A note on the asymptotic generalised variance for a moving average process

Relationships between the asymptotic generalised variance, of an r^th order moving average process, and the ordinary variance, of an associated r^th order autoregressive process, are established for r=1 and 2. This is extended to the case r=3, and a generalisation is suggested for all r>O, though this has not been checked beyond r=4.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

A note on asymptotic parametric prediction

This paper is devoted to asymptotic behaviour of plug-in statistical predictors obtained by replacing the unknown parameter in a conditional expectation by a suitable estimator. We derive the L²$L^2$-convergence rate and limit in distribution for the predictors. Applications to ARMA processes and diffusion processes are considered.     

A note on the asymptotic behaviour of empirical likelihood statistics

This paper develops some theoretical results about the asymptotic behaviour of the empirical likelihood and the empirical profile likelihood statistics, which originate from fairly general estimating functions. The results accommodate, within a unified framework, various situations potentially occurring in a wide range of applications. For this reason, they are potentially useful in several contexts, such as, for example, in inference for dependent data. We provide examples showing that known findings in literature about the asymptotic behaviour of some empirical likelihood statistics in time series models can be derived as particular cases of our results.     

A note on asymptotic linearity of a bank statistic in regression parameter

This note extends JureckovS's (1969) results to the case

where the initial randan variables nay not be identically distributed.     

A note on homogeneity tests of covariance matrices

In this note we propose two procedures for testing homogeneity of co-variance matrices that are both extensions of Hartley's (1940) test for equality of variances. The first is a two-stage procedure where the first step is a simple test for equality of the largest eigenvalues, and corresponding eigenvectors, of the covariance matrices. The second is based on projection pursuit and seems harder to apply in practice.     

A note on the asymptotic optimality of a two population sequential probability ratio test

It is shown that a two population sequential probability ratio test studied by a number of recent authors in the context of sequential medical trials is asymptotically optimal.     

A note on exact calculation of the non central hypergeometric distribution

Direct calculation of the non central hypergeometric (NH) distribution and its moments can present computational issues in both efficiency and accuracy. In response, several methods, both approximate and exact, for calculating the NH mean and variance have appeared in the literature. We add to this body of work, a straight-forward, exact method that is easily programed, efficient, and computationally stable. Specifically, by considering the logs of the values of the NH probability mass function (pmf) and then shifting the exponents so that, prior to normalization, the mode acquires a value of 1, concerns for overflow are eliminated.     

A note on sequential ML estimates and their asymptotic covariances

A marginal and sequential maximum likelihood estimation method is described which can be used instead of full information maximum likelihood estimation if the latter method is unfeasible. It is shown that the sequential procedure yields strongly consistent and asymptotically normal estimates under relatively general regularity conditions. It is shown that the covariance matrix of the sequential ML estimator does not coincide with the inverse of the Fisher information matrix. Hence, the corrected covariance matrix is derived. The application of the sequential procedure to the multivariate probit model with dichotomous, ordered categorical, single-sided censored and double-sided censored endogenous variables is included. This research was partially supported by a dissertation grant of theStudienstiftung des Deutschen Volkes. Comments and suggestions on earlier drafts by Gerhard Arminger, Giorgio Calzolari, Bernd Kortzen and an anonymous referee are gratefully acknowledged.     

A note on asymptotic expansions for the power of perturbed tests

For simplicity or tractability reasons one sometimes uses modified test statistics, which differ from the original ones up to O_p(a_n) terms with a_n→0. In this note, some technical conditions are provided under which a corresponding expansion for the powers of such perturbed tests holds. The necessity of some of these conditions is discussed and illustrated by examples. An application to invariant testing multivariate normality is presented.     

A note on the passage time of finite-state Markov chains

Consider a Markov chain with finite state {0, 1, …, d}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations: (1) the absorbing time of state d when the chain starts from any state i and absorbing at state d; (2) the passage time of any state i when the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly.     

A note on the asymptotic efficiency of the sobel–wald test

Let X₁,X₂, … be iid random variables with the pdf f(x,θ)=exp(θx?b(θ)) relative to a σ-finite measure μ, and consider the problem of deciding among three simple hypotheses H_i:θ=θ_i (1?i?3) subject to P(acceptH_i|θ_i)=1?α (1?i?3). A procedure similar to Sobel–Wald procedure is discussed and its asymptotic efficiency as compared with the best nonsequential test is obtained by finding the limit lim_a→0(E_iN(a)/n(a)), where N (a) is the stopping time of the proposed procedure and n(a) is the sample size of the best non-sequential test. It is shown that the same asymptotic limit holds for the original Sobel–Wald procedure. Specializing to N(θ,1) distribution it is found that lim_a→0$\text{(E}{}_{\mathrm{i}}\text{N(α)/n(α))=}\text{1}\text{4}$ (i=1,2) and lim_a→0 (E₃N(α)n(α))=δ²₁/4δ, where δ_i=(θ_i+1?θ_i) with 0<δ₁?δ₂. Also, the asymptotic efficiency evaluated when the X's have an exponential distribution.     

A note on the asymptotic distribution of mardia's measure of multivariate kurtosis

Weak convergence results are used to investigate asymptotic properties of Mardia's measure of multivariate kurtosis in the context of assessing multivariate normality.     

A note on the residual median process

We study the residual median process, defined as the median of those observations which are greater than a number t. Using appropriate limit theorems, it is shown that the stochastic process converges in law to a Gaussian process defined in terms of a Brownian bridge.     

A note on information theoretic measures

An axiomatic characterization of an information theoretic quantity is given using only two axioms. In particular, this quantity contain PEARSON'S chi-square statistic and some other useful measures     

A note on the asymptotic distribution of a Perron-type innovational outlier unit root test with a break

This note provides the asymptotic distribution of a Perron-type innovational outlier unit root test developed by Popp (J Stat Comput Sim 78:1145–1161, 2008) in case of a shift in the intercept for non-trending data. In Popp (J Stat Comput Sim 78:1145–1161, 2008), only critical values for finite samples based on Monte Carlo techniques are tabulated. Using similar arguments as in Zivot and Andrews (J Bus Econ Stat 10:251–270, 1992), weak convergence is shown for the test statistics.