Bootstrap approximation of nearest neighbor regression function estimates

Let (X, Y) be a random vector in the plane and denote by m(x) = (Y|X = x) the corresponding regression function. We show that the bootstrap approximation for the distribution of a smoothed nearest neighbor estimate of m(x) is valid. Also we compare, by Monte Carlo, confidence intervals which are obtained from both the normal and the bootstrap approximation.     

A smoothed bootstrap estimator for a studentized sample quantile

If the underlying distribution functionF is smooth it is known that the convergence rate of the standard bootstrap quantile estimator can be improved fromn ^–1/4 ton ^–1/2+, for arbitrary >0, by using a smoothed bootstrap. We show that a further significant improvement of this rate is achieved by studentizing by means of a kernel density estimate. As a consequence, it turns out that the smoothed bootstrap percentile-t method produces confidence intervals with critical points being second-order correct and having smaller length than competitors based on hybrid or on backwards critical points. Moreover, the percentile-t method for constructing one-sided or two-sided confidence intervals leads to coverage accuracies of ordern ^–1+, for arbitrary >0, in the case of analytic distribution functions.     

Resampling student'st-type statistics

The present paper establishes conditional and unconditional central limit theorems for various resampling procedures for thet-statistic. The results work under fairly general conditions and the underlying random variables need not to be independent. Specific examples are then them(n) (double) bootstrap out ofk(n) observations, the Bayesian bootstrap and two-samplet-type permutation statistics. In case whenm(n)/k(n) is bounded away from zero and infinity necessary and sufficient conditions for the conditional central limit law of the bootstrapt-statistics are established. For high resampling intensity whenm(n)/k(n) tends to infinity the following general result is obtained. Without further other assumptions the bootstrap makes the resampledt-statistic automatically normal. The results are based on a general conditional limit theorem for weighted resampling statistics which is of own interest.     

Unusual properties of bootstrap confidence intervals in regression problems

Summary We show that the percentile-t method, and one of the two percentile methods, have unusually good performance when employed to construct bootstrap confidence intervals in a regression setting. In the case of slope parameters, percentile-t produces two-sided intervals with coverage errorn ^-2, and one-sided intervals with coverage errorn ^-3/2, wheren is sample size. The errors are onlyn ^-1 in most other problems. One of the percentile methods produces critical points which are third-order correct for Efron's [11] relatively complex accelerated bias-corrected points.     

An evaluation of the multivariate dispersion charts with estimated parameters under non‐normality

Various charts such as |S|, W, and G are used for monitoring process dispersion. Most of these charts are based on the normality assumption, while exact distribution of the control statistic is unknown, and thus limiting distribution of control statistic is employed which is applicable for large sample sizes. In practice, the normality assumption of distribution might be violated, while it is not always possible to collect large sample size. Furthermore, to use control charts in practice, the in‐control state usually has to be estimated. Such estimation has a negative effect on the performance of control chart. Non‐parametric bootstrap control charts can be considered as an alternative when the distribution is unknown or a collection of large sample size is not possible or the process parameters are estimated from a Phase I data set. In this paper, non‐parametric bootstrap multivariate control charts |S|, W, and G are introduced, and their performances are compared against Shewhart‐type control charts. The proposed method is based on bootstrapping the data used for estimating the in‐control state. Simulation results show satisfactory performance for the bootstrap control charts. Ultimately, the proposed control charts are applied to a real case study.     

Consistency of general bootstrap methods for degenerate -type and -type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.     

On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations

The asymptotic distribution of some test criteria for a covariance matrix are derived under local alternatives. Except for the existence of some higher moments, no assumption as to the form of the distribution function is made. As an illustration, a case of t distribution included normal model is considered and the power of the likelihood ratio test and Nagao's test for sphericity, as described in Srivastava and Khatri and Anderson, is computed. Also, the power is computed using the bootstrap method. In the case of t distribution, the bootstrap approximation does not appear to be as good as the one obtained by the asymptotic expansion method.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

On some confidence intervals for estimating the mean of a skewed population

A number of methods are available in the literature to measure confidence intervals. Here, confidence intervals for estimating the population mean of a skewed distribution are considered. This note proposes two alternative confidence intervals, namely, Median t and Mad t, which are simple adjustments to the Student's t confidence interval. In order to compare the performance of these intervals, the following criteria are considered: (i) coverage probability; (ii) average width; and (iii) ratio of coverage to width. A simulation study has been undertaken to compare the performance of the intervals. The simulation study shows that for small sample size and moderate to highly skewed distributions, the proposed Median t performs the best in the sense of higher coverage, and the Mad t performs best in the sense of smaller confidence width. The proposed methods are very easy to calculate and are not overly computer-intensive, like Bootstrap confidence intervals. Some real-life examples have been considered that support the findings of the paper to some extent.     

Global singularity structures of weak solutions to 4-D semilinear dispersive wave equations

In this paper, we are concerned with the global singularity structures of weak solutions to 4-D semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere. Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions are C²−regular away from the focusing cone surface |x|=|t−1| and the outgoing cone surface |x|=t+1. This research was supported by the National Natural Science Foundation of China and the Doctoral Foundation of NEM of China.     

A controlled M / G / 1 workload process with an application to perishable inventory systems

We derive the stationary distribution of the regenerative process W(t), t ≥ 0, whose cycles behave like an M / G / 1 workload process terminating at the end of its first busy period or when it reaches or exceeds level 1, and restarting with some fixed workload . The result is used to obtain the overflow distribution of this controlled workload process; we derive and , where T is the duration of the first cycle. W(t) can be linked to a certain perishable inventory model, and we use our results to determine the distribution of the duration of an empty period.D. Perry was supported by a Mercator Fellowship of the Deutsche Forschungsgemeinschaft.     

Weighted bootstrap for U-statistics

In this paper we investigate the weighted bootstrap for U-statistics and its properties. Under very general choices of random weights and certain regularity conditions, we show that the weighted bootstrap method with U-statistics provides second-order accurate approximations to the distribution of U-statistics. We shall prove this via one-term Edgeworth expansions of weighted U-statistics.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.     

On the Beck‐Fiala conjecture for random set systems

Motivated by the Beck‐Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability ; and when |X| ? |Σ|^t the hereditary discrepancy of (X,Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.     

A process variability control chart

In this study a Shewhart type control chart namely the V _t chart, is proposed for improved monitoring of the process variability of a quality characteristic of interest Y. The proposed control chart is based on the ratio type estimator of the variance using a single auxiliary variable X. It is assumed that (Y, X) follows a bivariate normal distribution. The design structure of the V _t chart is developed for Phase-I quality control and its comparison is made with those of the S ² chart (a well-known Shewhart control chart) and the V _r chart (a Shewhart type control chart proposed by Riaz (Comput Stat, 2008a) used for the same purpose. It is observed that the proposed V _t chart outperforms the S ² and V _r charts, in terms of discriminatory power, for detecting moderate to large shifts in the process variability. It is observed that the performance of the V _t chart keeps improving with an increase in |ρ _yx | , where ρ _yx is the correlation between Y and X.     

A footnote to mean square value of DirichletL-series

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for } Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t^5/8 q^−1/8, which is an improvement in the range q |t| < q^11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for |t|q^9/5.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

L1 regression estimate and its bootstrap

We study the asymptotic distribution of the L ₁ regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution. This work was supported by J.C. Bose National Fellowship, Government of India