degree of plant coverage of the system (PCP) for a set of observed values. In the case of the ANN model, the confidence interval was derived via the bootstrap method, which is a resampling with replacement technique used to generate new samples from the original dataset. To the best of our knowledge, the derivation of confidence intervals using a combination of neural networks with the bootstrap method has not been used before, at least for predictions in horticulture. (3) Results: The ANN employed here consisted of one input layer with four neurons, one hidden layer with five neurons, and one output layer with one neuron. The comparison showed that the confidence interval obtained using ANN has a shorter length (and thus it is more accurate) than that obtained by the multiple linear regression model. The choice of the experimental module façade had a significant influence (of magnitude 1.9073) on the plant coverage percentage. An increase of one unit in soil humidity will determine an increase of almost 5.1% in plant coverage percentage, and an increase of 1 °C in soil temperature will determine a decrease of almost 1.21% in plant coverage percentage. The choice of the experimental module façade had a significant influence (of magnitude 1.9073) on the plant coverage percentage. (4) Conclusions: Although both methods showed to be useful in making predictions, the ANN method showed better predictive capabilities, at least when the performance is measured by the mean squared error. This fact may be useful when predicting the percentage of plant coverage of a green wall system with a higher degree of accuracy, in the case of organizing outdoor exhibitions or other similar projects.

Carbon Stock Prediction in Managed Forest Ecosystems Using Bayesian and Frequentist Geostatistical Techniques and New Generation Remote Sensing Metrics

The study compares the performance of a hierarchical Bayesian geostatistical methodology with a frequentist geostatistical approach, specifically, Kriging with External Drift (KED), for predicting C stock using prediction aides from the Landsat-8 and Sentinel-2 multispectral remote sensing platforms. The frequentist geostatistical approach’s reliance on the long-run frequency of repeated experiments for constructing confidence intervals is not always practical or feasible, as practitioners typically have access to a single dataset due to cost constraints on surveys and sampling. We evaluated two approaches for C stock prediction using two new generation multispectral remote sensing datasets because of the inherent uncertainty characterizing spatial prediction problems in the unsampled locations, as well as differences in how the Bayesian and frequentist geostatistical paradigms handle uncertainty. Information on C stock spectral prediction in the form of NDVI, SAVI, and EVI derived from multispectral remote sensing platforms, Landsat-8 and Sentinel-2, was used to build Bayesian and frequentist-based C stock predictive models in the sampled plantation forest ecosystem. Sentinel-2-based C stock predictive models outperform their Landsat-8 counterparts using both the Bayesian and frequentist inference approaches. However, the Bayesian-based Sentinel-2 C stock predictive model (RMSE = 0.17 MgCha−1) is more accurate than its frequentist-based Sentinel-2 (RMSE = 1.19 MgCha−1) C stock equivalent. The Sentinel-2 frequentist-based C stock predictive model gave the C stock prediction range of 1 ≤ MgCha−1 ≤ 290, whilst the Sentinel-2 Bayesian-based C stock predictive model resulted in the prediction range of 1 ≤ MgCha−1 ≤ 285. However, both the Bayesian and frequentist C stock predictive models built with the Landsat-8 sensor overpredicted the sampled C stock because the range of predicted values fell outside the range of the observed C stock values. As a result, we recommend and conclude that the Bayesian-based C stock prediction method, when it is combined with high-quality remote sensing data such as that of Sentinel-2, is an effective inferential statistical methodology for reporting C stock in managed plantation forest ecosystems.

Confidence intervals for the length of the receiver-operating characteristic curve based on a smooth estimator.

A good diagnostic test should show different behavior on both the positive and the negative populations. However, this is not enough for having a good classification system. The binary classification problem is a complex task, which implies to define decision criteria. The knowledge of the level of dissimilarity between the two involved distributions is not enough. We also have to know how to define those decision criteria. The length of the receiver-operating characteristic curve has been proposed as an index of the optimal discriminatory capacity of a biomarker. It is related not with the actual but with the optimal classification capacity of the considered diagnostic test. One particularity of this index is that its estimation should be based on parametric or smoothed models. We explore here the behavior of a kernel density estimator-based approximation for estimating the length of the receiver-operating characteristic curve. We prove the asymptotic distribution of the resulting statistic, propose a parametric bootstrap algorithm for confidence intervals construction, discuss the role that the bandwidth parameter plays in the quality of the provided estimations and, via Monte Carlo simulations, study its finite-sample behavior considering four different criteria for the bandwidth selection. The practical use of the length of the receiver-operating characteristic curve is illustrated through two real-world examples.

Confidence intervals of the process capability index Cpc$C_{pc}$ revisited via modified bootstrap technique and ROC curves

AbstractWe consider the process capability index (PCI), a widely used quality‐related statistic used to assess the quality of products and performance of monitored processes in various industries. It is widely known that the conventional PCIs perform well when the quality process being monitored has a normal distribution. Unfortunately, using the indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, , that can be used in both normal and non‐normal scenarios. The objective of this article is threefold: (i) We provide a corrected form of the confidence interval for . (ii) We compare the performance of three nonparametric bootstrap confidence intervals (BCIs) for . Specifically, the standard bootstrap, percentile bootstrap, and bias‐corrected percentile bootstrap. Under various distributional assumptions such as the normal, chi‐square, Student t, Laplace, and two‐parameter exponential distributions, the estimated coverage probabilities and average width of the confidence intervals and BCIs for are compared. (iii) The power of the respective bootstrap approaches is evaluated by using the equivalence relation between confidence interval construction and two‐sided hypothesis testing. We also provide the receiver operating characteristic curves to evaluate their performance. Finally, as an illustrative example, an actual data set related to groove dimensions (in inches) measured from a manufacturing process of ignition keys is re‐analyzed to illustrate the utility of the proposed methods.

Doubly Robust Capture-Recapture Methods for Estimating Population Size

Estimation of population size using incomplete lists has a long history across many biological and social sciences. For example, human rights groups often construct partial lists of victims of armed conflicts, to estimate the total number of victims. Earlier statistical methods for this setup often use parametric assumptions, or rely on suboptimal plug-in-type nonparametric estimators; but both approaches can lead to substantial bias, the former via model misspecification and the latter via smoothing. Under an identifying assumption that two lists are conditionally independent given measured covariates, we make several contributions. First, we derive the nonparametric efficiency bound for estimating the capture probability, which indicates the best possible performance of any estimator, and sheds light on the statistical limits of capture-recapture methods. Then we present a new estimator, that has a double robustness property new to capture-recapture, and is near-optimal in a nonasymptotic sense, under relatively mild nonparametric conditions. Next, we give a confidence interval construction method for total population size from generic capture probability estimators, and prove nonasymptotic near-validity. Finally, we apply them to estimate the number of killings and disappearances in Peru during its internal armed conflict between 1980 and 2000. Supplementary materials for this article are available online.

Uncertainty Quantification and Exploration for Reinforcement Learning

Quantify the uncertainty to decide and explore better In statistical inference, large-sample behavior and confidence interval construction are fundamental in assessing the error and reliability of estimated quantities with respect to the data noises. In the paper “Uncertainty Quantification and Exploration for Reinforcement Learning”, Dong, Lam, and Zhu study the large sample behavior in the classic setting of reinforcement learning. They derive appropriate large-sample asymptotic distributions for the state-action value function (Q-value) and optimal value function estimations when data are collected from the underlying Markov chain. This allows one to evaluate the assertiveness of performances among different decisions. The tight uncertainty quantification also facilitates the development of a pure exploration policy by maximizing the worst-case relative discrepancy among the estimated Q-values (ratio of the mean squared difference to the variance). This exploration policy aims to collect informative training data to maximize the probability of learning the optimal reward collecting policy, and it achieves good empirical performance.

Shortest Confidence Intervals for Weibull Modulus for Small Samples in Materials Reliability Analysis

The Weibull distribution has been widely used to model strength properties of brittle materials. Estimation of confidence intervals for Weibull shape parameter has been an important concern, since small sample sizes in materials science experiments bring about large intervals. Many methods have been proposed in the literature for constructing shorter intervals; the methods of maximum likelihood, least square, and Menon are among the most extensively studied methods. However, they all use an equal-tails approach. The pivotal quantities used for constructing confidence intervals have right-skewed and unimodal distributions, thus, they clearly do not produce the shortest intervals for a given confidence level in equal tail form. This study constructs the shortest confidence intervals for the three aforementioned methods and compares their performances by their equal-tails counterparts. To this end, a comprehensive simulation study has been conducted for the shape parameter values between 1 to 80 and the sample sizes between 3 to 20. The comparison criterion is chosen as the expected interval length. The results show that the shortest confidence intervals in each of three methods have yielded considerably narrower intervals. Further, the unknown parameter values are more centered in these intervals.

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.