On the asymptotic accuracy of the bootstrap under arbitrary resampling size

We study the order of convergence of the Kolmogorov-Smirnov distance for the bootstrap of the mean and the bootstrap of quantiles when an arbitrary bootstrap sample size is used. We see that for the bootstrap of the mean, the best order of the bootstrap sample is of the order ofn, wheren is the sample size. In the case of non-lattice distributions and the bootstrap of the sample mean; the bootstrap removes the effect of the skewness of the distribution only when the bootstrap sample equals the sample size. However, for the bootstrap of quantiles, the preferred order of the bootstrap sample isn ^2/3. For the bootstrap of quantiles, if the bootstrap sample is of ordern ² or bigger, the bootstrap is not consistent.     

The product-limit estimator and the bootstrap: Some asymptotic representations

Summary The product-limit estimator and its quantile process are represented as i.i.d. mean processes, with a remainder of ordern ^–3/4(logn)^3/4 a.s. Corresponding bootstrap versions of these representations are given, which can help one visualize how the bootstrap procedure operates in this set up.Research supported by NSF grants MCS-81-02341 and MCS 83-01082     

Simultaneous prediction regions

Summary Two new methods for constructing simultaneous prediction regions are the subject of this article. Both methods simultaneously assert a collection of prediction regions, one prediction region for each future observable of interest. Both methods have the same aims: to control the overall coverage probability of the simultaneous prediction region and to keep equal the coverage probabilities of the individual prediction statements that make up the simultaneous region. The latter property is called balance.The two approaches differ in their choice of critical values. For leading cases, the first method achieves the desired overall coverage probability and the desired balance up to errors of ordern ^–1, wheren is the size of the learning sample. The second method reduces both errors to ordern ^–2. Calculating critical values in the second approach usually relies on a bootstrap algorithm.If overall coverage probability and degree of balance are instead calculatedconditionally given the learning sample, the two methods show the same asymptotic performance. This result reflects intrinsic limits on the extent to which conditional coverage probabilities can be controlled in prediction.This research was supported in part by NSF Grant DMS-87-01426. Part of the work was done while the author was a guest of Sonderforschungsbereich 123 at Universität Heidelberg     

Exact convergence rate of bootstrap quantile variance estimator

Summary It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n ^-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n ^-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n ^-2/5.     

Variance estimation for sample quantiles using them out ofn bootstrap

We consider the problem of estimating the variance of a sample quantile calculated from a random sample of sizen. Ther-th-order kernel-smoothed bootstrap estimator is known to yield an impressively small relative error of orderO(n ^−r/(2r+1) ). It nevertheless requires strong smoothness conditions on the underlying density function, and has a performance very sensitive to the precise choice of the bandwidth. The unsmoothed bootstrap has a poorer relative error of orderO(n ^−1/4), but works for less smooth density functions. We investigate a modified form of the bootstrap, known as them out ofn bootstrap, and show that it yields a relative error of order smaller thanO(n ^−1/4) under the same smoothness conditions required by the conventional unsmoothed bootstrap on the density function, provided that the bootstrap sample sizem is of an appropriate order. The estimator permits exact, simulation-free, computation and has accuracy fairly insensitive to the precise choice ofm. A simulation study is reported to provide empirical comparison of the various methods. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7131/00P).     

Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

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.     

Higher order representations of the Robbins–Monro process

For quasi-linear regression functions, the Robbins–Monro process X_n is decomposed in a sum of a linear form and a quadratic form both defined in the observation errors. Under regularity conditions, the remainder term is of order O(n^−3/2) with respect to the L_p-norm. If a cubic form is added, the remainder term can be improved up to an order of O(n⁻²). As a corollary the expectation of X_n is expanded up to an error of order O(n⁻²). This is used to correct the bias of X_n up to an error of order O(n^−3/2 log n).     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

An asymptotically complete class of tests

Summary This paper investigates sequences of asymptotically similar critical regions {S _n >0},n, under the assumption that the test-statisticS _n admits a certain stochastic expansion. It is shown that for such test-sequences, first order efficiency implies second order efficiency (i.e. efficiency up to an error termo(n ^–1/2)). Moreover, the asymptotic power functions of first order efficient test-sequences are determined up to an error termo(n ^–1), and a class of critical regions is specified which is minimal essentially complete up too(n ^–1).The results of this paper rest upon the technique of Edgeworth-expansions and are, therefore, restricted to continuous probability distributions.     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Let F(s, t) = P(X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that

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, where {ζ_i(s, t)} is i.i.d. mean zero process and R_n(s, t) is of the order O((n⁻¹log n)^3/4) a.s. uniformly on compact sets. Weak convergence of the process {n⁻¹ Σ_{i = 1}ⁿ ζ_i(s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator is also modified so that the modified estimator is closer to the true survival function than in supnorm.     

On the Estimation of Jump Points in Smooth Curves

Two-step methods are suggested for obtaining optimal performance in the problem of estimating jump points in smooth curves. The first step is based on a kernel-type diagnostic, and the second on local least-squares. In the case of a sample of size n the exact convergence rate is n ^{– 1}, rather than n ^{– 1 +} (for some > 0) in the context of recent one-step methods based purely on kernels, or n ^{– 1} (log n)^{1 +} for recent techniques based on wavelets. Relatively mild assumptions are required of the error distribution. Under more stringent conditions the kernel-based step in our algorithm may be used by itself to produce an estimator with exact convergence rate n ^{– 1} (log n)^1/2. Our techniques also enjoy good numerical performance, even in complex settings, and so offer a viable practical alternative to existing techniques, as well as providing theoretical optimality.     

Central limit theorems in D[0, 1]

Summary Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X _n} of independent copies of X, let S _n=X₁++X_n. Conditions which are either necessary or sufficient for the weak convergence of n ^–1/2(S _n–ES_n) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part II: Clenshaw–Lord Type Approximants

Laurent–Padé (Chebyshev) rational approximants P _m (w,w ^–1)/Q _n (w,w ^–1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P _m /Q _n matches that of a given function f(w,w ^–1) up to terms of order w ^±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ^±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P _m matches that of Q _n f up to terms of order w ^±(m+n), but based on knowledge of the series coefficients of f up to terms in w ^±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ₀ against the alternative A:θ ≠ θ₀. For this problem we propose a class of tests , which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ² type asymptotic expansion of the distribution of T up to order n⁻¹, where n is the sample size. Also we derive the χ² type asymptotic expansion of the distribution of T under the sequence of alternatives A_n: θ = θ₀ + /√n, ε > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.