Bootstrap approaches for estimation and confidence intervals of long memory processes

In this work, we investigate an alternative bootstrap approach based on a result of Ramsey [F.L. Ramsey, Characterization of the partial autocorrelation function, Ann. Statist. 2 (1974), pp. 1296–1301] and on the Durbin–Levinson algorithm to obtain a surrogate series from linear Gaussian processes with long range dependence. We compare this bootstrap method with other existing procedures in a wide Monte Carlo experiment by estimating, parametrically and semi-parametrically, the memory parameter d. We consider Gaussian and non-Gaussian processes to prove the robustness of the method to deviations from normality. The approach is also useful to estimate confidence intervals for the memory parameter d by improving the coverage level of the interval.     

Using jackknife in nonlinear models - confidence regions for a function of the structural parameter

A method of constructing jackknife confidence regions for a function of the structural parameter of a nonlinear model is investigated. The method is available for nonlinear regression models as well as for models with errors in the variables. Properties are discussed in comparison with traditional methods. This is supported by a simulation study.     

Nonparametric standard errors and confidence intervals

We investigate several nonparametric methods; the bootstrap, the jackknife, the delta method, and other related techniques. The first and simplest goal is the assignment of nonparametric standard errors to a real-valued statistic. More ambitiously, we consider setting nonparametric confidence intervals for a real-valued parameter. Building on the well understood case of confidence intervals for the median, some hopeful evidence is presented that such a theory may be possible.     

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.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

Small sample confidence intervals for survival functions under the proportional hazards model

We develop a saddlepoint-based method for generating small sample confidence bands for the population surviival function from the Kaplan-Meier (KM), the product limit (PL), and Abdushukurov-Cheng-Lin (ACL) survival function estimators, under the proportional hazards model. In the process we derive the exact distribution of these estimators and developed mid-ppopulation tolerance bands for said estimators. Our saddlepoint method depends upon the Mellin transform of the zero-truncated survival estimator which we derive for the KM, PL, and ACL estimators. These transforms are inverted via saddlepoint approximations to yield highly accurate approximations to the cumulative distribution functions of the respective cumulative hazard function estimators and these distribution functions are then inverted to produce our saddlepoint confidence bands. For the KM, PL and ACL estimators we compare our saddlepoint confidence bands with those obtained from competing large sample methods as well as those obtained from the exact distribution. In our simulation studies we found that the saddlepoint confidence bands are very close to the confidence bands derived from the exact distribution, while being much easier to compute, and outperform the competing large sample methods in terms of coverage probability.     

Repeated confidence intervals and prediction intervals using stochastic curtailment

Jennlson and Turnbull (1984,1989) proposed procedures for repeated confidence intervals for parameters of interest In a clinical trial monitored with group sequential methods. These methods are extended for use with stochastic curtailment procedures for two samples in the estimation of differences of means, differences of proportions, odds ratios, and hazard ratios. Methods are described for constructing 1) confidence intervals for these estimates at repeated times In the course of a trial, and 2) prediction intervals for predicted estimates at the end of a trial. Specific examples from several clinical trials are presented.     

Large sample confidence intervals for the skewness parameter of the skew-normal distribution based on Fisher's transformation

The skew-normal model is a class of distributions that extends the Gaussian family by including a skewness parameter. This model presents some inferential problems linked to the estimation of the skewness parameter. In particular its maximum likelihood estimator can be infinite especially for moderate sample sizes and is not clear how to calculate confidence intervals for this parameter. In this work, we show how these inferential problems can be solved if we are interested in the distribution of extreme statistics of two random variables with joint normal distribution. Such situations are not uncommon in applications, especially in medical and environmental contexts, where it can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Loperfido [7 Loperfido, N. 2002. Statistical implications of selectively reported inferential results. Statist. Probab. Lett., 56: 13–22. [Crossref], [Web of Science ®] , [Google Scholar]], proves that such extreme statistics have a skew-normal distribution with skewness parameter that can be expressed as a function of the correlation coefficient between the two initial variables. It is then possible, using some theoretical results involving the correlation coefficient, to find approximate confidence intervals for the parameter of skewness. These theoretical intervals are then compared with parametric bootstrap intervals by means of a simulation study. Two applications are given using real data.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Shortest confidence and prediction intervals for the log-normal

We provide the shortest prediction interval for X, and the shortest confidence interval for the median of X, when X has the log-normal distribution for both the case σ², the variance of log X, known and unknown. Tables are given to assist the practitioner in constructing these intervals. A real-life example is provided to illustrate the results.     

Efficient computation of the performance of bootstrap and jackknife estimators of the variance of L-statistics

Calculation of the bootstrap and the jackknife estimators of the variance of a statistic often relies on approximation techniques because the exact values are difficult if not impossible to obtain analytically. For the special case where the statistic is a linear combination of order statistics we propose to calculate the exact values combinatorically, thus completely eliminating the second-stage simulation error.     

Comparing confidence intervals for the logarithmic series distribution

It is assumed that a small random sample of fixed size n is drawn from a logarithmic series distribution with parameter θ and that it is desired to estimate θ by means of a two-sided confidence interval. In this note Crow's system of confidence intervals is compared, in shortness of intervals, with Clopper and Pearson's, and the corresponding randomized counterparts.     

Reliable confidence intervals for assessing normal process incapability

This study aims to provide a reliable confidence interval for assessing the process incapability index [C_pp]. The concept of the generalized pivotal quantities is utilized for constructing the generalized confidence interval for [C_pp]. And, simulations are performed for demonstrating our proposed method and one existent method. The results show that the empirical confidences of these two methods are significantly affected by the degree of process departure. Therefore, we suggest the practitioners to select proper one for capability testing purpose based on the information of degree of process departure.     

On the lengths of t-based confidence intervals

Confidence interval is a basic type of interval estimation in statistics. When dealing with samples from a normal population with the unknown mean and the variance, the traditional method to construct t-based confidence intervals for the mean parameter is to treat the n sampled units as n groups and build the intervals. Here we propose a generalized method. We first divide them into several equal-sized groups and then calculate the confidence intervals with the mean values of these groups. If we define “better” in terms of the expected length of the confidence interval, then the first method is better because the expected length of the confidence interval obtained from the first method is shorter. We prove this intuition theoretically. We also specify when the elements in each group are correlated, the first method is invalid, while the second can give us correct results in terms of the coverage probability. We illustrate this with analytical expressions. In practice, when the data set is extremely large and distributed in several data centers, the second method is a good tool to get confidence intervals, in both independent and correlated cases. Some simulations and real data analyses are presented to verify our theoretical results.     

Short confidence intervals for variance components

Four approximate methods are proposed to construct confidence intervals for the estimation of variance components in unbalanced mixed models. The first three methods are modifications of the Wald, arithmetic and harmonic mean procedures, see Harville and Fenech (1985), while the fourth is an adaptive approach, combining the arithmetic and harmonic mean procedures. The performances of the proposed methods were assessed by a Monte Carlo simulation study. It was found that the intervals based on Wald's method maintained the nominal confidence levels across all designs and values of the parameters under study. On the other hand, the arithmetic (harmonic) mean method performed well for small (large) values of the variance component, relative to the error variance component. The adaptive procedure performed rather well except for extremely unbalanced designs. Further, compared with equal tails intervals, the intervals which use special tables, e.g., Table 678 of Tate and Klett (1959), provided adequate coverage while having much shorter lengths and are thus recommended for use in practice.     

The conditional level of confidence intervals for the normal variance and applications

Suppose we observe two independent random vectors each having a multivariate normal distribution with covariance matrix known up to an unknown scale factor σ . The first random vector has a known mean vector while the second has an unknown mean vector. Interest centers around finding confidence intervals for σ² with confidence coefficient 1 ? α. Standard results show that, when we only observe the first random vector, an optimal (i.e., smallest length) confidence interval C, based on the well-known chi- squared statistic, can be constructed for σ² . When we additionally observe the second random vector, the confidence interval C is no longer optimal for estimating σ². One criterion useful for detecting the non-optimality of a confidence interval C concerns whether C admits positively or negatively biased relevant subsets. This criterion has recently received a good deal of attention. It is shown here that under some conditions the confidence interval C admits positively biased relevant subsets.

Applications of this result to the construction of ‘better‘ unconditional confidence intervals for σ² are presented. Some simulation results are given to indicate the typical extent of improvement attained.