Computation of the mean value and variance functions in geometric process

Geometric process (GP) is widely used as a non-stationary stochastic model in reliability analysis. In many of applications related with GP its mean value and variance functions are needed. Since there are no analytical forms of these functions in a lot of situations their computations are of importance. In this study, a numerical approximation and Monte Carlo estimation method based on the convolutions of distribution functions have been proposed for both the mean value and variance functions.     

A quadratic approximation for Jackknife estimators of the variance of sample mean functions

Jackknife estimators of the variance of estimators which are functions of the sample mean are considered. A quadratic approximation of them is proposed and compared with a linear approximation by Monte Carlo experiments carried out by statistical software Minitab.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

Bootstrap variance and bias estimation in linear models

Let θ be a nonlinear function of the regression parameters and θ be its estimator based on the least-squares method. This paper studies the bootstrap estimators of the variance and bias of θ. The bootstrap estimators are shown to be consistent and asymptotically unbiased under some conditions. Asymptotic orders of the mean squared errors of the bootstrap estimators are also obtained. The bootstrap and the classical linearization method are compared in a simulation study. Discussions about when to use the bootstrap are given.     

Improved shrinkage estimators for the mean vector of a scale mixture of normals with unknown variance

The problem of estimating the mean θ of a not necessarily normal p-variate (p > 3) distribution with unknown covariance matrix of the form σ²A (A a known diagonal matrix) on the basis of n_i > 2 observations on each coordinate X_t (1 < i < p) is considered. It is argued that the class of scale (or variance) mixtures of normal distributions is a reasonable class to study. Assuming the loss function is quadratic, a large class of improved shrinkage estimators is developed in the case of a balanced design. We generalize results of Berger and Strawderman for one observation in the known-variance case. This methodology also permits the development of a new class of minimax shrinkage estimators of the mean of a p-variate normal distribution for an unbalanced design. Numerical calculations show that the improvements in risk can be substantial.     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.     

A test for the variance residual life

In this paper we develop a test based on the empirical distribution function for the alternative representing 'decreasing variance residual life1 property. The test is consistent with asymptotically normal test statistic and is shown to perform well in the Pitman's asymptotic relative efficiency sense.     

On the procedure of hodges and lehmann for testing composite hypotheses of the mean of a normal distribution with unknown variance

In 1954 Hodges and Lehmann gave a test procedure for testing the hypothesis that the mean of an identically independently normally distributed random sample with unknown variance is contained within a certain interval [μ1, μ2]. The test is similar on the boundary of the zero-hypothesis and superior in power to the composite t-test usually applied to this problem. However Hodges and Lehmann could prove the unbiasedness of their test only for the special case that the sample consists of two elements. From numerical computations they guessed that unbiasedness would be valid for arbitrary sample sizes. This question is discussed here and partially answered.     

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox-type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time-transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co-workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson-type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple-infection data taken from a clinical study on chronic granulomatous disease is also provided.     

Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

The author proves that Wold‐type decompositions with strong orthogonal prediction innovations exist in smooth, reflexive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. His theory includes as special cases all previous Wold‐type decompositions of discrete time processes, completely characterizes when non‐linear heavy‐tailed processes obtain a strong‐orthogonal moving average representation, and easily promotes a theory of non‐linear impulse response functions for infinite‐variance processes. The author exemplifies his theory by developing a non‐linear impulse response function for smooth transition threshold processes, and discusses how to test decomposition innovations for strong orthogonality and whether the proposed model represents the best predictor. A data set on currency exchange rates allows him to illustrate his methodology.     

Globally intensity-reweighted estimators for K- and pair correlation functions

We introduce new estimators of the inhomogeneous K-function and the pair correlation function of a spatial point process as well as the cross K-function and the cross pair correlation function of a bivariate spatial point process under the assumption of second-order intensity-reweighted stationarity. These estimators rely on a ‘global’ normalisation factor which depends on an aggregation of the intensity function, while the existing estimators depend ‘locally’ on the intensity function at the individual observed points. The advantages of our new global estimators over the existing local estimators are demonstrated by theoretical considerations and a simulation study.     

On ratio estimators in two phase sampling under size stratification and estimation over two successive occasions

Adopting size stratification when the auxiliary character is approximated by a continuous uniform distribution, a double sampling ratio strategy has been suggested. this sampling strategy has been compared with some of the known sampling strategies. the applicability of this strategy to sampling on two successive occasions has also been investigated     

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

Abstract

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.     

Invariance principles and almost sure central limit theorem for the error variance estimator in linear models

A number of score statistics are derived for a heterogeneous spatial Poisson process which has a composite intensity. The intensity consists of a 'background' process which is estimated

from a control point process by kernel density estimation. The parametric form of the composite intensity yields score tests for particular spatial effects. A numerical example concerning respiratory cancer mortality is given.     

Dimension reduction for the conditional mean and variance functions in time series

This paper deals with the nonparametric estimation of the mean and variance functions of univariate time series data. We propose a nonparametric dimension reduction technique for both mean and variance functions of time series. This method does not require any model specification and instead we seek directions in both the mean and variance functions such that the conditional distribution of the current observation given the vector of past observations is the same as that of the current observation given a few linear combinations of the past observations without loss of inferential information. The directions of the mean and variance functions are estimated by maximizing the Kullback–Leibler distance function. The consistency of the proposed estimators is established. A computational procedure is introduced to detect lags of the conditional mean and variance functions in practice. Numerical examples and simulation studies are performed to illustrate and evaluate the performance of the proposed estimators.     

Modified tests for change points in variance in the possible presence of mean breaks

This paper considers the detection problem of variance changes for the time series involving abrupt and/or smooth breaks in mean. Often, in these situations, the tests of choice are based on cumulative sum of squares statistics. We show that the test statistics are not robust in the presence of broken mean and their sizes suffer severe distortions. The adjusted residual-based method is then proposed to eliminate these deficiencies and makes a significant improvement. Finally, simulation results confirm the validity of these modified test statistics, and an empirical data analysis using some stock price series from the Shanghai Stock Exchange is reported.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L ^p norm (p≥; 1) between the variance and this estimator. The method of the proof consists in writing the L ^p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.