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High-frequency sampled multivariate continuous time autoregressive moving average processes are investigated. We obtain asymptotic expansion for the spectral density of the sampled MCARMA process $$(Y_{n\varDelta })_{n \in {\mathbb {Z}}}$$ as $$\varDelta \downarrow 0$$ , where $$(Y_t)_{t \in {\mathbb {R}}}$$ is an MCARMA process. We show that the properly filtered process is a vector moving average process, and determine the asymptotic moving average representation of it, thus generalizing the univariate results to the multivariate model. The determination of the moving average representation of the filtered process, important for the analysis of high-frequency data, is difficult for any fixed positive $$\varDelta $$ . However, the results established here provide a useful and insightful approximation when $$\varDelta $$ is very small.

We show that in the case of Fay–Herriot model for small area estimation, there is an estimator of the variance of the random effects so that the resulting EBLUP is the best in the sense that it minimizes the leading term in the asymptotic expansion of the mean squared error (MSE) of the EBLUP. In particular, in the balanced case, i.e., when the sampling variances are equal, this best EBLUP has the minimal MSE in the exact sense. We also propose a modified Prasad–Rao MSE estimator which is second-order unbiased and show that it is less biased than the jackknife MSE estimator in a suitable sense in the balanced case. A real data example is discussed.

The paper revisits the problem of selection of priors for regular one-parameter family of distributions. The goal is to find some “objective” or “default” prior by approximate maximization of the distance between the prior and the posterior under a general divergence criterion as introduced by Amari (Ann Stat 10:357–387, 1982) and Cressie and Read (J R Stat Soc Ser B 46:440–464, 1984). The maximization is based on an asymptotic expansion of this distance. The Kullback–Leibler, Bhattacharyya–Hellinger and Chi-square divergence are special cases of this general divergence criterion. It is shown that with the exception of one particular case, namely the Chi-square divergence, the general divergence criterion yields Jeffreys’ prior. For the Chi-square divergence, we obtain a prior different from that of Jeffreys and also from that of Clarke and Sun (Sankhya Ser A 59:215–231, 1997).

We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The model considered in the paper is very general as we do not impose any preliminary restrictions on the probabilistic nature of the observation process and cover a wide class of nonlinear recursive procedures. In this paper we study asymptotic behaviour of the recursive estimators. The results of the paper can be used to determine the form of a recursive procedure which is expected to have the same asymptotic properties as the corresponding non-recursive one defined as a solution of the corresponding estimating equation., Comment: 30 pages with 1 postscript figure

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.

We consider the nonparametric regression model with long memory data that are not necessarily Gaussian and provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet-based mean regression function estimators. We show this MISE expansion, when the underlying mean regression function is only piecewise smooth, is the same as analogous expansion for the kernel estimators. However, for the kernel estimators, this MISE expansion generally fails if an additional smoothness assumption is absent.

The e-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 e-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.

We consider a boundary crossing probability of a Brownian bridgeB0 and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB0(t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values.

An asymptotic expansion of the logarithm of the likelihood ratio for Markov dependent observation is obtained. A functional limit theorem for the likelihood ratio is proved, which gives a way to study limiting distributions of the likelihood ratio based on stopping times, in particular, that of sequential probability ratio test.

Model selection procedures, based on a simple cross-validation technique and on suitable predictive densities, are taken into account. In particular, the selection criterion involving the estimative predictive density is recalled and a procedure based on the approximate p* predictive density is defined. This new model selection procedure, compared with some other well-known techniques on the basis of the squared prediction error, gives satisfactory results. Moreover, higher-order asymptotic expansions for the selection statistics based on the estimative and the approximate p* predictive densities are derived, whenever a natural exponential model is assumed. These approximations correspond to meaningful modifications of the Akaike's model selection statistic.

Let (X1,...,X,...,Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2c1(d, θ0) and the second order term is of the form n4/2-1c2(d, θ0>, w), with an error term that is o(nd/2-1). We identify the functions c1 and c2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Classification between two populations dealing with both continuous and binary variables is handled by splitting the problem into different locations. Given the location specified by the values of the binary variables, discrimination is performed using the continuous variables. The location probability model with homoscedastic across location conditional dispersion matrices is adopted for this problem. In this paper, we consider presence of continuous covariates with heterogeneous location conditional dispersion matrices. The continuous covariates have equal location specific mean in both populations. Conditional homoscedasticity fails when strong interaction between the continuous and binary variables is present. A plug-in covariance adjusted rule is constructed and its asymptotic distribution is derived. An asymptotic expansion for the overall error rate is given. The result is extended to include binary covariates.

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form sδ/gr - 1, where ρ(> 0) and δ(= ±1) are parameters indicating the types of expansion. For δ = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis (δ = -1) or by numerical computation (δ = 1).

We study the asymptotic behaviour of the posterior distributions for a one-parameter family of discontinuous densities. It is shown that a suitably centered and normalized posterior converges almost surely to an exponential limit in the total variation norm. Further, asymptotic expansions for the density, distribution function, moments and quantiles of the posterior are also obtained. It is to be noted that, in view of the results of Ghosh et al. (1994, Statistical Decision Theory and Related Topics V, 183-199, Springer, New York) and Ghosal et al. (1995, Ann. Statist., 23, 2145-2152), the nonregular cases considered here are essentially the only ones for which the posterior distributions converge. The results obtained here are also supported by a simulation experiment.

Suppose thatX n =(X1,...Xn) is a collection ofm-dimensional random vectorsXi forming a stochastic process with a parameter ϑ. Let\(\hat \theta \) be the MLE of ϑ. We assume that a transformationA(\(\hat \theta \)) of\(\hat \theta \) has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k≥2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.

The minimum risk point estimation for the mean is addressed for a natural exponential family (NEF) that also has a power variance function (PVF) under a loss function given by the squared error plus linear cost. An appropriate accelerated version of the full purely sequential methodology of Bose and Boukai (1993b, submitted) is proposed along the lines of Mukhopadhyay (1993a, Tech. Report, No. 93-27, Department of Statistics, University of Connecticut) in order to achieve operational savings. The main result provides the asymptotic second-order expansion of the regret function associated with the accelerated sequential estimator of the population mean.

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor ex- pansions, rearranging terms according to their asymptotic order. The resulting expansions are called here ezpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, ex- pected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation produces invariant ezpected/observed expansions. Let jc = {p~ : co Eft c_ ~a} be a parametric family of probability distributions defined on a sample space/12 and dominated by a a-finite measure #. The param- eter space ~t is assumed to be an open non-empty subset of ~a. Let us denote by p(x; co), x ~ 2(, the density of P~ with respect to # and by l(a:) = logp(x; co) the log likelihood function based on the sample data x. We assume that the log likeli- hood function is a smooth function of the parameter and that the usual additional regularity conditions hold ensuring, in particular, that the maximum likelihood * This research was partially supported by the Italian National Research Council grant

Examples of exact expressions for the moments (mainly of the mean) of functions of sample moments are given. These provide checks on alternative developments such as asymptotic series for n→∞, and simulation processes. Exact expressions are given for the mean of the square of the sample coefficient of variation, particularly in uniform sampling; Frullani integrals studied by G. H. Hardy arise. It should be kept in mind that exact results for (joint) moment generating functions (mgfs) are of interest as they produce a means of obtaining exact results for (cross) moments—including moments with negative indices. Thus an exact expression for the joint mgf of the 1st two noncentral moments can be used to obtain the mean of the (c.v.)2 (but not for the mean of the c.υ.). A general expression is given for the moment generating function of the sample variance. The limitations of Fisher's symbolic formula for the characteristic function of sample moments (or more general statistics) are noted.

This paper deals with Watson statistic T w and likelihood ratio (LR) statistic T l for testing hypothesis H 0s: μ ∈ V (a given s-dimensional subspace) based on a sample of size n from a p-variate Langevin distribution M p(μ, κ). Asymptotic expansions of the null and non-null distributions of T w and T l are obtained when n is large. Asymptotic expressions of those powers are also obtained. It is shown that the powers of them are coincident up to the order n -1 when κ is unknown.

Our aim is to investigate a way to characterize the elements of a statistical manifold (the metric and the family of connections) using invariance properties suggested by Le Cam's theory of experiments. We distinguish the case where the statistical manifold is flat. Then, there naturally exists an entropy and it is proven that experiment invariance is equivalent to entropy invariance. If the statistical manifold is not flat, we introduce a notion of local invariance of selected order associated to the asymptotic (on n observations, n tending to infinity) expansion of the power of the Neymann Pearson test in a contiguous neighborough of some point. This invariance provides a substantial number of morphisms. This was not always true for the entropy invariance: particularly, the case of Gaussian experiments is investigated where it can be proven that entropy invariance does not characterize a metric or a family of connections.

In the present note, asymptotic expansions for conditional and unconditional distributions of the score vector are derived. Our aim is to consider these expansions in the light of differential geometry, particularly the theory of derivative strings. Expansions for the distributions of the maximum likelihood estimator are obtained from those for the score vector via transformation, with a view to interpreting from the standpoint of differential geometry the various terms entering the expansions.

Stein's two-stage procedure produces a t-test which can realize a prescribed power against a given alternative, regardless of the unknown variance of the underlying normal distribution. This is achieved by determining the size of a second sample on the basis of a variance estimate derived from the first sample. In the paper we introduce a nonparametric competitor of this classical procedure by replacing the t-test by a rank test. For rank tests, the most precise information available are asymptotic expansions for their power to order n -1, where n is the sample size. Using results on combinations of rank tests for sub-samples, we obtain the same level of precision for the two-stage case. In this way we can determine the size of the additional sample to the natural order and moreover compare the nonparametric and the classical procedure in terms of expected additional numbers of observations required.

The technical validity of the saddlepoint technique for approximating the resampling distribution of the sample mean of i.i.d. and non-i.i.d. random variables is examined. The relative errors are shown to occur at the same rate as in parametric analysis. Disereteness in resampling problems is accounted for. The uniformity of the errors is also explored.