Confidence intervals centred on bootstrap smoothed estimators

SummaryThis paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ= aTβ where a is specified. When, as a first step, a data‐based variable selection (e.g. minimum Akaike information criterion) is used to select a model, it is common statistical practice to then carry out inference about θ, using the same data, based on the (false) assumption that the selected model had been provided a priori. The paper considers a confidence interval for θ with nominal coverage 1 ‐ α constructed on this (false) assumption, and calls this the naive 1 ‐ α confidence interval. The minimum coverage probability of this confidence interval can be calculated for simple variable selection procedures involving only a single variable. However, the kinds of variable selection procedures used in practice are typically much more complicated. For the real‐life data presented in this paper, there are 20 variables each of which is to be either included or not, leading to 220 different models. The coverage probability at any given value of the parameters provides an upper bound on the minimum coverage probability of the naive confidence interval. This paper derives a new Monte Carlo simulation estimator of the coverage probability, which uses conditioning for variance reduction. For these real‐life data, the gain in efficiency of this Monte Carlo simulation due to conditioning ranged from 2 to 6. The paper also presents a simple one‐dimensional search strategy for parameter values at which the coverage probability is relatively small. For these real‐life data, this search leads to parameter values for which the coverage probability of the naive 0.95 confidence interval is 0.79 for variable selection using the Akaike information criterion and 0.70 for variable selection using Bayes information criterion, showing that these confidence intervals are completely inadequate.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

The difference of two independent binomial proportions is frequently of interest in biomedical research. The interval estimation may be an important tool for the inferential problem. Many confidence intervals have been proposed. They can be classified into the class of exact confidence intervals or the class of approximate confidence intervals. Ore may prefer exact confidence interval s in that they guarantee the minimum coverage probability greater than the nominal confidence level. However, someone, for example Agresti and Coull (1998) claims that "approximation is better than exact." It seems that when sample size is large, the approximate interval is more preferable to the exact interval. However, the choice is not clear when sample, size is small. In this note, an exact confidence and an approximate confidence interval, which were recommended by Santner et al. (2007) and Lee (2006b), respectively, are compared in terms of the coverage probability and the expected length.

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.

This thesis proposes some confidence intervals for the mean of a positively skewed distribution. The following confidence intervals are considered: Student-t, Johnson-t, median-t, mad-t, bootstrap-t, BCA, T1 , T3 and six new confidence intervals, the median bootstrap-t, mad bootstrap-t, median T1, mad T1 , median T3 and the mad T3. A simulation study has been conducted and average widths, coefficient of variation of widths, and coverage probabilities were recorded and compared across confidence intervals. To compare confidence intervals, the width and coverage probabilities were compared so that smaller widths indicated a better confidence interval when coverage probabilities were the same. Results showed that the median T1 and median T3 outperformed other confidence intervals in terms of coverage probability and the mad bootstrap-t, mad-t, and mad T3 outperformed others in terms of width. Some real life data are considered to illustrate the findings of the thesis.

This paper investigates the performance of the confidence intervals for a single inverse mean and the difference of two inverse means of normal distributions with unknown coefficients of variation (CVs). The confidence intervals for the inverse mean with unknown coefficient of variation (CV) were constructed based on the generalized confidence interval (GCI) approach and the large sample approach. The generalized confidence interval and large sample confidence interval for the inverse mean with unknown CV were compared with the generalized confidence interval for the inverse mean of Niwitpong and Wongkhao [5]. Moreover, the confidence intervals for the difference of inverse means with unknown CVs were constructed using the GCI approach, large sample approach and method of variance estimates recovery (MOVER) approach and then compared with existing confidence interval for the difference of inverse means based on the GCI approach of Niwitpong and Wongkhao [6]. The coverage probability and average length of the confidence intervals were evaluated by a Monte Carlo simulation. Carrying out the simulation studies, the results showed that the generalized confidence interval provides the best confidence interval for the inverse mean with unknown CV. The generalized confidence interval and the MOVER confidence interval for the difference of inverse means with unknown CVs perform well in terms of the coverage probability and average length. Finally, two real datasets are analyzed to illustrate the proposed confidence intervals.

We consider a linear regression model with regression parameters (θ1,...,θp) and error variance parameter σ2. Our aim is to find a confidence interval with minimum coverage probability 1 − α for a parameter of interest θ1 in the presence of nuisance parameters (θ2,...,θp,σ2). We consider two confidence intervals, the first of which is the standard confidence interval for θ1 with coverage probability 1 − α. The second confidence interval for θ1 is obtained after a variable selection procedure has been applied to θp. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 − α. The confidence intervals are compared using a risk function that is defined as a scaled version of the expected length of the confidence interval. We show that, subject to certain conditions including that [(dimension of response vector) − p] is small, the second confidence interval is preferable to the first when we anticipate (without being certain) that |θp|/σ is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.

A conditional approach to robustness is described and applied to interval estimation for a location parameter. Two new types of confidence interval estimators using two-dimensional numerical integrations within location-and-scale configurations emerge. Strong confidence intervals have robust conditional coverage probabilities. Bioptimal confidence intervals are as short as possible and have robust overall coverage probabilities. Robust intervals compromising the Gaussian and the slash distributions serve as examples. A comparison with a variety of other confidence intervals shows the nonrobustness of rank-based interval estimators. For samples of size 10, robust intervals based on M-estimators are clearly superior. Tuning constants must be chosen carefully for interval estimation. An interval based on Tukey's biweight, for example, should be used with tuning constants of about 7, 8, and 11 for samples of size 20, 10, and 5, respectively.

PurposeConfidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance.Design/methodology/approachThe construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation.FindingsThe GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR.Originality/valueThis research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance.

If it hasn’t failed, does it work? On ‘the worst we can expect’ from observational trial results, with reference to airway management devices

The research on statistical inference after data-driven model selection can be traced as far back as Koopmans (1949). The intensive research on modern model selection methods for high-dimensional data over the past three decades revived the interest in statistical inference after model selection. In recent years, there has been a surge of articles on statistical inference after model selection and now a rather vast literature exists on this topic. Our manuscript aims at presenting a holistic review of post-model-selection inference in linear regression models, while also incorporating perspectives from high-dimensional inference in these models. We first give a simulated example motivating the necessity for valid statistical inference after model selection. We then provide theoretical insights explaining the phenomena observed in the example. This is done through a literature survey on the post-selection sampling distribution of regression parameter estimators and properties of coverage probabilities of naïve confidence intervals. Categorized according to two types of estimation targets, namely the population- and projection-based regression coefficients, we present a review of recent uncertainty assessment methods. We also discuss possible pros and cons for the confidence intervals constructed by different methods.

The shrinkage parameters in the ridge regression model have been extensively discussed and compared in the literature. Typically, the mean square error is used as the primary criterion for comparison. However, it does not fully explain the inferential performance of the estimator. This paper aims to examine 18 ridge regression regularization parameters based on their coverage probability and confidence interval widths using a simulation approach under various conditions. The results reveal that even though most estimators exhibit narrower confidence intervals compared to ordinary least squares, the shrinkage parameters that demonstrate a lower mean square error do not consistently maintain a coverage probability of 95. Additionally, increasing collinearity widens the width of the confidence interval. This paper studies the impact of multicollinearity on confidence interval coverage in linear regression models and provides information for researchers interested in inference based on confidence intervals.

This paper compares the ordinary unweighted average, weighted average, and maximum likelihood methods for estimating a common bioactivity from multiple parallel line bioassays. Some of these or similar methods are also used in meta-analysis. Based on a simulation study, these methods are assessed by comparing coverage probabilities of the true relative bioactivity and the length of the confidence intervals computed for these methods. The ordinary unweighted average method outperforms all statistical methods by consistently giving the best coverage probability but with somewhat wider confidence intervals. The weighted average methods give good coverage and smaller confidence intervals when combining homogeneous bioactivities. For heterogeneous bioactivities, these methods work well when a liberal significance level for testing homogeneity of bioactivities is used. The maximum likelihood methods gave good coverage when homogeneous bioactivities were considered. Overall, the preferred methods are the ordinary unweighted average and two weighted average methods that were specifically developed for bioassays.

For regression analysis of doubly truncated data, we propose two empirical likelihood (EL) inference approaches, called non-smooth EL and non-smooth Jackknife EL (JEL), to make inference about regression parameters based on the generalized estimating equations of existing weighted rank estimators. The limiting distributions of non-smooth log-EL and log-JEL ratios statistics are derived and non-smooth EL, and JEL confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed approaches with the random weighting (RW) approach. The simulation results demonstrate that the non-smooth EL and JEL confidence intervals have better performances than the RW confidence intervals based on coverage probability and average length of confidence intervals of regression parameters when the dependent variable is subject to the double truncation. A real data example is provided to illustrate the proposed approaches.