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1.
We consider the one-term Edgeworth expansion for finite-population L-statistics. We provide an explicit formula for the Edgeworth correction term and give sufficient conditions for the validity of the expansion that are expressed in terms of the weight function defining the statistics and moment conditions.  相似文献   

2.
In this article, an unconstrained Taylor series expansion is constructed for scalar-valued functions of vector-valued arguments that are subject to nonlinear equality constraints. The expansion is made possible by first reparameterizing the constrained argument in terms of identified and implicit parameters and then expanding the function solely in terms of the identified parameters. Matrix expressions are given for the derivatives of the function with respect to the identified parameters. The expansion is employed to construct an unconstrained Newton algorithm for optimizing the function subject to constraints.Parameters in statistical models often are estimated by solving statistical estimating equations. It is shown how the unconstrained Newton algorithm can be employed to solve constrained estimating equations. Also, the unconstrained Taylor series is adapted to construct Edgeworth expansions of scalar functions of the constrained estimators. The Edgeworth expansion is illustrated on maximum likelihood estimators in an exploratory factor analysis model in which an oblique rotation is applied after Kaiser row-normalization of the factor loading matrix. A simulation study illustrates the superiority of the two-term Edgeworth approximation compared to the asymptotic normal approximation when sampling from multivariate normal or nonnormal distributions.  相似文献   

3.
The asymptotic expansions of the distributions of the sums of independent identically distributed random variables are given by Edgeworth type expansions when moments do not necessarily exist, but when the density can be approximated by rational functions. Supported in part by the Sakkokai Foundation.  相似文献   

4.
We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.  相似文献   

5.
We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators, we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.  相似文献   

6.
For symmetric asymptotically linear statistics based on simple random samples, we construct a one–term empirical Edgeworth expansion, where the moments defining the true Edgeworth expansion are replaced by their jackknife estimators. In order to establish the validity of the empirical Edgeworth expansion (in probability) we prove the consistency of the jackknife estimators.  相似文献   

7.
For symmetric asymptotically linear statistics based on simple random samples, we construct the one-term empirical Edgeworth expansion, where the moments defining the true Edgeworth expansion are replaced by their jackknife estimators. In order to establish the validity of the empirical Edgeworth expansion (in probability), we prove the consistency of the jackknife estimators.  相似文献   

8.
The asymptotic behavior of the parametric bootstrap estimator of the sampling distribution of a maximum likelihood estimator is investigated in a simple lattice case, integer valued random variables whose distributions form an exponential family. The expected value of the bootstrap estimator is compared with an Edgeworth expansion, less the continuity correction.  相似文献   

9.
Summary In order to construct a higher-order asymptotic theory of statistical inference, it is useful to know the Edgeworth expansions of the distributions of related statistics. Based on the differential-geometrical method, the Edgeworth expansions are performed up to the third-order terms for the joint distribution of any efficient estimators and complementary (approximate) ancillary statistics in the case of curved exponential family. The marginal and conditional distributions are also obtained. The roles and meanings of geometrical quantities are elucidated by the geometrical interpretation of the Edgeworth expansions. The results of the present paper provide an indispensable tool for constructing the differential-geometrical theory of statistics.  相似文献   

10.
Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.  相似文献   

11.
Summary An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution. The Institute of Statistical mathematics  相似文献   

12.
Moment independent sensitivity index is widely concerned and used since it can reflect the influence of model input uncertainty on the entire distribution of model output instead of a specific moment. In this paper, a novel analytical expression to estimate the Borgonovo moment independent sensitivity index is derived by use of the Gaussian radial basis function and the Edgeworth expansion. Firstly, the analytical expressions of the unconditional and conditional first four-order moments are established by the training points and the widths of the Gaussian radial basis function. Secondly, the Edgeworth expansion is used to express the unconditional and conditional probability density functions of model output by the unconditional and conditional first four-order moments, respectively. Finally, the index can be readily computed by measuring the shifts between the obtained unconditional and conditional probability density functions of model output, where this process doesn't need any extra calls of model evaluation. The computational cost of the proposed method is independent of the dimensionality of model inputs and it only depends on the training points and the widths which are involved in the Gaussian radial basis function meta-model. Results of several case studies demonstrate the effectiveness of the proposed method.  相似文献   

13.
14.
Summary Approximate formulae using a large number of terms of Edgeworth type asymptotic expansions for the distributions of statistics often produce spurious oscillations and give poor fits to the exact distribution functions in parts of the tails. A general method for suppressing these oscillations and evoking more accurate approximations is introduced here. This work was supported in part by Ministry of Education Grant 59530016 and 60530017. The Institute of Statistical Mathematics  相似文献   

15.
For the censored simple linear regression model, we establish a oneterm Edgeworth expansion for the Koul, Susarla and Van Ryzin type estimator of the regression coefficient. Our approach is to represent the estimator of the regression coefficient as an asymptoticU-statistic plus some ignorable terms and hence apply the known results on the Edgeworth expansions for asymptoticU-statistic. The counting process and martingale techniques are used to provide the proof of the main results.  相似文献   

16.
The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503–520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.  相似文献   

17.
The Edgeworth expansion for distributions of extreme values   总被引:3,自引:0,他引:3  
We present necessary and sufficient conditions of Edgeworth expansion for distributions of extreme values. As a corollary, rates of the uniform convergence for distributions of extreme values are obtained.  相似文献   

18.
Summary By direct probabilistic argument one term of an Edgeworth type asymptotic expansion is obtained for certain first passage distributions for random walks. These results provide partial justification for and extensions of approximations suggested earlier as a heuristic consequence of Laplace transform calculations.Research supported by ONR Contract N00014-77-C-0306, NSF Grant MCS77-16974, and by the Humboldt Stiftung  相似文献   

19.
Edgeworth expansions and saddlepoint approximations for the distributions of estimators of certain eigenfunctions of covariance and correlation matrices are developed. These expansions depend on second-, third-, and fourth-order moments of the sample covariance matrix. Expressions for and estimators of these moments are obtained. The expansions and moment expressions are used to construct second-order accurate confidence intervals for the eigenfunctions. The expansions are illustrated and the results of a small simulation study that evaluates the finite-sample performance of the confidence intervals are reported.  相似文献   

20.
In this paper we investigate scaling properties of risky asset returns and make a strong case (1) against the need for multifractal models and (2) in favor of the requirement of heavy tailed distributions. Amongst the standard empirical properties of risky asset returns are an autocorrelation function for the returns which dies away rapidly and is statistically insignificant beyond a few lags, and also autocorrelation functions of squares and absolute values of returns which die away very slowly, persisting over years, or even decades. Together these indicate that, assuming returns come from a stationary process, they are not independent, but at most short-range dependent, while various functions of the returns are long-range dependent. These scaling properties are well known, although commonly ignored for modeling convenience. However, much more can be inferred from the scaling properties of the returns. It turns out that the empirical scaling functions are initially linear and ultimately concave, which is strongly suggestive of returns distributions with infinite low order moments or alternatively that multifractal behavior is a modeling requirement. Modifications of the commonly used models cannot readily meet these requirements. The evidence will be presented and its significance discussed, along with a class of models which can incorporate the empirically observed features.  相似文献   

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