Higher order properties of the wild bootstrap under misspecification

We examine the higher order properties of the wild bootstrap (Wu, 1986) in a linear regression model with stochastic regressors. We find that the ability of the wild bootstrap to provide a higher order refinement is contingent upon whether the errors are mean independent of the regressors or merely uncorrelated with them. In the latter case, the wild bootstrap may fail to match some of the terms in an Edgeworth expansion of the full sample test statistic. Nonetheless, we show that the wild bootstrap still has a lower maximal asymptotic risk as an estimator of the true distribution than a normal approximation, in shrinking neighborhoods of properly specified models. To assess the practical implications of this result we conduct a Monte Carlo study contrasting the performance of the wild bootstrap with a normal approximation and the traditional nonparametric bootstrap.     

Exploring the relation between the r* approximation and the Edgeworth expansion

In this paper we study the relation between the r* saddlepoint approximation and the Edgeworth expansion when quite general assumptions for the statistic under consideration are fulfilled. We will show that the two term Edgeworth expansion approximates the r* formula up to an O(n ^?3/2) remainder; this provides a new way of looking at the order of the error of the r* approximation. This finding will be used to inspect the close connection between the r* formula and the Edgeworth B adjustment introduced in Phillips (Biometrika 65:91–98, 1978). We will show that, whenever an Edgeworth expansion exists, this adjustment approximates both the distribution function of the statistic and the r* formula to the same order degree as the Edgeworth expansion. Some numerical examples for the sample mean and U-statistics are given in order to shed light on the theoretical discussion.     

Edgeworth expansions for realized volatility and related estimators

This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we derive Edgeworth expansions for such estimators. The expansions are developed in the framework of small-noise asymptotics. The results have application to Cornish–Fisher inversion and help setting intervals more accurately than those relying on normal distribution.     

Approximating the distribution of the two-stage least squares estimator when the concentration parameter is small

This paper presents a new approximation to the exact sampling distribution of the instrumental variables estimator in simultaneous equations models. It differs from many of the approximations currently available, Edgeworth expansions for example, in that it is specifically designed to work well when the concentration parameter is small. The approximation is remarkable in that simultaneously: (i) it has an extremely simple final form; (ii) in situations for which it is designed it is typically much more accurate than is the large sample normal approximation; and (iii) it is able to capture most of those stylized facts that characterize lack of identification and weak instrument scenarios. The development leading to the approximation is also novel in that it introduces techniques of some independent interest not seen in this literature hitherto.     

Heuristic Approximation to Cramér-von Mises Type Statistics

We derive a formal expansion for a distribution in terms of another distribution. As a particular case we get the formal Edgeworth expansion. The heuristic procedure that we present is used to obtain approximations for distribution functions of the Cramér-von Mises and Watson goodness-of-fit statistics. Finally we compare our results with some obtained in the literature.     

Bootstrapping rank statistics

The bootstrap, which provides powerful approximations for many classes of statistics, is studied for simple linear rank statistics employing bounded and smooth score functions. To verify consistency we view a rank statistic as a statistic induced by a statistical functional ψ which is evaluated at a pair of dependent signed measures. Thus, we can apply the von Mises method to verify asymptotic results for the bootstrap. The strong consistency of the bootstrap distribution estimator is derived for the bootstrap based on resampling from the original data. Further, the residual bootstrap is studied. The accuracy of the bootstrap approximations for small sample sizes is studied by simulations. The simulations indicate that the bootstrap provides better results than a normal approximation.     

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

Inference in nonlinear models is usually based on the asymptotic normal distribution, based on linearizing the model. The accuracy of this approximation can in many cases be improved by a reparametrization. Systematic methods for doing this will be described. Sometimes a saddlepoint approximation can be used, and this offers several advantages compared to the asymptotic distribution and the Edgeworth expansion. The improved methods are unfortunately not commonly used. It will be discussed why this is so. The methods will be illustrated by a series of examples.     

Approximate distribution of the maximum deviation of histograms

Summary We consider a histogram, based on order statistics, and density estimators which are closely related to the histogram.When investigating the distribution of the maximum absolute deviation of density estimators it turns out that an approximation by the distribution of the largest absolute value of a normal sample is asymptotically considerably better than an approximation by the limit distribution (which is the extreme value distribution). For the one-side deviation, a corresponding approximation is less accurate. The accuracy can be improved by using an asymptotic expansion.     

Exact linear rank tests for two‐sample equivalence problems with continuous data

The present paper introduces a methodology for the semiparametric or non‐parametric two‐sample equivalence problem when the effects are specified by statistical functionals. The mean relative risk functional of two populations is given by the average of the time‐dependent risk. This functional is a meaningful non‐parametric quantity, which is invariant under strictly monotone transformations of the data. In the case of proportional hazard models, the functional determines just the proportional hazard risk factor. It is shown that an equivalence test of the type of the two‐sample Savage rank test is appropriate for this functional. Under proportional hazards, this test can be carried out as an exact level α test. It also works quite well under other semiparametric models. Similar results are presented for a Wilcoxon rank‐sum test for equivalence based on the Mann–Whitney functional given by the relative treatment effect.     

Macro‐integration with inequality constraints: an application to the integration of transport and trade statistics

Macro‐integration is the process of combining data from several sources at an aggregate level. We review a Bayesian approach to macro‐integration with special emphasis on the inclusion of inequality constraints. In particular, an approximate method of dealing with inequality constraints within the linear macro‐integration framework is proposed. This method is based on a normal approximation to the truncated multivariate normal distribution. The framework is then applied to the integration of international trade statistics and transport statistics. By combining these data sources, transit flows can be derived as differences between specific transport and trade flows. Two methods of imposing the inequality restrictions that transit flows must be non‐negative are compared. Moreover, the figures are improved by imposing the equality constraints that aggregates of incoming and outgoing transit flows must be equal.     

Record values from generalized Pareto distribution and associated inference

In this paper, we derive exact explicit expressions for the single, double, triple and quadruple moments of the upper record values from a generalized Pareto distribution. We then use these expressions to compute the mean, variance, and the coefficients of skewness and kurtosis of certain linear functions of record values. Finally, we develop approximate confidence intervals for the location and scale parameters of the generalized Pareto distribution using the Edgeworth approximation and compare them with the intervals constructed through Monte Carlo simulations. Received: June 1999     

Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

基于Wilcoxon秩和检验的多元非参数控制图

在统计过程控制(SPC)中,对多元数据的监测仍然是一个重要且具有挑战性的问题。当缺乏或有限的关于潜在过程分布的认知时,特别是当过程测量是多变量的时候,非参数控制图在统计过程控制(SPC)中是有用的。文章基于Wilcoxon秩和检验结合广义加权移动平均(GWMA)控制方案来制定图表统计量,得到一个新的多元非参数控制图,用于监测多元数据的位置参数变化。文章的理论和数值研究表明,所提出的控制图能够为任意数据分布位置偏移检测提供令人满意的性能。     

One-Sided Coverage Intervals for a Proportion Estimated from a Stratified Simple Random Sample

Using an Edgeworth expansion to speed up the asymptotics, we develop one-sided coverage intervals for a proportion based on a stratified simple random sample. To this end, we assume the values of the population units are generated from independent random variables with a common mean within each stratum. These stratum means, in turn, may either be free to vary or are assumed to be equal. The more general assumption is equivalent to a model-free randomization-based framework when finite population correction is ignored. Unlike when an Edgeworth expansion is used to construct one-sided intervals under simple random sampling, it is necessary to estimate the variance of the estimator for the population proportion when the stratum means are allowed to differ. As a result, there may be accuracy gains from replacing the normal  z -score in the Edgeworth expansion with a  t -score.     

Local asymptotic normality for randomly censored models with applications to rank tests

For randomly right censored models we study the asymptotic behaviour of linear (rank) statistics under local alternatives. The results can be used to evaluate the asymptotic power of the corresponding tests. For instance we treat the question how to choose the best scores in order to derive asymptotically optimal (rank) tests under certain alternatives.     

Hermite polynomial based expansion of European option prices

We seek a closed-form series approximation of European option prices under a variety of diffusion models. The proposed convergent series are derived using the Hermite polynomial approach. Departing from the usual option pricing routine in the literature, our model assumptions have no requirements for affine dynamics or explicit characteristic functions. Moreover, convergent expansions provide a distinct insight into how and on which order the model parameters affect option prices, in contrast with small-time asymptotic expansions in the literature. With closed-form expansions, we explicitly translate model features into option prices, such as mean-reverting drift and self-exciting or skewed jumps. Numerical examples illustrate the accuracy of this approach and its advantage over alternative expansion methods.     

On Tests for a Normal Mean with Known Coefficient of Variation

Hinkley (1977) derived two tests for testing the mean of a normal distribution with known coefficient of variation (c.v.) for right alternatives. They are the locally most powerful (LMP) and the conditional tests based on the ancillary statistic for μ. In this paper, the likelihood ratio (LR) and Wald tests are derived for the one‐ and two‐sided alternatives, as well as the two‐sided version of the LMP test. The performances of these tests are compared with those of the classical t, sign and Wilcoxon signed rank tests. The latter three tests do not use the information on c.v. Normal approximation is used to approximate the null distribution of the test statistics except for the t test. Simulation results indicate that all the tests maintain the type‐I error rates, that is, the attained level is close to the nominal level of significance of the tests. The power functions of the tests are estimated through simulation. The power comparison indicates that for one‐sided alternatives the LMP test is the best test whereas for the two‐sided alternatives the LR or the Wald test is the best test. The t, sign and Wilcoxon signed rank tests have lower power than the LMP, LR and Wald tests at various alternative values of μ. The power difference is quite large in several simulation configurations. Further, it is observed that the t, sign and Wilcoxon signed rank tests have considerably lower power even for the alternatives which are far away from the null hypothesis when the c.v. is large. To study the sensitivity of the tests for the violation of the normality assumption, the type I error rates are estimated on the observations of lognormal, gamma and uniform distributions. The newly derived tests maintain the type I error rates for moderate values of c.v.     

Nonparametric methods for paired samples

The small sample and asymptotic properties of nonparametric tests for paired sampled are examined. Linear rank statistics are compared with the paired t-test and the Wilcoxon-signed-rank test in simulation studies. From a minimax point of view the linear rank statistics turn out to be the best. Moreover, it is illustrated that the Wilcoxon-signed-rank test should not be used if it is not clear that the differences of the pairs have a symmetric distribution.     

Analytic moments for GJR-GARCH (1, 1) processes

For a GJR-GARCH(1, 1) specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moments for the most commonly used GARCH models are stated as special cases. We also derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments. A simulation study using these analytic moments produces approximate predictive distributions which are free from the bias affecting simulations. An empirical study using almost 30 years of daily equity index, exchange rate and interest rate data applies Johnson SU and Edgeworth expansion distribution fitting to our closed-form formulae for higher moments of returns.     

A new approximation of the posterior distribution of the log–odds ratio

In this paper, the posterior density of the log–odds ratio is studied. It is assumed that the observations have a multinomial distribution and that the prior on the multinomial parameters is a Dirichlet density. Several approximations currently available are reviewed. Under certain conditions on the prior parameters of the Dirichlet density, it is shown that the posterior moments can be computed exactly. A new approximation, similar to the Edgeworth expansion is also proposed. Using a numerical example, the different methods of approximation of posterior density are compared.