Sequential fixed-width confidence bands for kernel regression estimation

We consider a random design model based on independent and identically distributed (iid) pairs of observations (Xi, Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non-parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least 1−. Here, d(> 0) and 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel estimators of nonparametric regression with data-driven bandwidths.

The sample size was optimized using the purely and two-stage sequential procedure together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accuracy. The numerical results indicate that the confidence bands based on the local linear estimator have the best performance than those constructed by using Nadaraya-Watson estimator. However both estimators are shown to have asymptotically correct coverage properties.

Sample size determination for kernel regression estimation using sequential fixed-width confidence bands

We consider a random design model based on independent and identically distributed pairs of observations (Xi, Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non-parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least 1−. Here, d(> 0) and 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel estimators of nonparametric regression with data-driven bandwidths. The sample size was optimized using the purely and two-stage sequential procedures together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accuracy. The numerical results indicate that the confi dence bands based on the local linear estimator have the better performance than those constructed by using Nadaraya-Watson estimator. However both estimators are shown to have asymptotically correct coverage properties.

Estimating process capability index Cpm using a bootstrap sequential sampling procedure

Construction of a confidence interval for process capability index CPM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index CPM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.

Two-stage sequential procedure for nonparametric regression estimation

In nonparametric statistics the functional form of the relationship between the response variable and its associated predictor variables is unspecified but it is assumed to be a smooth function. We develop a procedure for constructing a fixed width confidence interval for the predicted value at a specified point of the independent variable. The optimal sample size for constructing this interval is obtained using a two stage sequential procedure which relies on some asymptotic properties of the Nadaraya--Watson and local linear estimators. Finally, a large scale simulation study demonstrates the applicability of the developed procedure for small and moderate sample sizes. The procedure developed here should find wide applicability since many practical problems which arise in industry involve estimating an unknown function.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a complicated target distribution via simple ergodic averages. A fundamental question in MCMC applications is when should the sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the MCMC sampling the first time the width of a confidence interval based on the ergodic averages is less than a user-specified value. Hence calculating Monte Carlo standard errors is a critical step in assessing the output of the simulation. In particular, we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We describe sufficient conditions for the strong consistency and asymptotic normality of both methods and investigate their finite sample properties in a variety of examples.

Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε. © 2008 Elsevier B.V. All rights reserved.

Point-wise confidence interval estimation by neural networks: A comparative study based on automotive engine calibration.

In developing neural network techniques for real world applications it is still very rare to see estimates of confidence placed on the neural network predictions. This is a major deficiency, especially in safety-critical systems. In this paper we explore three distinct methods of producing point-wise confidence intervals using neural networks. We compare and contrast Bayesian, Gaussian Process and Predictive error bars evaluated on real data. The problem domain is concerned with the calibration of a real automotive engine management system for both air-fuel ratio determination and on-line ignition timing. This problem requires real-time control and is a good candidate for exploring the use of confidence predictions due to its safety-critical nature.

Conditional Confidence Interval for the Scale Parameter of a Weibull Distribution

2000 Mathematics Subject Classification: 62F25, 62F03.

Optimal local polynomial regression of noisy time varying signals

We address the problem of local-polynomial modeling of smooth time-varying signals with unknown functional form, in the presence of additive noise. The problem formulation is in the time domain and the polynomial coefficients are estimated in the pointwise minimum mean square error (PMMSE) sense. The choice of the window length for local modeling introduces a bias-variance tradeoff, which we solve optimally by using the intersection-of-confidence-intervals (ICI) technique. The combination of the local polynomial model and the ICI technique gives rise to an adaptive signal model equipped with a time-varying PMMSE-optimal window length whose performance is superior to that obtained by using a fixed window length. We also evaluate the sensitivity of the ICI technique with respect to the confidence interval width. Simulation results on electrocardiogram (ECG) signals show that at 0dB signal-to-noise ratio (SNR), one can achieve about 12dB improvement in SNR. Monte-Carlo performance analysis shows that the performance is comparable to the basic wavelet techniques. For 0 dB SNR, the adaptive window technique yields about 2-3dB higher SNR than wavelet regression techniques and for SNRs greater than 12dB, the wavelet techniques yield about 2dB higher SNR.

Determinants and two-year change in anterior chamber angle width in a Chinese population

Objective: To study the population distribution and longitudinal changes in anterior chamber angle width and its determinants among Chinese adults. Design: Prospective cohort, population-based study. Participants: Persons aged 35 years or more residing in Guangzhou, China, who had not previously undergone incisional or laser eye surgery. Methods: In December 2008 and December 2010, all subjects underwent automated keratometry, and a random 50% sample had anterior segment optical coherence tomography with measurement of angle-opening distance at 500 μm (AOD500), angle recess area (ARA), iris thickness at 750 μm (IT750), iris curvature, pupil diameter, corneal thickness, anterior chamber width (ACW), lens vault (LV), and lens thickness (LT) and measurement of axial length (AL) and anterior chamber depth (ACD) by partial coherence laser interferometry. Main Outcome Measures: Baseline and 2-year change in AOD500 and ARA in the right eye. Results: A total of 745 subjects were present for full biometric testing in both 2008 and 2010 (mean age at baseline, 52.2 years; standard deviation [SD], 11.5 years; 53.7% were female). Test completion rates in 2010 varied from 77.3% (AOD500: 576/745) to 100% (AL). Mean AOD500 decreased from 0.25 mm (SD, 0.13 mm) in 2008 to 0.21 mm (SD, 13 mm) in 2010 (difference, -0.04; 95% confidence interval [CI], -0.05 to -0.03). The ARA decreased from 21.5±3.73 10^-2 mm² to 21.0±3.64 10 ^-2 mm² (difference, -0.46; 95% CI, -0.52 to -0.41). The decrease in both was most pronounced among younger subjects and those with baseline AOD500 in the widest quartile at baseline. The following baseline variables were significantly associated with a greater 2-year decrease in both AOD500 and ARA: deeper ACD, steeper iris curvature, smaller LV, greater ARA, and greater AOD500. By using simple regression models, we could explain 52% to 58% and 93% of variation in baseline AOD500 and ARA, respectively, but only 27% and 16% of variation in 2-year change in AOD500 and ARA, respectively. Conclusions: Younger persons and those with the least crowded anterior chambers at baseline have the largest 2-year decreases in AOD500 and ARA. The ability to predict change in angle width based on demographic and biometric factors is relatively poor, which may have implications for screening. Financial Disclosure(s): The author(s) have no proprietary or commercial interest in any materials discussed in this article. © 2012 American Academy of Ophthalmology.

A systematic review of the performance of ceramic and metal implant abutments supporting fixed implant reconstructions

OBJECTIVES: The objective of this systematic review was to assess the 5-year survival rates and incidences of complications associated with ceramic abutments and to compare them with those of metal abutments. METHODS: An electronic Medline search complemented by manual searching was conducted to identify randomized-controlled clinical trials, and prospective and retrospective studies providing information on ceramic and metal abutments with a mean follow-up time of at least 3 years. Patients had to have been examined clinically at the follow-up visit. Assessment of the identified studies and data abstraction was performed independently by three reviewers. Failure rates were analyzed using standard and random-effects Poisson regression models to obtain summary estimates of 5-year survival proportions. RESULTS: Twenty-nine clinical and 22 laboratory studies were selected from an initial yield of 7136 titles and data were extracted. The estimated 5-year survival rate of ceramic abutments was 99.1% [95% confidence interval (CI): 93.8-99.9%] and 97.4% (95% CI: 96-98.3%) for metal abutments. The estimated cumulative incidence of technical complications after 5 years was 6.9% (95% CI: 3.5-13.4%) for ceramic abutments and 15.9% (95% CI: 11.6-21.5%) for metal abutments. Abutment screw loosening was the most frequent technical problem, occurring at an estimated cumulative incidence after 5 years of 5.1% (95% CI: 3.3-7.7%). All-ceramic crowns supported by ceramic abutments exhibited similar annual fracture rates as metal-ceramic crowns supported by metal abutments. The cumulative incidence of biological complications after 5 years was estimated at 5.2% (95% CI: 0.4-52%) for ceramic and 7.7% (95% CI: 4.7-12.5%) for metal abutments. Esthetic complications tended to be more frequent at metal abutments. A meta-analysis of the laboratory data was impossible due to the non-standardized test methods of the studies included. CONCLUSION: The 5-year survival rates estimated from annual failure rates appeared to be similar for ceramic and metal abutments. The information included in this review did not provide evidence for differences of the technical and biological outcomes of ceramic and metal abutments. However, the information for ceramic abutments was limited in the number of studies and abutments analyzed as well as the accrued follow-up time. Standardized methods for the analysis of abutment strength are needed.