The local power of the gradient test

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n ^−1/2, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

The asymptotic power of rank tests under scale-al ternatives including contaminated distributions

Summary The alternative hypothesis of translated scale for the classical non-parametric hypothesis of equality of two distribution functions in the two-sample problem is extended to a scale-alternative including contamination. The asymptotic power of rank tests and the two-sampleF-test under contiguous sequences of the alternatives is derived and asymptotic relative efficiency of these rank tests with respect to theF-test is investigated. It is found that some of the rank tests have reasonably high asymptotic powers satisfied enough.     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

Asymptotic Expansion for the Null Distribution of the <Emphasis Type="Italic">F</Emphasis>-statistic in One-way ANOVA under Non-normality

In this paper we derive the asymptotic expansion of the null distribution of the F-statistic in one-way ANOVA under non-normality. The asymptotic framework is when the number of treatments is moderate but sample size per treatment (replication size) is small. This kind of asymptotics will be relevant, for example, to agricultural screening trials where large number of cultivars are compared with few replications per cultivar. There is also a huge potential for the application of this kind of asymptotics in microarray experiments. Based on the asymptotic expansion we will devise a transformation that speeds up the convergence to the limiting distribution. The results indicate that the approximation based on limiting distribution are unsatisfactory unless number of treatments is very large. Our numerical investigations reveal that our asymptotic expansion performs better than other methods in the literature when there is skewness in the data or even when the data comes from a symmetric distribution with heavy tails.     

Cramér-von Mises statistics based on the sample quantile function and estimated parameters

The estimated weighted empirical quantile process is introduced, and under mild regularity conditions is shown to converge weakly in L²(0, 1) to a Gaussian process. This leads to an elementary approach to the derivation of the asymptotic null distribution of Cramér-von Mises type statistics for testing a composite null hypothesis based on the sample quantile function and estimated parameters. Special emphasis is given to the location/scale composite null hypothesis.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

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.     

Semi-Classical Green Kernel Asymptotics for the Dirac Operator

We consider a semi-classical Dirac operator in d ∈ ? spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

On the goodness-of-fit testing for a switching diffusion process

We study the asymptotic behavior of the Cramér–von Mises type statistic in the goodness-of-fit hypotheses testing problem for ergodic diffusion processes. The basic (simple) hypothesis is defined by the stochastic differential equation with sign-type trend coefficient and known diffusion coefficient. It is shown that the limit distribution of the proposed test statistic (under hypothesis) is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process and show that the eigenfunctions in this expansion are expressed in terms of Bessel functions. This representation for the limit statistic allows us to approximate the threshold.     

Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property

The ε-Markov process is a general model of stochastic processes which includes nonlinear time series models, diffusion processes with jumps, and many point processes. With a view to applications to the higher-order statistical inference, we will consider a functional of the ε-Markov process admitting a stochastic expansion. Arbitrary order asymptotic expansion of the distribution will be presented under a strong mixing condition. Applying these results, the third order asymptotic expansion of theM-estimator for a general stochastic process will be derived. The Malliavin calculus plays an essential role in this article. We illustrate how to make the Malliavin operator in several concrete examples. We will also show that the thirdorder expansion formula (Sakamoto and Yoshida (1998, ISM Cooperative Research Report, No. 107, 53–60; 1999, unpublished)) of the maximum likelihood estimator for a diffusion process can be obtained as an example of our result.     

ARCH(0,1)系数中位无偏估计分布的渐近展开(英文)

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH(0,1)coefficient.The exact distribution of the estimator can be easily derived,however its practical calculations are too heavy to implement, even though the middle range of sample sizes.Since the estimator is shown to have asymptotic normality,asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements.Accuracies of expansion formulas are evaluated numerically,and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations.Derived expansion are applied to testing hypothesis of stationarity,and an implementation for a real data set is illustrated.     

ARCH(0,1)系数中位无偏估计分布的渐近展开

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH（0,1） coefficient. The exact distribution of the estimator can be easily derived, however its practical calculations are too heavy to implement, even though the middle range of sample sizes. Since the estimator is shown to have asymptotic normality, asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements. Accuracies of expansion formulas are evaluated numerically, and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations. Derived expansion are applied to testing hypothesis of stationarity, and an implementation for a real data set is illustrated.     

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R^a, as multinomial goodness-of-fit statistics. Each R^a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R^a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R^a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R^a. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

A test for additional information in canonical correlation analysis

Summary In canonical correlation analysis a hypothesis concerning the relevance of a subset of variables from each of the two given variable sets is formulated. The likelihood ratio statistic for the hypothesis and an asymptotic expansion for its null distribution are obtained. In discriminant analysis various alternative forms of a hypothesis concerning the relevance of a specified variable subset are also discussed.     

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition. Received: 7 September 1997 / Revised version: 17 March 1999     

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.     

An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T² distribution up to n^-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.