Two-sided Intervals Research Articles

Exact Boundary Correction Methods for Multivariate Kernel Density Estimation

This paper develops a method to obtain multivariate kernel functions for density estimation problems in which the density function is defined on compact support. If domain-specific knowledge requires certain conditions to be satisfied at the boundary of the support of an unknown density, the proposed method incorporates the information contained in the boundary conditions into the kernel density estimators. The proposed method provides an exact kernel function that satisfies the boundary conditions, even for small samples. Existing methods primarily deal with a one-sided boundary in a one-dimensional problem. We consider density in a two-sided interval and extend it to a multi-dimensional problem.

The Automated Bias-Corrected and Accelerated Bootstrap Confidence Intervals for Risk Measures

Different approaches to determining two-sided interval estimators for risk measures such as Value-at-Risk (VaR) and conditional tail expectation (CTE) when modeling loss data exist in the actuarial literature. Two contrasting methods can be distinguished: a nonparametric one not relying on distributional assumptions or a fully parametric one relying on standard asymptotic theory to apply. We complement these approaches and take advantage of currently available computer power to propose the bias-corrected and accelerated (BCA) confidence intervals for VaR and CTE. The BCA confidence intervals allow the use of a parametric model but do not require standard asymptotic theory to apply. We outline the details to determine interval estimators for these three different approaches using general computational tools as well as with analytical formulas when assuming the truncated Lognormal distribution as a parametric model for insurance loss data. An extensive simulation study is performed to assess the performance of the proposed BCA method in comparison to the two alternative methods. A real dataset of left-truncated insurance losses is employed to illustrate the implementation of the BCA-VaR and BCA-CTE interval estimators in practice when using the truncated Lognormal distribution for modeling the loss data.

Solution types of two-sided interval linear system and their application on interval linear programming problems

Two-sided interval linear systems contain variables on both sides of equations or inequalities. We propose six types of solutions (weak, strong, tolerance, control, left-localized and right-localized solutions) to the system of equations together with four types of solutions (weak, strong, tolerance and control solutions) to the system of inequalities. We prove the necessary and sufficient conditions for checking their solvabilities in the form of systems of linear inequalities depending on the sign of variables. As a result, an interval linear programming problem with nonnegative variables and two-sided interval linear constraints can be solved by a standard linear programming approach.

Empirical weighted Bayesian tolerance intervals

ABSTRACT Bayesian statistics has been widely utilized as an approach that can incorporate prior knowledge into statistical inference. Tolerance intervals (TI) are the most commonly used statistical methods for product quality assurance. There are two main Bayesian approaches for calculating statistical tolerance intervals: Hamada and Wolfinger. A simulation-based approach was implemented to compare two-sided Wolfinger, Hamada, and frequentist tolerance intervals which control the probability content at a specified level of confidence. As sample sizes increase, compared to frequentist, Hamada TI become more conservative while Wolfinger TI are more liberal. To address this issue, we propose an empirical weighted Bayesian TI approach that is a compromise between Hamada and Wolfinger approaches. The proposed Bayesian TI result in narrower limits in certain scenarios while ensuring the confidence content coverage remains comparable to frequentist.

Use of a two-sided tolerance interval in the design and evaluation of biosimilarity in clinical studies.

In assessing biosimilarity between two products, the question to ask is always "How similar is similar?" Traditionally, the equivalence of the means between products is the primary consideration in a clinical trial. This study suggests an alternative assessment for testing a certain percentage of the population of differences lying within a prespecified interval. In doing so, the accuracy and precision are assessed simultaneously by judging whether a two-sided tolerance interval falls within a prespecified acceptance range. We further derive an asymptotic distribution of the tolerance limits to determine the sample size for achieving a targeted level of power. Our numerical study shows that the proposed two-sided tolerance interval test controls the type I error rate and provides sufficient power. A real example is presented to illustrate our proposed approach.

Approximate two-sided tolerance interval for sample variances

Tolerance limits for variances are useful in quality assessments when the focus is on the precision of a quality characteristic. Two-sided tolerance intervals (limits) provide insight into a process degradation as well as improvement, in terms of process variability. Sarmiento, Chakraborti, and Epprecht constructed the exact two-sided tolerance intervals for the population of sample variances, assuming normality of the data. The required tolerance factors cannot be expressed in a closed-form and their computation is complex, depending on the numerical solutions of a system of three nonlinear equations. Motivated by this, from a practical point of view, we consider a simpler, approximate tolerance interval based on the approximate tolerance interval for the gamma distribution, which uses the Wilson–Hilferty approximation. The required tolerance factors for the proposed interval are readily obtained using existing tables and software and therefore can be implemented more easily in practice. The performance of the proposed tolerance interval is compared with that of the exact interval in terms of accuracy and robustness in simulation studies. In addition, the tolerance intervals are illustrated with a dataset from a real application. A summary and some conclusions are offered. It is seen that the proposed approximate tolerance intervals are fairly accurate, reasonably robust and being much simpler to calculate, can be useful in practical applications.

AE solutions to two-sided interval linear systems over max-plus algebra

This paper introduces a concept of AE solutions to two-sided interval max-plus linear systems, a rather general concept which includes many known concepts of solutions to interval systems, in particular, weak, strong, tolerance and control solutions as its special cases. We state full characterizations of AE solutions for the two-sided interval max-plus systems, including both linear inequalities and linear equations. Moreover, we provide a specific example to illustrate an efficient method of finding the AE solution set.

Simple Approach to Calculate Random Effects Model Tolerance Intervals to Set Release and Shelf-Life Specification Limits of Pharmaceutical Products.

Tolerance intervals are used to statistically derive the acceptance limits to which drugs must conform upon manufacture (release) and throughout shelf-life. The single measurement per lot in release data and repeated measurements per lot longitudinally for stability data have to be considered in the calculation. Methods for the one-way random effects model by Hoffman and Kringle (HK) for two-sided intervals and Hoffman (H) for one-sided limits are extended to a random-intercept, fixed-slope model in this paper. The performance of HK and H was evaluated via simulation by varying the following factors: (a) magnitude of stability trend over time, (b) sample size, (c) percentage of lot-to-lot contribution to total variation, (d) targeted proportion, and (e) data inclusion. The performance metrics are average width (for two-sided) or average limit (for one-sided) and attained confidence level. HK and H maintained nominal confidence levels as originally developed, but H is too conservative (i.e., achieved confidence level exceeds the nominal level) in some situations. The HK method adapted for an attribute that changes over time performed comparably to the more computationally intensive generalized pivotal quantity and Bayesian posterior predictive methods. Mathematical formulas and example calculations as implemented using R statistical software functions are provided to assist practitioners in implementing the methods. The calculations for the proposed approach can also be easily performed in a spreadsheet given basic regression output from a statistical software package. Microsoft Excel spreadsheets are available from the authors upon request.LAY ABSTRACT: Tolerance intervals (a measure of what can be expected from the manufacturing process) calculated from attribute measurements of drug product lots are one of the factors considered when establishing acceptance limits to ensure drug product quality. The methods often used to calculate tolerance intervals when there are multiple measurements per lot and the attribute changes over time are either lacking in statistical rigor or statistically rigorous but computationally intensive to implement. The latter type requires simulations that have to be programmed using specialized statistical software, because closed-form mathematical formulas are not available. As a consequence, some quality practitioners and applied statisticians involved in setting acceptance limits may be hindered in using such computationally intensive methods. This paper aims to address this need by proposing an approach that is statistically rigorous yet simple enough to implement using spreadsheets. The approach builds upon previously published works developed for attributes that do not change over time and adapts the cited works for attributes that change over time. The proposed approach is demonstrated to have good statistical properties and compares favorably against the more computationally intensive alternative methods. The paper provides closed-form mathematical formulas, example data, and illustrative calculations as implemented in programmed R functions to facilitate implementation by practitioners. Alternatively, the calculations can be performed without requiring complex programming/simulation using Microsoft Excel spreadsheets that can be requested from the authors.

Confidence Bounds and Narrowest Reliable Intervals in D-Posterior Approach

Several new methods of confidence and of asymptotically confidence limits in the dposterior approach is proposed. For the so-called reliable two-sided intervals, close in construction to Bayesian intervals, the method of constructing the narrowest intervals is given.

Predicting the Probability of Exceeding Critical System Thresholds

In this paper we show how regression modelling can be combined with a special kind of data transformation technique that improves model precision and produces several “preliminary” estimates of the target value. These preliminary estimates can be used for interval estimates of the target value as well as for predicting the probability that it has or will exceed arbitrary predefined thresholds. Our approach can be combined with various regression models and applied in many domains that need to estimate the probability of system malfunctions or other hazardous states brought about by system variables exceeding critical safety thresholds. We rigorously derive the formulas for the probability of crossing an upper bound and a lower bound both separately (one-sided intervals) and together (a two-sided interval), and verify the approach experimentally on a real dataset from the electric power industry. Problems in programming 2018; 2-3: 189-196

An algorithm for solving two-sided interval system of max-plus linear equations

We study six types of solutions (weak, strong, tolerance, control, Left-localized, and Right-localized solutions) to a two-sided interval system of max-plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions. These conditions are in the form of two-sided systems of max-plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems. Therefore, we could work on the interval inclusion linear systems, instead of directly solving these six types of solutions themselves. An optimization problem with the two-sided interval system of max-plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems.

An approach for sample size determination of average bioequivalence based on interval estimation.

In 1992, the US Food and Drug Administration declared that two drugs demonstrate average bioequivalence (ABE) if the log-transformed mean difference of pharmacokinetic responses lies in (-0.223, 0.223). The most widely used approach for assessing ABE is the two one-sided tests procedure. More specifically, ABE is concluded when a 100(1 - 2α) % confidence interval for mean difference falls within (-0.223, 0.223). As known, bioequivalent studies are usually conducted by crossover design. However, in the case that the half-life of a drug is long, a parallel design for the bioequivalent study may be preferred. In this study, a two-sided interval estimation - such as Satterthwaite's, Cochran-Cox's, or Howe's approximations - is used for assessing parallel ABE. We show that the asymptotic joint distribution of the lower and upper confidence limits is bivariate normal, and thus the sample size can be calculated based on the asymptotic power so that the confidence interval falls within (-0.223, 0.223). Simulation studies also show that the proposed method achieves sufficient empirical power. A real example is provided to illustrate the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

On the Laplace transforms of the first exit times in one-dimensional non-affine jump–diffusion models

We compute the Laplace transforms of the first exit times for certain one-dimensional jump–diffusion processes from two-sided intervals. The method of proof is based on the solutions of the associated integro-differential boundary value problems for the corresponding value functions. We consider jump–diffusion processes solving stochastic differential equations driven by Brownian motions and several independent compound Poisson processes with multi-exponential jumps. The results are illustrated on the non-affine pure jump analogues of certain mean-reverting or diverting diffusion processes which represent closed-form solutions of the appropriate stochastic differential equations.

High Confidence Intervals Applied to Aircraft Altitude Prediction

This paper describes the application of high-confidence-interval prediction methods to the aircraft trajectory prediction problem, more specifically to the altitude prediction during climb. We are interested in methods for finding two-sided intervals that contain, with a specified confidence, at least a desired proportion of the conditional distribution of the response variable. This paper introduces two-sided Bonferroni-quantile confidence intervals, which is a new method for obtaining high-confidence two-sided intervals in quantile regression. This paper also uses the Bonferroni inequality to propose a new method for obtaining tolerance intervals in least squares regression. The latter has the advantages of being reliable, fast, and easy to calculate. We compare physical point-mass models to the introduced models on an air traffic management data set composed of traffic at major French airports. Experimental results show that the proposed interval prediction models perform significantly better than the conventional point-mass model currently used in most trajectory predictors. When comparing with a recent state-of-the-art point-mass model with adaptive mass estimation, the proposed methods give altitude intervals that are slightly wider but more reliable.

Exact confidence intervals for the relative risk and the odds ratio.

For comparison of proportions, there are three commonly used measurements: the difference, the relative risk, and the odds ratio. Significant effort has been spent on exact confidence intervals for the difference. In this article, we focus on the relative risk and the odds ratio when data are collected from a matched-pairs design or a two-arm independent binomial experiment. Exact one-sided and two-sided confidence intervals are proposed for each configuration of two measurements and two types of data. The one-sided intervals are constructed using an inductive order, they are the smallest under the order, and are admissible under the set inclusion criterion. The two-sided intervals are the intersection of two one-sided intervals. R codes are developed to implement the intervals. Supplementary materials for this article are available online.

Sample Size Determination for a Two-Sided Multiple Components Tolerance Interval

In the conventional concept, the variance of a tolerance interval from the measurements is a single component, and the sample size for quality control process was estimated by the variance of a single component. However, we can find examples in recent about several components that could vary in their measurements, so an approximate method must be found to modify the conventional tolerance interval. In our paper, we develop an approach to calculate the sample size for a two-sided tolerance interval including several components in the variance from the measurements. An example is presented to illustrate our proposed method.

On Bayes predictor of times to failure of Type-II progressively censored sample

The purpose of this paper is Bayesian prediction, robust Bayesian prediction and two-sided predictive interval for both one-sample and two-sample prediction problems. In one-sample prediction problem, we discuss the prediction of the life-lengths (j=1, 2, … , ri; i=1, 2, … , m) of all censored units in all m stages of censoring, where denotes the jth-order statistics out of ri removed units at stage i. While in two-sample prediction problem, we discuss the prediction of the life-lengths (T1, … , Ts) which is a second independent ordered sample of size s of future observation from the same population of Y, where Y is the progressively Type-II right-censored order statistics. Finally, we conduct a simulation study, and consider an illustrative example for our proposed predictors.

Equivalence Assessment for Interchangeability Based on Two-Sided Tests

Interchangeability was originally developed in order to assess drug bioequivalence beyond average bioequivalence. In 2003, the Food and Drug Administration (FDA) published a Guidance documenting the procedures on using in vivo bioequivalence crossover trial to assess interchangeability between test and reference products. In general, this FDA Guidance describes interchangeability in terms of population and individual bioequivalence. The Guidance procedures were criticized for their lack of sampling distribution of the test statistics. As a result, the critical points were generated from simulation studies without adjusting for sample size. Further more, they lack consistency with average bioequivalence required in the 1992 FDA Guidance. Alternative interchangeability or interchangeability procedures were proposed to measure the probability of individual response difference under two treatments within prespecified lower and upper limits. Interchangeability is claimed if this probability is greater than a prespecified threshold. Tse et al. (2006) proposed an approximate distribution of the estimated probability based on the second-order Taylor expansion. For the same interchangeability probability hypothesis, Liu and Chow (1997) and Tsong and Shen (2007) also proposed a tolerance interval-based approach that can be extended to clinical trials with parallel arm design under the normality assumption. In this article, we first generalized the two-sided tolerance interval based interchangeability without equal sample size and variance assumption. We also derived a power function for the proposed method, and performed simulation studies to compare the type I error rate, power, and sample size between the Tse approximated test and the generalized tolerance interval approach for interchangeability assessment.

Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables

Tolerance intervals for discrete variables are widely used, especially in industrial applications. However, there is no thorough treatment of tolerance intervals when sampling without replacement. This paper proposes methods for constructing one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. Equal-tailed tolerance intervals (i.e., tolerance intervals that control the percentages in both tails) are studied followed by a small adjustment to the nominal coverage level to obtain tolerance intervals that control a specified inner percentage of the sampled distribution. The tolerance interval calculations implicitly use confidence bounds for M, the unknown number of elements possessing a certain attribute in the finite population of size N. Three different methods for obtaining such confidence bounds are suggested: a large sample approach, an approach with a continuity correction, and an exact method based on nonrandomization. The intervals are examined for desirable coverage probabilities and expected widths. The methods are also illustrated using some examples.

Sample size determination for quality control process using a multiple components tolerance interval

ABSTRACTIn the past, a tolerance interval was used for the statistical quality control process on raw materials and/or the final product. In the traditional concept of the tolerance interval, the variance from the measurements is a single component. However, we can find examples about several components that could vary in their measurements, so an approximate method must be found to modify the traditional tolerance interval. Now we employ a tolerance interval considering multiple components in the variance from the measurements to deal with quality control process. In our paper, the proposed method is used to solve the sample size determination for a two-sided tolerance interval approach considering multiple components on the variance of measurements.