Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 2014 is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post‐model‐selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.     

SOME PROPERTIES OF PROFILE BOOTSTRAP CONFIDENCE INTERVALS

We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ₀ in the presence of a (possibly infinite dimensional) nuisance parameter ψ₀. It is supposed that the value taken by θ₀ does not restrict the value that ψ₀ may take and vice-versa. Given a sensible estimate ψ_n of ψ₀, profile bootstrap confidence interval for θ₀ is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ₀=ψ_n. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ₀ and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity.     

On the bootstrap in cube root asymptotics

The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

A weighted bootstrap approach to bootstrap iteration

The operation of resampling from a bootstrap resample, encountered in applications of the double bootstrap, maybe viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample. This suggests an approximate approach to double-bootstrap Monte Carlo simulation, where weighted bootstrap methods are used to circumvent much of the labour involved in compounded Monte Carlo approximation. In the case of distribution estimation or, equivalently, confidence interval calibration, the new method may be used to reduce the computational labour. Moreover, the method produces the same order of magnitude of coverage error for confidence intervals, or level error for hypothesis tests, as a full application of the double bootstrap.     

Better nonparametric bootstrap confidence intervals for the correlation coefficient

We respond to criticism leveled at bootstrap confidence intervals for the correlation coefficient by recent authors by arguing that in the correlation coefficient case, non–standard methods should be employed. We propose two such methods. The first is a bootstrap coverage coorection algorithm using iterated bootstrap techniques (Hall, 1986; Beran, 1987a; Hall and Martin, 1988) applied to ordinary percentile–method intervals (Efron, 1979), giving intervals with high coverage accuracy and stable lengths and endpoints. The simulation study carried out for this method gives results for sample sizes 8, 10, and 12 in three parent populations. The second technique involves the construction of percentile–t bootstrap confidence intervals for a transformed correlation coefficient, followed by an inversion of the transformation, to obtain “transformed percentile–t” intervals for the correlation coefficient. In particular, Fisher's z–transformation is used, and nonparametric delta method and jackknife variance estimates are used to Studentize the transformed correlation coefficient, with the jackknife–Studentized transformed percentile–t interval yielding the better coverage accuracy, in general. Percentile–t intervals constructed without first using the transformation perform very poorly, having large expected lengths and erratically fluctuating endpoints. The simulation study illustrating this technique gives results for sample sizes 10, 15 and 20 in four parent populations. Our techniques provide confidence intervals for the correlation coefficient which have good coverage accuracy (unlike ordinary percentile intervals), and stable lengths and endpoints (unlike ordinary percentile–t intervals).     

Bootstrap in functional linear regression

We have considered the functional linear model with scalar response and functional explanatory variable. One of the most popular methodologies for estimating the model parameter is based on functional principal components analysis (FPCA). In recent literature, weak convergence for a wide class of FPCA-type estimates has been proved, and consequently asymptotic confidence sets can be built. In this paper, we have proposed an alternative approach in order to obtain pointwise confidence intervals by means of a bootstrap procedure, for which we have obtained its asymptotic validity. Besides, a simulation study allows us to compare the practical behaviour of asymptotic and bootstrap confidence intervals in terms of coverage rates for different sample sizes.     

A Comparison of Approaches to Interval Estimation of Mediated Effects in Linear Regression Models

A novel approach based on the concepts of a generalized pivotal quantity (GPQ) is developed to construct confidence intervals for the mediated effect. Thereafter, its performance is compared with six interval estimation approaches in terms of empirical coverage probability and expected length via simulation and two real examples. The results show that the GPQ-based and bootstrap percentile methods outperform other methods when mediated effects exist in small and medium samples. Moreover, the GPQ-based method exhibits a more stable performance in small and non-normal samples. A discussion on how to choose the best interval estimation method for mediated effects is presented.     

The p-values for one-sided hypothesis testing in univariate linear calibration

In this article, we focus on the one-sided hypothesis testing for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved. The observations of the response variable corresponding to known values of the explanatory variable are used to make inferences on a single unknown value of the explanatory variable. We apply the generalized inference to the calibration problem, and take the generalized p-value as the test statistic to develop a new p-value for one-sided hypothesis testing, which we refer to as the one-sided posterior predictive p-value. The behavior of the one-sided posterior predictive p-value is numerically compared with that of the generalized p-value, and simulations show that the proposed p-value is quite satisfactory in the frequentist performance.     

ON THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER VARIABLE SELECTION

This paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ= a^Tβ 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.     

EMPIRICAL LIKELIHOOD METHODS FOR THE GINI INDEX

The Gini index and its generalizations have been used extensively for measuring inequality and poverty in the social sciences. Recently, interval estimation based on nonparametric statistics has been proposed in the literature, for example the naive bootstrap method, the iterated bootstrap method and the bootstrap method via a pivotal statistic. In this paper, we propose empirical likelihood methods to construct confidence intervals for the Gini index or the difference of two Gini indices. Simulation studies show that the proposed empirical likelihood method performs slightly worse than the bootstrap method based on a pivotal statistic in terms of coverage accuracy, but it requires less computation. However, the bootstrap calibration of the empirical likelihood method performs better than the bootstrap method based on a pivotal statistic.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

Simultaneous confidence interval for assessing non‐inferiority with assay sensitivity in a three‐arm trial with binary endpoints

A three‐arm trial including an experimental treatment, an active reference treatment and a placebo is often used to assess the non‐inferiority (NI) with assay sensitivity of an experimental treatment. Various hypothesis‐test‐based approaches via a fraction or pre‐specified margin have been proposed to assess the NI with assay sensitivity in a three‐arm trial. There is little work done on confidence interval in a three‐arm trial. This paper develops a hybrid approach to construct simultaneous confidence interval for assessing NI and assay sensitivity in a three‐arm trial. For comparison, we present normal‐approximation‐based and bootstrap‐resampling‐based simultaneous confidence intervals. Simulation studies evidence that the hybrid approach with the Wilson score statistic performs better than other approaches in terms of empirical coverage probability and mesial‐non‐coverage probability. An example is used to illustrate the proposed approaches.     

Exact Short Confidence Intervals from Discrete Data

Suppose that X is a discrete random variable whose possible values are {0, 1, 2,⋯} and whose probability mass function belongs to a family indexed by the scalar parameter θ . This paper presents a new algorithm for finding a 1 − α confidence interval for θ based on X which possesses the following three properties: (i) the infimum over θ of the coverage probability is 1 − α ; (ii) the confidence interval cannot be shortened without violating the coverage requirement; (iii) the lower and upper endpoints of the confidence intervals are increasing functions of the observed value x . This algorithm is applied to the particular case that X has a negative binomial distribution.     

Uniformly most powerful unbiased test in univariate linear calibration

In this paper, we deal with the hypothesis testing problems for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved, and the observations of the response variable corresponding to known values of the explanatory variable are used for making inferences concerning a single unknown value of the explanatory variable. The uniformly most powerful unbiased tests for both one-sided and two-sided hypotheses are constructed and verified. The power behaviour of the proposed tests is numerically compared with that of the existing method, and simulations show that the proposed tests make the powers improved.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.