Multiscale Gaussian convolution algorithm for estimate of Gaussian mixture model

Abstract

This paper introduces a multiscale Gaussian convolution model of Gaussian mixture (MGC-GMM) via the convolution of the GMM and a multiscale Gaussian window function. It is found that the MGC-GMM is still a Gaussian mixture model, and its parameters can be mapped back to the parameters of the GMM. Meanwhile, the multiscale probability density function (MPDF) of the MGC-GMM can be viewed as the mathematical expectation of a random process induced by the Gaussian window function and the GMM, which can be directly estimated by the use of sample data. Based on the estimated MPDF, a novel algorithm denoted by the MGC is proposed for the selection of model and the parameter estimates of the GMM, where the component number and the means of the GMM are respectively determined by the number and the locations of the maximum points of the MPDF, and the numerical algorithms for the weight and variance parameters of the GMM are derived. The MGC is suitable for the GMM with diagonal covariance matrices. A MGC-EM algorithm is also presented for the generalized GMM, where the GMM is estimated using the EM algorithm by taking the estimates from the MGC as initial parameters of the GMM model. The proposed algorithms are tested via a series of simulated sample sets from the given GMM models, and the results show that the proposed algorithms can effectively estimate the GMM model.     

A distribution-free rank-like test for scale with unequal population locations

The properties of a distribution-free rank-like test proposed by Moses (1963) for the twosample scale problem is studied and a modification of the test using Savage scores is proposed. It is shown that this rank-like test is superior to commonly used rank tests for scale in that it:(1) does not require the estimation of any location or centrality parameter, (2) does not require equal or known location parameters, (3) is robust for skewed data, (4) is resolving and (5) has some significant power advantages. The test is shown to be asymptotically normal, and asymptotic relative efficiencies are calculated. Power properties, studied via simulation, indicate that the test is especially well suited for testing for equality of scale when the data are sampled from skewed populations with unequal medians. Extensions to the J-sample problem are indicated.     

About an adaptively weighted Kaplan-Meier estimate

The minimum averaged mean squared error nonparametric adaptive weights use data from m possibly different populations to infer about one population of interest. The definition of these weights is based on the properties of the empirical distribution function. We use the Kaplan-Meier estimate to let the weights accommodate right-censored data and use them to define the weighted Kaplan-Meier estimate. The proposed estimate is smoother than the usual Kaplan-Meier estimate and converges uniformly in probability to the target distribution. Simulations show that the performances of the weighted Kaplan-Meier estimate on finite samples exceed that of the usual Kaplan-Meier estimate. A case study is also presented.     

空间误差自相关随机前沿模型及其估计

将空间计量经济学的思想引入随机前沿分析，构建了基于横截面数据的空间误差自相关随机前沿模型，推导出模型的似然函数以求得参数估计，并给出了各生产单元技术效率的估计。     

Analysis of mixed outcomes: misclassified binary responses and measurement error in covariates

This paper considers regression models for mixed binary and continuous outcomes, when the true predictor is measured with error and the binary responses are subject to classification errors. The focus of the paper is to study the effects of these errors on the estimates of the model parameters and also to propose a model that incorporates both these errors. The proposed model results in a substantial improvement in the estimates as shown by extensive simulation studies.     

Estimating the variances of ratio estimates in complex sample surveys with two primary sampling units per stratum—a comparison of balanced replication and jackknife techniques

The balanced half-sample and jackknife variance estimation techniques are used to estimate the variance of the combined ratio estimate. An empirical sampling study is conducted using computer-generated populations to investigate the variance, bias and mean square error of these variance estimators and results are compared to theoretical results derived elsewhere for the linear case. Results indicate that either the balanced half-sample or jackknife method may be used effectively for estimating the variance of the combined ratio estimate.     

A robust nonparametric estimation of the autoregression function under an ergodic hypothesis

The authors propose a family of robust nonparametric estimators for regression or autoregression functions based on kernel methods. They show the strong uniform consistency of these estimators under a general ergodicity condition when the data are unbounded and range over suitably increasing sequences of compact sets. They give some implications of these results for stating the prediction in Markovian processes with finite order and show, through simulation, the efficiency of the predictors they propose.     

Estimation of variance components in the mixed effects models: A comparison between analysis of variance and spectral decomposition

The mixed effects models with two variance components are often used to analyze longitudinal data. For these models, we compare two approaches to estimating the variance components, the analysis of variance approach and the spectral decomposition approach. We establish a necessary and sufficient condition for the two approaches to yield identical estimates, and some sufficient conditions for the superiority of one approach over the other, under the mean squared error criterion. Applications of the methods to circular models and longitudinal data are discussed. Furthermore, simulation results indicate that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.     

Asymptotic study of the change-point mle in multivariate Gaussian families under contiguous alternatives

The problem of estimating an unknown change-point in the mean vector or covariance matrix of a sequence of independent multivariate Gaussian random variables is considered. Adapting the estimation methodology that Hinkley pursued for the case of abrupt changes, we develop theory for deriving the asymptotic distribution of the maximum likelihood estimator of the change-point when the amount of change is a function of the sample size and goes to zero in a smooth fashion as the sample size goes to infinity, yielding a contiguous change-point model. Simulations have been performed to illustrate the closeness of the asymptotic distribution with the empirical distribution, and to evaluate its robustness to departures from normality for reasonable sample sizes as well as parameter changes. Finally, we apply the methodology to estimate the change-point in the daily log-returns data of BLS (BellSouth) and VZ (Verizon) from NYSE.     

Non-parametric estimation of lifetime distribution of competing risk models when censoring times are missing

There are situations in the analysis of failure time or lifetime data where the censoring times of unfailed units are missing. The non-parametric estimator of the lifetime distribution for such data is available in literature. In this paper we consider an extension of this situation to the univariate and bivariate competing risk setups. The maximum likelihood and simple moment estimators of cause specific distribution functions in both univariate and bivariate situations are developed. A simulation study is carried out to assess the performance of the estimators. Finally, we illustrate the method with real data set.