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.     

Small-sample intervals for regression

For the general linear regression model Y = Xη + e, we construct small-sample exponentially tilted empirical confidence intervals for a linear parameter 6 = a^Tη and for nonlinear functions of η. The coverage error for the intervals is O_p(1/n), as shown in Tingley and Field (1990). The technique, though sample-based, does not require bootstrap resampling. The first step is calculation of an estimate for η. We have used a Mallows estimate. The algorithm applies whenever η is estimated as the solution of a system of equations having expected value 0. We include calculations of the relative efficiency of the estimator (compared with the classical least-squares estimate). The intervals are compared with asymptotic intervals as found, for example, in Hampel et at. (1986). We demonstrate that the procedure gives sensible intervals for small samples.     

Adjusted Confidence Bands in Nonparametric Regression

Suppose we have {(x _i , y _i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.     

Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M‐estimation

The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well‐known open problem. In this paper, we provide a first theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter, which immediately implies the consistency of t‐type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the L₁ maximal inequality condition required in the latter to the L_p maximal inequality condition for p≥1. The general L_p multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, for example, non‐parametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.     

Bootstrap test of goodness of fit to a linear model when errors are correlated

Given the regression model Y_i = m(x_i) +ε_i (x_i ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {ε_i} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: H_o : m ε {mθ(.) = A^t(.)θ : θ ε ? ? R^q} with A a functional from R to R^q. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d²([mcirc]_n, A^t(.)θ_n), between [mcirc]_n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in H_o that is closest to m_n in terms of this distance. The consistency of the bootstrap distribution of D and θ_n is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d².     

Combining unbiased estimates —a further examination of some old estimators

This paper considers the problem of combining k unbiased estimates, x _i of a parameter,μ, where each estimate, x _i is the average of n _i + l independent normal observations with unknown mean, μ, and unknown variance, σ _i ². The behavior of several commonly used estimators of μ is studied by means of an empirical sampling study, and the empirical results of this experiment are interpreted in terms of previous theoretical results. Finally, some extrapolations are made as to how these estimators might behave under varying conditions, and some new estimators are proposed which might have higher efficiencies under certain conditions than those which are generally used.     

Best exact nonparametric confidence intervals for quantiles

Well-known nonparametric confidence intervals for quantiles are of the form (X _{i : n} , X _{j : n} ) with suitably chosen order statistics X _{i : n} and X _{j : n} , but typically their coverage levels differ from those prescribed. It appears that the coverage level of the confidence interval of the form (X _{i : n} , X _{j : n} ) with random indices I and J can be rendered equal, exactly to any predetermined level γ?∈?(0, 1). Best in the sense of minimum E(J???I), i.e., ‘the shortest’, two-sided confidence intervals are constructed. If no two-sided confidence interval exists for a given γ, the most accurate one-sided confidence intervals are constructed.     

Percentile-t Bootstrap Confidence Intervals for Period Using Periodogram

The magnitude of light intensity of many stars varies over time in a periodic way. Therefore, estimation of period and making inference about this parameter are of great interest in astronomy. The periodogram can be used to estimate period, properly. Bootstrap confidence intervals for period suggested here, are based on using the periodogram and constructed by percentile-t methods. We prove that the equal-tailed percentile-t bootstrap confidence intervals for period have an error of order n ^?1. We also show that the symmetric percentile-t bootstrap confidence intervals reduce the error to order n ^?2, and hence have a better performance. Finally, we assess the theoretical results by conducting a simulation study, compare the results with the coverages of percentile bootstrap confidence intervals for period and then analyze a real data set related to the eclipsing system R Canis Majoris collected by Shiraz Biruni Observatory.     

Estimation of P(Y < X) for progressively first-failure-censored generalized inverted exponential distribution

In this article, we consider the problem of estimation of the stress–strength parameter δ?=?P(Y?<?X) based on progressively first-failure-censored samples, when X and Y both follow two-parameter generalized inverted exponential distribution with different and unknown shape and scale parameters. The maximum likelihood estimator of δ and its asymptotic confidence interval based on observed Fisher information are constructed. Two parametric bootstrap boot-p and boot-t confidence intervals are proposed. We also apply Markov Chain Monte Carlo techniques to carry out Bayes estimation procedures. Bayes estimate under squared error loss function and the HPD credible interval of δ are obtained using informative and non-informative priors. A Monte Carlo simulation study is carried out for comparing the proposed methods of estimation. Finally, the methods developed are illustrated with a couple of real data examples.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

THE CALCULATION OF CONFIDENCE REGIONS FOR EIGENVECTORS

An explicit algorithm is given for constructing a confidence region on the appropriate unit sphere for an eigenvector, given a large sample. It is assumed the eigenvector corresponds to the largest eigenvalue of Exx^T, a matrix with distinct eigenvalues, and that the estimation uses n^-1S?x_ix_ix^T. While the theory is generally applicable, the writer has in mind the special case where ∥x∥= 1 i.e. directional data.     

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.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

A Variation on the Coupon Collecting Problem

The classical coupon collector's problem is considered, where each new coupon collected is type i with probability p_i , ∑ ⁿ _{i = 1} p_i = 1. Suppose coupons are collected in a sequence of independent trials. An expression is developed for the probability that all coupon types i, i ≠ j, have been collected prior to collecting r ? 1 coupons of type j in the sequence of trials. Given two different coupon subsets A, B of {1, 2, …, n}, the foregoing is then generalized to an expression for the probability that s ? 1 copies of A appear in the sequence of trials before r ? 1 copies of B. Some computational considerations are discussed.     

Nonparametric Estimation of a Regression Function: Limiting Distribution2

Consider the regression model Y_i= g(x_i) + e_i, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x₀ < x₁ < … < x_n≤ 1 are chosen so that max_1≤i≤n(x_i-x_{i- 1}) = 0(n^-1), and where {e_i} are i.i.d. with Ee₁= 0 and Var e₁ - s?². In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g_1n of Cheng & Lin (1979), g_2n of Clark (1977), and g_3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &_in, i = 1, 2, 3, and that of the isotonic estimator g**_n are considered.     

Some results on bootstrap prediction intervals

We investigate the construction of a BC_a-type bootstrap procedure for setting approximate prediction intervals for an efficient estimator θ_m of a scalar parameter θ, based on a future sample of size m. The results are also extended to nonparametric situations, which can be used to form bootstrap prediction intervals for a large class of statistics. These intervals are transformation-respecting and range-preserving. The asymptotic performance of our procedure is assessed by allowing both the past and future sample sizes to tend to infinity. The resulting intervals are then shown to be second-order correct and second-order accurate. These second-order properties are established in terms of min(m, n), and not the past sample size n alone.     

On the accuracy of the bootstrap approximation of the joint distribution of sample quantiles

It is proved that the accuracy of the bootstrap approximation of the joint distribution of sample quantiles lies between O(n^?1/4) and O(n^?1/4 a_n), where (log(n))1/2=O(a_n). As an application, we investigated confidence intervals based on the bootstrap.