This paper considers two Brownian motions in a situation where one is correlated to the other with a slight delay. We study the problem of estimating the time lag parameter between these Brownian motions from their high-frequency observa tions, which are possibly subject to measurement errors. The measurement errors are assumed to be i.i.d., centered Gaussian and independent of the latent processes. We investigate the asymptotic structure of the likelihood ratio process for this model when the lag parameter is asymptotically infinitesimal. We show that the structure of the limit experiment depends on the level of the measurement errors: If the measurement errors locally dominate the latent Brownian motions, the model enjoys the LAN property. Otherwise, the limit experiment does not result in typical ones appearing in the literature. We also discuss the efficient estimation of the lag parameter to highlight the statistical implications.

This paper proposes nonparametric tests for the null hypothesis that a treatment has a zero effect on the conditional variance for all subpopulations characterized by the values of the covariates. Rather than the mean of an outcome, which measures the extent to which a treatment changes the level of the outcome, researchers are sometimes interested in how the treatment affects the dispersion of the outcome. We use the variance to measure dispersion and estimate the conditional variances using the series method. We provide a test rule that compares a Wald-type test statistic with the critical value of a chi-squared distribution. We also construct a normalized test statistic that is asymptotically standard normal under the null hypothesis. We illustrate the usefulness of the proposed test by Monte Carlo simulations and an empirical example that investigates the effect of unionism on wage dispersion.

We consider selecting of the linear and the quadratic discriminant functions in two normal populations. We do not know which of two discriminant functions lowers the expected probability of misclassification. When difference of the covariance matrices is large, it is known that the expected probability of misclassification of the quadratic discriminant functions is smaller than that of linear discriminant function. Therefore, we should consider only the selection when the difference between covariance matrices is small. In this paper we suggest a selection method using asymptotic expansion for the linear and the quadratic discriminant functions when the difference between the covariance matrices is small.

In order to use historical data in the design of sample surveys with a Bayesian approach, the information from the historical data must be expressed as a prior distribution. Then, the best prior distribution for the parameter of interest is a predictive distribution. The density function of the predictive distribution generally is not available in an analytical form. From the perspective of practical use, Schmidli et al. (2014) proposed an approximation for the predictive distribution using a mixture of conjugate prior distributions. Their method relies on random numbers drawn from the predictive distribution. However, if the population distribution includes a nuisance parameter, their method becomes impractical. We propose a new approximation method that does not rely on these simulated numbers. Our approximation instead minimizes the mean squared error between the exact Bayes estimator and the one corresponding to the approximated predictive distribution.

The Ewens sampling formula is well-known as a distribution of a random partition of the set of integers {1, 2, ... , n}. We give the condition that the number K_n of distinct components of the formula converges to the shifted Poisson distribution. Based on this convergence, we give the new approximations to the distribution of K_n, which are different from the approximations by Arratia et al. (2000, 2003). The formers are better than the latters. This is shown by comparing the bounds for the total variation distances between the distributions of the approximations and the distribution of K_n. Several examples are given to illustrate the results.

In this article, two estimation approaches based on age-specific parametric model have been proposed and a comparative study between them has been studied. We assume that outcome variable follows a parametric model, but the parameters are smooth function of time (age). Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the parameters at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. We derived asymptotic properties such as asymptotic biases,variances and mean squared error (MSE) for the local polynomial smoothed estimator and kernel smoothing estimator for the parameter of the time-variant parametric model. A mathematical relationship is established between two asymptotic MSEs. Mathematical relationship between two smoothing estimators has also been established. Applications of our two-step estimation method have been demonstrated through a large demographic study to estimate fecundability. Theoretical results on coverage of bootstrap confidence intervals for these smoothing estimators have been derived. Finite sample properties of our procedures are investigated by a simulation study.

This study proposes a spatial extension of time series autoregressive conditional heteroskedasticity (ARCH) models to those for areal data. We call the spatially extended ARCH models as spatial ARCH (S-ARCH) models. S-ARCH models specify conditional variances given surrounding observations, which constitutes a good contrast with time series ARCH models that specify conditional variances given past observations. We estimate the parameters of S-ARCH models by a two-step procedure of least squares and the quasi maximum likelihood estimation, which are validated to be consistent and asymptotically normal. We demonstrate the empirical properties by simulation studies and real data analysis of land price data in Tokyo areas.

Fisher’s “Nile” example is a classic which involves a bivariate random variable (X, Y) having a joint probability density function given by f(x, y; θ) = exp(−θx − θ⁻¹y), 0 < x, y < ∞, where θ > 0 is a single unknown parameter. We develop bounded-length confidence interval estimations for P_θ(X > a) with a preassigned confidence coefficient using both purely sequential and two-stage methodologies. We show: (i) Both methodologies enjoy asymptotic first-order efficiency and asymptotic consistency properties; (ii) Both methodologies enjoy second-order efficiency properties. After presenting substantial theoretical investigations, we have also implemented extensive sets of computer simulations to empirically validate the theoretical properties.

In this paper, we discuss two-sample tests for high-dimension, non-Gaussian data. We suppose that two classes have a strongly spiked eigenvalue model. First, we investigate the noise space for high-dimension, non-Gaussian data. A two-sample test is proposed by using the cross-data-matrix (CDM) methodology and its power is derived under some regularity conditions when the dimension is very large. We discuss the validity of assumptions. We check the performance of the proposed two-sample test procedure by simulations. Finally, we demonstrate the proposed two-sample test in actual data analyses.