Build Confidence Intervals Research Articles

Identification-Robust Estimation and Testing of the Zero-Beta CAPM

We propose exact simulation-based procedures for: (i) testing mean-variance efficiency when the zero-beta rate is unknown; (ii) building confidence intervals for the zero-beta rate. On observing that this parameter may be weakly identified, we propose likelihood-ratio-type tests as well as Fieller-type procedures based on a Hotelling-HAC statistic, which are robust to weak identification and allow for non-Gaussian distributions including parametric GARCH structures. The Fieller-Hotelling-HAC procedure also accounts (asymptotically) for general forms of heteroskedasticity and autocorrelation. We propose confidence sets for the zero-beta rate based on inverting exact tests for this parameter; for both procedures proposed, these sets can be interpreted as multivariate extensions of the classic Fieller method for inference on ratios. The exact distribution of likelihood-ratio-type statistics for testing efficiency is studied under both the null and the alternative hypotheses. The relevant nuisance parameter structure is established and finite-sample bound procedures are proposed, which extend and improve available Gaussian-specific bounds. Finite-sample distributional invariance results are also demonstrated analytically for the HAC statistic proposed by MacKinlay and Richardson (1991) . We study invariance to portfolio repacking for the tests and confidence sets proposed. The statistical properties of the proposed methods are analysed through a Monte Carlo study and compared with alternative available methods. Empirical results on NYSE returns show that exact confidence sets are very different from asymptotic ones, and allowing for non-Gaussian distributions affects inference results. Simulation and empirical evidence suggests that likelihood-ratio-type statistics--with p-values corrected using the Maximized Monte Carlo test method--are generally preferable to their multivariate Fieller-Hotelling-HAC counterparts from the viewpoints of size control and power. Copyright 2013, Oxford University Press.

The errors of our ways: taking account of error in computer-aided drug design to build confidence intervals for our next 25 years

The future of the advancement as well as the reputation of computer-aided drug design will be guided by a more thorough understanding of the domain of applicability of our methods and the errors and confidence intervals of their results. The implications of error in current force fields applied to drug design are given are given as an example. Even as our science advances and our hardware become increasingly more capable, our software will be perhaps the most important aspect in this realization. Some recommendations for the future are provided. Education of users is essential for proper use and interpretation of computational results in the future.

Remarks on Confidence Intervals for Self‐Similarity Parameter of a Subfractional Brownian Motion

We first present two convergence results about the second‐order quadratic variations of the subfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self‐similarity parameter associated with one‐dimensional subfractional Brownian motion.

Some Aspects of the Discrete Wavelet Analysis of Bivariate Spectra for Business Cycle Synchronisation

Abstract The paper considers some of the issues emerging from the discrete wavelet analysis of popular bivariate spectral quantities such as the coherence and phase spectra and the frequency-dependent time delay. The approach utilised here is based on the maximal overlap discrete Hilbert wavelet transform (MODHWT). Firstly, via a broad set of simulation experiments, we examine the small and large sample properties of two wavelet estimators of the scale-dependent time delay. The estimators are the wavelet cross-correlator and the wavelet phase angle-based estimator. Our results provide some practical guidelines for the empirical examination of short- and medium-term lead-lag relations for octave frequency bands. Further, we point out a deficiency in the implementation of the MODHWT and suggest using a modified implementation scheme, which was proposed earlier in the context of the dual-tree complex wavelet transform. In addition, we show how MODHWT-based wavelet quantities can serve to approximate the Fourier bivariate spectra and discuss issues connected with building confidence intervals for them. The discrete wavelet analysis of coherence and phase angle is illustrated with a scale-dependent examination of business cycle synchronisation between 11 euro zone countries. The study is supplemented by a wavelet analysis of the variance and covariance of the euro zone business cycles. The empirical examination underlines the good localisation properties and high computational efficiency of the wavelet transformations applied and provides new arguments in favour of the endogeneity hypothesis of the optimum currency area criteria as well as the wavelet evidence on dating the Great Moderation in the euro zone.

Homogeneity tests among groups for microsatellite data

We propose a homogeneity test among groups on a quadratic distance measure. The underlying mutation process in the microsatellite loci is studied using the stepwise mutation model. Asymptotic normality of the test statistic is proved under very mild regularity conditions. Resampling methods, such as jackknife, are used in the application to build confidence intervals for the difference in allelic variation between and within groups. The method is applied in a real data to test whether there are differences in the distribution of the repeated sequence among groups defined by ethnicity and alcoholism index (ALDX1).

Variance estimation and confidence intervals for the standardized mortality ratio with application to the assessment of a cancer screening program

The effect of a cancer screening program can be measured through the standardized mortality ratio (SMR) statistic. The numerator of the SMR is the observed number of deaths from the screened disease among participants in the screening program, whereas the denominator of the SMR is an estimate of the expected number of deaths in these participants under the assumption that the screening program has no effect. In this article, we propose a variance estimator for the denominator of the SMR when this expected number of deaths is estimated with Sasieni's method. We give both a general formula for this variance as well as formulas for specific disease incidence and survival estimators. We show how this new variance estimator can be used to build confidence intervals for the SMR. We investigate the coverage properties of various types of confidence intervals by simulation and find that intervals that make use of the proposed variance estimator perform well. We illustrate the method by applying it to the Québec Breast Cancer Screening program.

Potenciální produkt, mezera výstupu a míra nejistoty spojená s jejich určením při použití Hodrick-Prescottova filtru

In various fields of macroeconomic modelling, researchers often face the problem of decomposing time series into trend component and cycle fluctuations. While there are several potentially useful methods to perform the task in question, Hodrick-Prescott (HP) fi lter seems to have remained (despite some serious criticism) the most popular approach over the past decade. In this article I propose a straightforward and easy-to-implement bootstrap procedure for building pointwise and simultaneous confidence intervals around point estimates produced by HP filter. The principle of proposed method can be described as follows: first, we use maximum entropy bootstrap (Vinod, 2004, 2006) to approximate ensemble from which original time series is drawn and then apply the HP filter directly to each bootstrap replication. If necessary, the proposed method can be adapted to allow for uncertainty in the smoothing parameter. Practical usefulness of our approach is demonstrated with an application to the GDP data. Results are encouraging - obtained confi dence intervals for the trend and cyclical component are overall plausible thus supplying a researcher with some measure of uncertainty related to HP filtering. Finally, we demonstrate that a former approach to build confidence intervals for HP filter (Gallego and Johnson, 2005) leads to erratic inference for cycle due to the shape-destroying block bootstrap sampling.

On Some Problems in Discrete Wavelet Analysis of Bivariate Spectra with an Application to Business Cycle Synchronization in the Euro Zone

The paper considers some of the problems emerging from discrete wavelet analysis of popular bivariate spectral quantities like the coherence and phase spectra and the frequency-dependent time delay. The approach taken here, introduced by Whitcher and Craigmile (2004), is based on the maximal overlap discrete Hilbert wavelet transform (MODHWT). Firstly, we point at a deficiency in the implementation of the MODHWT and suggest using a modified implementation scheme resembling the one applied in the context of the dual-tree complex wavelet transform of Kingsbury (see Selesnick et al., 2005). Secondly, via a broad set of simulation experiments we examine small and large sample properties of two wavelet estimators of the scale- dependent time delay. The estimators are: the wavelet cross-correlator and the wavelet phase angle-based estimator. Our results provide some practical guidelines for empirical examination of short- and medium-term lead-lag relations for octave frequency bands. Besides, we show how the MODHWT-based wavelet quantities can serve to approximate the Fourier bivariate spectra and discuss certain issues connected with building confidence intervals for them. The discrete wavelet analysis of coherence and phase angle is illustrated with a scale-dependent examination of business cycle synchronization between 11 euro zone member countries. The study is supplemented with wavelet analysis of variance and covariance of the euro zone business cycles. The empirical examination underlines good localization properties and high computational efficiency of the wavelet transformations applied, and provides new arguments in favour of the endogeneity hypothesis of the optimum currency area criteria as well as a wavelet evidence on dating the Great Moderation in the euro zone.

RMediation: An R package for mediation analysis confidence intervals

This article describes the RMediation package,which offers various methods for building confidence intervals (CIs) for mediated effects. The mediated effect is the product of two regression coefficients. The distribution-of-the-product method has the best statistical performance of existing methods for building CIs for the mediated effect. RMediation produces CIs using methods based on the distribution of product, Monte Carlo simulations, and an asymptotic normal distribution. Furthermore, RMediation generates percentiles, quantiles, and the plot of the distribution and CI for the mediated effect. An existing program, called PRODCLIN, published in Behavior Research Methods, has been widely cited and used by researchers to build accurate CIs. PRODCLIN has several limitations: The program is somewhat cumbersome to access and yields no result for several cases. RMediation described herein is based on the widely available R software, includes several capabilities not available in PRODCLIN, and provides accurate results that PRODCLIN could not.

Identification-Robust Estimation and Testing of the Zero-Beta CAPM

We propose exact simulation-based procedures for: (i) testing mean-variance efficiency when the zero-beta rate is unknown, and (ii) building confidence intervals for the zero-beta rate. On observing that this parameter may be weakly identified, we propose LR-type statistics as well as heteroskedascity and autocorrelation corrected (HAC) Wald-type procedures, which are robust to weak identification and allow for non-Gaussian distributions including parametric GARCH structures. In particular, we propose confidence sets for the zero-beta rate based on “inverting” exact tests for this parameter; these sets provide a multivariate extension of Fieller’s technique for inference on ratios. The exact distribution of LR-type statistics for testing efficiency is studied under both the null and the alternative hypotheses. The relevant nuisance parameter structure is established and finite-sample bound procedures are proposed, which extend and improve available Gaussianspecific bounds. Furthermore, we study the invariance to portfolio repacking property for tests and confidence sets proposed. The statistical properties of available and proposed methods are analyzed via aMonte Carlo study. Empirical results on NYSE returns show that exact confidence sets are very different from the asymptotic ones, and allowing for non-Gaussian distributions affects inference results. Simulation and empirical results suggest that LR-type statistics - with p-values corrected using the Maximized Monte Carlo test method - are generally preferable to their Wald-HAC counterparts from the viewpoints of size control and power.

On Capability Indices for Multivariate Autocorrelated Processes

In this paper the effects of the autocorrelation on some multivariate capability indices commonly used for independent processes are discussed and a correction is proposed. Some results are shown for VARMA(1,1) and VAR(1) time series processes under the multivariate normality assumption and the proportion of non-conforming units is calculated for some bivariate VAR(1) models. An extension of Veevers capability index for non-centered processes is also a subject addressed in this paper. An example of application in blast charcoal furnace pig iron process is presented and bootstrap is used to build confidence intervals for its true capability value as well as to evaluate the performance of the capability estimators. Similar as to what is already known for univariate processes the results showed that autocorrelation has a large impact in the multivariate capabilities indices. This paper also shows that some care should be taken when using Niverthi and Dey’s capabilities indices since they are very sensitive to any deviations from the process means to the specification means up to a point that a capable process might be considered non-capable.

Selecting a binary Markov model for a precipitation process

This paper uses rth-order categorical Markov chains to model the probability of precipitation. Several stationary and non-stationary high-order Markov models are proposed and compared using BIC. The number of parameters increases exponentially by adding the Markov order. Several classes of high-order Markov models are proposed which their increase of number of parameters are modest. For example models that use the number of precipitation days in a period prior to date, temperature of the previous day and sines/cosines periodic functions (to model the seasonality) are considered. The theory of partial likelihood is used to estimate the parameters. Parsimonious non-stationary first order Markov models with few seasonal terms are found optimal using BIC and temperature does not turn out to be a useful covariate. However BIC seems to underestimate the number of seasonal terms. We have also compared the results with AIC in some cases which tends to pick parsimonious models with more seasonal terms and higher order. We also show that ignoring seasonal terms result in picking higher order Markov chains. Finally we apply the methods to build confidence intervals for the probability of periods with no precipitation or low number of precipitation days in Calgary using historical data from 1980 to 2000.

Comparing Interval Estimators for Reliability in a Dependent Set-up

In this paper some procedures for building confidence intervals for the reliability in stress-strength models are discussed and empirically compared. The particular case of a bivariate normal setup is considered. The confidence intervals suggested are obtained employing approximations or asymptotic properties of maximum likelihood estimators. The coverage and the precision of these intervals are empirically checked through a simulation study. An application to real paired data is also provided. Keywords—approximate estimators, asymptotic theory, confidence interval, Monte Carlo simulations, stress-strength, variance estimation.

Asymptotic distribution of conical-hull estimators of directional edges

Nonparametric data envelopment analysis (DEA) estimators have been widely applied in analysis of productive efficiency. Typically they are defined in terms of convex-hulls of the observed combinations of $\mathrm{inputs}\times\mathrm{outputs}$ in a sample of enterprises. The shape of the convex-hull relies on a hypothesis on the shape of the technology, defined as the boundary of the set of technically attainable points in the $\mathrm{inputs}\times\mathrm{outputs}$ space. So far, only the statistical properties of the smallest convex polyhedron enveloping the data points has been considered which corresponds to a situation where the technology presents variable returns-to-scale (VRS). This paper analyzes the case where the most common constant returns-to-scale (CRS) hypothesis is assumed. Here the DEA is defined as the smallest conical-hull with vertex at the origin enveloping the cloud of observed points. In this paper we determine the asymptotic properties of this estimator, showing that the rate of convergence is better than for the VRS estimator. We derive also its asymptotic sampling distribution with a practical way to simulate it. This allows to define a bias-corrected estimator and to build confidence intervals for the frontier. We compare in a simulated example the bias-corrected estimator with the original conical-hull estimator and show its superiority in terms of median squared error.

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Yi, Zi), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( +d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and cg(·), dg(·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density fZ, the class \({\mathcal{G}}\), the kernel K and the functions cg(·), dg(·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.

Investigating the effect of modeling single-vehicle and multi-vehicle crashes separately on confidence intervals of Poisson–gamma models

Crash prediction models still constitute one of the primary tools for estimating traffic safety. These statistical models play a vital role in various types of safety studies. With a few exceptions, they have often been employed to estimate the number of crashes per unit of time for an entire highway segment or intersection, without distinguishing the influence different sub-groups have on crash risk. The two most important sub-groups that have been identified in the literature are single- and multi-vehicle crashes. Recently, some researchers have noted that developing two distinct models for these two categories of crashes provides better predicting performance than developing models combining both crash categories together. Thus, there is a need to determine whether a significant difference exists for the computation of confidence intervals when a single model is applied rather than two distinct models for single- and multi-vehicle crashes. Building confidence intervals have many important applications in highway safety. This paper investigates the effect of modeling single- and multi-vehicle (head-on and rear-end only) crashes separately versus modeling them together on the prediction of confidence intervals of Poisson–gamma models. Confidence intervals were calculated for total (all severities) crash models and fatal and severe injury crash models. The data used for the comparison analysis were collected on Texas multilane undivided highways for the years 1997–2001. This study shows that modeling single- and multi-vehicle crashes separately predicts larger confidence intervals than modeling them together as a single model. This difference is much larger for fatal and injury crash models than for models for all severity levels. Furthermore, it is found that the single- and multi-vehicle crashes are not independent. Thus, a joint (bivariate) model which accounts for correlation between single- and multi-vehicle crashes is developed and it predicts wider confidence intervals than a univariate model for all severities. Finally, the simulation results show that separate models predict values that are closer to the true confidence intervals, and thus this research supports previous studies that recommended modeling single- and multi-vehicle crashes separately for analyzing highway segments.

Covariances between regression coefficient estimates in a single mediator model.

This study presents formulae for the covariances between parameter estimates in a single mediator model. These covariances are necessary to build confidence intervals (CI) for effect size measures in mediation studies. We first analytically derived the covariances between the parameter estimates in a single mediator model. Using the derived covariances, we computed the multivariate-delta standard errors, and built the 95% CIs for the effect size measures. A simulation study evaluated the accuracy of the standard errors as well as the Type I error, power, and coverage of the CIs using various parameter values and sample sizes. Finally, we presented a numerical example and a SAS MACRO that calculates the CIs for the effect size measures.

The Use of a Clinical Utility Index to Compare Insomnia Compounds: A Quantitative Basis for Benefit–Risk Assessment

The use of a clinical utility index (CUI) was proposed in order to compare two calcium channel alpha2delta ligands that were in development for the treatment of insomnia. The important attributes included in the CUI were two measures of residual sedation and five measures of efficacy (wake after sleep onset, sleep quality, sleep latency, and sleep stages (stage 1 and stages 3-4)). Dose-response analyses were conducted on each end point, and a sensitivity analysis was conducted to determine a clinically meaningful difference in CUI. Nonparametric bootstrap parameters were used to build confidence intervals (CIs). Peak CUI (80% CI) was 0.345 (0.25-0.43), observed at a dose of approximately 30 mg with the lead compound and 0.436 (0.35-0.52) observed at >600-mg dose for the backup. Although CUI was slightly greater for the backup, peak CUI values were observed at doses that were not considered viable, and therefore development of the ligand was discontinued. The use of the CUI allowed an efficient, quantitative, and transparent decision.

Use of sensitivity analysis to aid interpretation of a probabilistic Bacillus cereus spore lag time model applied to heat-treated chilled foods (REPFEDs)

The microbiological safety and quality of REfrigerated Processed Foods of Extended Durability (REPFEDs) relies on a combination of mild heat treatment and refrigeration, sometimes in combination with other inhibitory agents that are not effective when used alone. In this context, the output of a probabilistic model predicting the lag time of heat-treated Bacillus cereus spores under realistic heat-treatment profile and chilled supply-chain conditions, has been investigated using a sensitivity analysis technique. Indeed, knowing that there was uncertainty in the model (e.g. due to lack of data to build the model input probability density function), the objective of the analysis was to evaluate if the variability associated with some inputs (e.g. the consumers' refrigerator temperature values reported in Europe and US markets were different) had a significant impact on the model output, i.e. on the expected lag time of heat-treated B. cereus spores in REPFEDs. To do so, the uncertainty and variability associated with the various model inputs have been identified and then separated using a second order Monte Carlo decomposition. Concerning the variability, there was a significant difference between the chilled supply-chains (Europe, US) and between the raw material groups (low, medium or high contamination levels). For example, in the European market, after a heat treatment of 90 °C for 10 min, with a high raw material contamination level, the predicted 5th percentile of the lag time was 17 days, while it was 35 days with a low raw material contamination level. This was confirmed with an ANOVA. The impact of the uncertainty on the lag time has been illustrated graphically by building confidence intervals around its 5th percentile. A sensitivity analysis based upon uncertainty and variability decomposition is clearly a complex and time consuming exercise; however, it provides a greater confidence (greater transparency and better understanding) in the model output when making food safety decisions (e.g. determining the safe shelf-life of REPFEDs).

Approximate and generalized confidence bands for the mean and mode functions of the lognormal diffusion process

Approximate and generalized confidence bands for the mean and mode functions of the univariate lognormal diffusion process are obtained. To this end, the already existing methods for building confidence intervals for the mean of the lognormal distribution have been suitably adapted. Moreover, a new method has been proposed. The bands obtained from the above procedures are compared through a simulation study and the comparisons are made both in terms of coverage errors and average widths. This comparative study allows to choose the most appropriate confidence band for each particular case in practical situations. This is shown in an application to a real data set.