Interval Estimation Method Research Articles

Expected and empirical coverages of different methods for generating noncentral t confidence intervals for a standardized mean difference

Different methods have been suggested for calculating "exact" confidence intervals for a standardized mean difference using the noncentral t distributions. Two methods are provided in Hedges and Olkin (1985, "H") and Steiger and Fouladi (1997, "S"). Either method can be used with a biased estimator, d, or an unbiased estimator, g, of the population standardized mean difference (methods abbreviated Hd, Hg, Sd, and Sg). Coverages of each method were calculated from theory and estimated from simulations. Average coverages of 95% confidence intervals across a wide range of effect sizes and across sample sizes from 5 to 89 per group were always between 85 and 98% for all methods, and all were between 94 and 96% with sample sizes greater than 40 per group. The best interval estimation was the Sd method, which always produced confidence intervals close to 95% at all effect sizes and sample sizes. The next best was the Hg method, which produced consistent coverages across all effect sizes, although coverage was reduced to 93-94% at sample sizes in the range 5-15. The Hd method was worse with small sample sizes, yielding coverages as low as 86% at n = 5. The Sg method produced widely different coverages as a function of effect size when the sample size was small (93-97%). Researchers using small sample sizes are advised to use either the Steiger & Fouladi method with d or the Hedges & Olkin method with g as an interval estimation method.

Prediction interval estimation of landslide displacement using adaptive chicken swarm optimization-tuned support vector machines

Accurate and reliable displacement prediction is vital for developing landslide early warning system, since displacement variations directly reveal the evolution of landslide. The past decade has seen the rapid development of displacement prediction technology. Most of the existing studies are based on deterministic point prediction models, with the goal of obtaining high-precision predictions. However, these pointwise predictions do not give any indication of credibility, which makes it difficult to make accurate decisions for subsequent landslide treatment. To address this problem, a prediction interval estimation method, instead of point prediction, for landslide displacement is proposed. In the methodology, double exponential smoothing is employed to deal with the nonlinear cumulative displacement, while two support vector machines are proposed to directly generate the displacement lower boundary and upper boundary. To obtain optimal model parameters, the adaptive chicken swarm optimization using chaotic mapping and adaptive inertia weight strategies is proposed to minimize a modified prediction interval-based objective function. A real-world landslide displacement data set is used to demonstrate the proposed method. Experimental results show that it is a promising tool for the construction of high-quality prediction intervals, which can inform the landslide treatment-related decision-making.

Critical quality attribute assessment of big brand traditional Chinese medicine: quality control method of Huangjing Zanyu Capsules based on chemical properties of characteristic components

The chemical properties of characteristic components are significant to the manufacturing quality control of big brand traditional Chinese medicine. In this study, the Huangjing Zanyu Capsules were used as the research carrier to determine the content of five characteristic components including icraiin, emodin, schisandrin A, 2,3,5,4'-tetrahydroxystilbene-2-O-β-D-glucoside, and osthole simultaneously by high-performance liquid chromatography(HPLC). The results showed that the chemical properties of five cha-racteristic components had a good linear relationship(r>0.999 9) within the quantitative range; the relative standard deviations(RSD) was 0.11%-2.0% and 0.25%-2.8% respectively for intra-day and inter-day precision; the RSD of repeatability was 1.8%-2.6%; the RSD of stability within 48 hours was 0.19%-2.8%, and the average recovery rate was 95.52%-100.1%, all meeting the requirements of pharmaceutical quantitative analysis. Additionally, the interval estimation method was used to directly reflect the distribution of samples with abnormal chemical properties of characteristic components, and the results showed ten samples were detected beyound the 95% control line of confidence level. Multivariate statistical process control(MSPC) method was used to monitor the abnormal samples of Huangjing Zanyu Capsules collectively, and the results showed that two samples were beyond the 95% control line of Hotelling's T~2 and three samples beyond the 95% control line of squared prediction error(SPE), indicating consistent quality control of Huangjing Zanyu Capsules. In conclusion, the proposed method is not only accurate and efficient but also a compensation for the traditional single-component quality control method, providing a scientific basis for the quality control in manufacturing process of Huangjing Zanyu Capsules. Furthermore, it could also serve as a reference method for the quality control in manufacturing big brand traditional Chinese medicine.

Estimating Time: Comparing the Accuracy of Estimation Methods for Interval Timing

In interval timing experiments, motor reproduction is the predominant method used when participants are asked to estimate an interval. However, it is unknown how its accuracy, precision and efficiency compare to alternative methods, such as indicating the duration by spatial estimation on a timeline. In two experiments, we compared different interval estimation methods. In the first experiment, participants were asked to reproduce an interval by means of motor reproduction, timeline estimation, or verbal estimation. We found that, on average, verbal estimates were more accurate and precise than line estimates and motor reproductions. However, we found a bias towards familiar whole second units when giving verbal estimates. Motor reproductions were more precise, but not more accurate than timeline estimates. In the second experiment, we used a more complex task: Participants were presented a stream of digits and one target letter and were subsequently asked to reproduce both the interval to target onset and the duration of the total stream by means of motor reproduction and timeline estimation. We found that motor reproductions were more accurate, but not more precise than timeline estimates. In both experiments, timeline estimates had the lowest reaction times. Overall, our results suggest that the transformation of time into space has only a relatively minor cost. In addition, they show that each estimation method comes with its own advantages, and that the choice of estimation method depends on choices in the experimental design: for example, when using durations with integer durations verbal estimates are superior, yet when testing long durations, motor reproductions are time intensive making timeline estimates a more sensible choice.

Unbiased Estimates and Confidence Intervals for Riverine Loads

AbstractEstimating the uncertainty in annual riverine constituent loads, which is the mass that passes through a river cross‐section into a receiving water body, using infrequent water quality (WQ) observations is a difficult and unsolved task. Therefore, we propose an unbiased point estimation method and interval estimation method for river loads based on the rating curve (RC) method using importance sampling and the bootstrap method, respectively. In this paper, we first statistically explain the unbiasedness of load estimates using the proposed method. Second, the effectiveness of point and interval estimates by the proposed method is demonstrated for river loads from a small catchment and from large watersheds based on discharge and WQ data of solutes, nutrients, and suspended sediments with 10‐min to daily intervals. The results show that the proposed method provides unbiased estimates and appropriate coverage of confidence intervals regardless of the RC model used. The results also reveal that the dominant cause of bias in load estimates based on ordinary RC methods, such as the Loadest model, is not due to the poor simulation of observed loading rates by the RCs or because of the nonnormality of the regression residuals but rather improper sampling strategies. The proposed method is currently not feasible for WQ monitoring sites in large rivers due to the unmanageability of missing observations or censored data and an inefficient sampling strategy, although the requirement of unbiased estimation explained here can aid in scheduling high‐flow sampling for monitoring sites.

Prediction Interval Estimation Methods for Artificial Neural Network (ANN)-Based Modeling of the Hydro-Climatic Processes, a Review

Despite the wide applications of artificial neural networks (ANNs) in modeling hydro-climatic processes, quantification of the ANNs’ performance is a significant matter. Sustainable management of water resources requires information about the amount of uncertainty involved in the modeling results, which is a guide for proper decision making. Therefore, in recent years, uncertainty analysis of ANN modeling has attracted noticeable attention. Prediction intervals (PIs) are one of the prevalent tools for uncertainty quantification. This review paper has focused on the different techniques of PI development in the field of hydrology and climatology modeling. The implementation of each method was discussed, and their pros and cons were investigated. In addition, some suggestions are provided for future studies. This review paper was prepared via PRISMA (preferred reporting items for systematic reviews and meta-analyses) methodology.

On estimation of stress-strength reliability using lower record values from odd generalized exponential: Exponential distribution

This research paper aims to find the estimated values closest to the true values of the reliability function under lower record values, and to know how to obtain these estimated values using point estimation methods or interval estimation methods. This helps researchers later in obtaining values of the reliability function in theory and then applying them to reality which makes it easier for the researcher to access the missing data for long periods such as weather. We evaluated the stress-strength model of reliability based on point and interval estimation for reliability under lower records by using Odd Generalize Exponential-Exponential distribution (OGEE) which has an important role in the lifetime of data. After that, we compared the estimated values of reliability with the real values of it. We analyzed the data obtained by the simulation method and the real data in order to reach certain results. The Numerical results for estimated values of reliability supported with graphical illustrations. The results of both simulated data and real data gave us the same coverage.

Prediction Interval Estimation of Aeroengine Remaining Useful Life Based on Bidirectional Long Short-Term Memory Network

Reliable and accurate aero-engine remaining useful life (RUL) prediction plays a key role in aero-engine prognostics and health management (PHM) system. However, due to the epistemic uncertainties associated with aero-engine systems, prediction errors are unavoidable and sometimes significant in traditional deterministic point prediction methods. To improve the accuracy and credibility of RUL prediction, a novel prediction interval (PI) estimation method is proposed to quantify the uncertainties in RUL prediction. The proposed method involves the data clustering, mathematical statistical analysis and deep learning techniques, and is achieved through offline and online phases. In the offline phase, an enhanced fuzzy c-means algorithm (FCM) is proposed to divide the aero-engine health status into several discrete states. After labeling the health state of each sampling point, PIs are computed for them. This step is achieved by the empirical distributions of errors associated with all instances belonging to the health state under consideration. In the online phase, a bidirectional long short-term memory (Bi-LSTM) network is employed to estimate the boundaries of point prediction, and thus the PI of aero-engine RUL is generated. The aero-engine degradation dataset from NASA is used to validate the proposed RUL PI estimation method. The results obtained indicate that the proposed method is a promising tool for providing reliable aero-engine RUL interval estimates, which can inform maintenance-related decisions.

Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors

The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.

Reduced-order observer-based interval estimation for discrete-time linear systems

This paper proposes an interval estimation method for discrete-time linear systems by designing robust reduced-order observer with reachability analysis. First, the discrete-time linear system is transformed into a reduced-order system via a special equivalent transformation, which depends on the orthogonal procedure on the output matrix. Then, a kind of robust reduced-order observer is developed such that the states of the original discrete-time system can be indirectly estimated, where the technique is used to attenuate the effects of disturbances. Based on the estimated states provided by the reduced-order observer, the interval estimation can be obtained by reachability analysis. A simulation example is given to illustrate that the proposed method of interval estimation has better performance than the full-order observer-based interval estimation.

Confidence Intervals for a Population Size Based on Capture-Recapture Data

Capture-recapture is a popular sampling method to estimate the total number of individuals in a population. This method is also used to estimate the size of a target population based on several incomplete records/databases of individuals. In this context, a simple approximate confidence interval (CI) based on the hypergeometric distribution is proposed. The proposed CI is compared with a popular approximate CI, likelihood CI and an exact admissible CI in terms of coverage probability and precision. Our numerical study indicates that the proposed CI is very satisfactory in terms of coverage probability, better than the popular approximate CI, and much shorter than the admissible CI. The interval estimation method is illustrated using a few examples with epidemiological data.

Potentially Resectable mCRC-treated with Cetuximab Combined with Chemotherapy.

Surgical resection of a potentially resectable metastatic colorectal cancer (mCRC) may lead to additional clinical benefits for the patients. In some cases, patients with initially unresectable lesions can be converted to resectable ones after induction of chemotherapy; and these patients are primarily treated with fluorouracil-based systemic chemotherapy preoperatively. However, the optimal protocol for neoadjuvant therapy has not been determined yet, and it remains a source of controversy about whether systemic chemotherapy combined with cetuximab can increase the surgical resection rate and obtain more clinical benefits. Metastatic CRC patients that received chemotherapy combined with cetuximab were compared with those undergoing chemotherapy alone. The hazard ratio (HR) and odds ratio (OR) were used as the efficacy indicators. The 95% confidence interval (95% CI) was used as interval estimation method. Seven studies including 1,895 patients were selected. Compared with patients undergoing chemotherapy, the patients receiving systemic chemotherapy combined with cetuximab of the R0 resection was not improved (OR=1.25; 95% CI, 0.76-2.06; p=0.08); The progression-free survival (PFS) of the patients received chemotherapy combined with cetuximab was slightly longer than those received chemotherapy alone (HR=0.88; 95% CI, 0.77-1; p=0.005); and the overall survival (OS) of patients undergoing systemic chemotherapy combined with cetuximab was not prolonged relative to that in patients receiving chemotherapy alone (HR=0.98; 95% CI, 0.86-1.11; p=0.04). Compared with patients receiving chemotherapy alone, the surgical resection rate and PFS are not increased in patients undergoing chemotherapy combined with cetuximab, but the OS is slightly prolonged. Key Words: Chemotherapy, Cetuximab, Colorectal cancer, Metastatic tumor, Surgery.

Estimating Postmortem Interval by Protein Chip Detection Technology Combined with Multidimensional Statistics

Objective To obtain the protein expression profile of rat liver tissue after death by the 2100 bioanalyzer combined with protein chip, and infer the relationship between protein expression profile and postmortem interval. Methods Rats were killed by abdominal anesthesia and placed at 16 ℃. Water-soluble proteins in liver tissues were extracted at 14 time points after death. The expression profile data of proteins with relative molecular weight of 14 000-230 000 were obtained using protein chip, and principal component analysis （PCA）, partial least squares-discriminant analysis （PLS-DA） and Fisher discriminant were used to analyze the data. Results According to the changes of protein expression profile, the postmortem interval was divided into group A （0 d）, group B （1-9 d）, group C （12-30 d） according to the result of PLS-DA. The prediction accuracy of the training set and test set of the model were all 100.0%, and the internal cross-validation of the training set was 100.0% according to Fisher discriminant. The Fisher discriminant model at each time point of group B and C was established to narrow the time window of postmortem interval estimation. The prediction accuracy of the training set and test set were all 100.0%, and the internal cross-validation accuracy of the training set was 100.0% in group B. The prediction accuracy of the training set and test set were respectively 95.2% and 78.6% in group C, and the internal cross-validation of the training set was 88.1%. Conclusion Protein chip detection technology can quickly and easily obtain the expression profile of water-soluble proteins of rat liver tissue with a relative molecular weight of 14 000-230 000 at different time points after death. PLS-DA and Fisher discriminant models are established to classify and predict the postmortem interval, in order to provide new ideas and methods for postmortem interval estimation.

Event-triggered Control of Positive Switched Systems with Actuator Saturation and Time-delay

This paper investigates the event-triggered control of positive switched systems with randomly occurring actuator saturation and time-delay, where the actuator saturation and time-delay obey different Bernoulli distributions. First, an event-triggering condition is constructed based on a 1-norm inequality. Under the presented event-triggering scheme, an interval estimation method is utilized to deal with the error term of the systems. Using a co-positive Lyapunov functional, the event-triggered controller and the cone attraction domain gain matrices are designed via matrix decomposition techniques. The positivity and stability of the resulting closed-loop systems are reached by guaranteeing the positivity of the lower bound of the systems and the stability of the upper bound of the systems, respectively. The proposed approach is developed for interval and polytopic uncertain systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the theoretical findings.

A copula-based Markov chain model for serially dependent event times with a dependent terminal event

Copula modeling for serial dependence has been extensively discussed in a time series context. However, fitting copula-based Markov models for serially dependent survival data is challenging due to the complex censoring mechanisms. The purpose of this paper is to develop likelihood-based methods for fitting a copula-based Markov chain model to serially dependent event times that are dependently censored by a terminal event, such as death. We propose a novel copula-based Markov chain model for describing serial dependence in recurrent event times. We also apply another copula model for handling dependent censoring. Due to the complex likelihood function with the two copulas, we propose a two-stage estimation method under Weibull distributions for fitting the survival data. The asymptotic normality of the proposed estimator is established through the theory of estimating functions. We propose a jackknife method for interval estimates, which is shown to be asymptotically consistent. To select suitable copulas for a given dataset, we propose a model selection method according to the 2nd stage likelihood. We conduct simulation studies to assess the performance of the proposed methods. For illustration, we analyze survival data from colorectal cancer patients. We implement the proposed methods in our original R package “Copula.Markov.survival” that is made available in CRAN ( https://cran.r-project.org/ ).

The use of the bootstrap method for the assessment of investment effectiveness and risk – the case of confidence intervals estimation for the Sharpe ratio and TailVaR

This paper is aimed at presenting application of bootstrap interval estimation methods to the assessment of financial investment’s effectiveness and risk. At first, we give an overview of various methods of bootstrap confidence interval estimation, i.e. bootstrap-t interval, percentile interval and BCa interval. Then, bootstrap confidence interval estimation methods are used to estimate confidence intervals for the Sharpe ratio and TailVaR of the Warsaw Stock Exchange sectoral indices. The results show that the bootstrap confidence intervals of different types are quite similarly positioned for each of the analysed index and measure. Taking into the account the locations of confidence intervals for both the Sharpe ratio and TailVaR, the real estate sector tends to be the most advantageous from the investor’s viewpoint.

Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model

Doubly-truncated data arise in many fields, including economics, engineering, medicine, and astronomy. This article develops likelihood-based inference methods for lifetime distributions under the log-location-scale model and the accelerated failure time model based on doubly-truncated data. These parametric models are practically useful, but the methodologies to fit these models to doubly-truncated data are missing. We develop algorithms for obtaining the maximum likelihood estimator under both models, and propose several types of interval estimation methods. Furthermore, we show that the confidence band for the cumulative distribution function has closed-form expressions. We conduct simulations to examine the accuracy of the proposed methods. We illustrate our proposed methods by real data from a field reliability study, called the Equipment-S data.

Comparing Methods of Uncertainty Estimation in Optimal Fingerprinting

AbstractThe performances of different uncertainty estimates in multipattern optimal fingerprinting are compared. The comparisons are performed in a perfect model context using the Canadian Earth System Model Large Ensemble to provide a clean comparison in the absence of model error or inconsistencies in internal variability. Performance is quantified in terms of the accuracy (correct coverage) of the confidence intervals for the fingerprint scaling factors. Sensitivity to temporal and spatial smoothing is tested. The most accurate method for interval estimation differs between variables, with a maximum likelihood method being most accurate for temperature and a bootstrap method being most accurate for precipitation. These differences can only partially be explained in terms of a relevant measure of signal‐to‐noise ratio.

Common Medical and Statistical Problems: The Dilemma of the Sample Size Calculation for Sensitivity and Specificity Estimation

Sample size calculation in biomedical practice is typically based on the problematic Wald method for a binomial proportion, with potentially dangerous consequences. This work highlights the need of incorporating the concept of conditional probability in sample size determination to avoid reduced sample sizes that lead to inadequate confidence intervals. Therefore, new definitions are proposed for coverage probability and expected length of confidence intervals for conditional probabilities, like sensitivity and specificity. The new definitions were used to assess seven confidence interval estimation methods. In order to determine the sample size, two procedures—an optimal one, based on the new definitions, and an approximation—were developed for each estimation method. Our findings confirm the similarity of the approximated sample sizes to the optimal ones. R code is provided to disseminate these methodological advances and translate them into biomedical practice.

Study on the Risk Early Warning Index System of Typical Chemical Production Process Based on Three Dimensional Factors

Prevention and control in advance embodies the essence of risk management, and risk early warning index system is the key link of implementing risk early warning. In order to construct risk early warning index system of typical chemical production process, the risk identification, analysis and early warning theory are closely combined in this paper, the accident risk factors are extended to three aspects, as possibility, severity and sensitivity. Based on the common accidents of typical chemical production process, the risk factors of the production process are comprehensively analysed and the risk early warning index system is put forward based on three dimensional factors. Through the index screening methods for interval estimation, the primary warning indicators are screened. Finally, the risk early warning index system of typical chemical production process is determined.