EMPIRICAL BAYES TEST PROBLEMS OF VARIANCE COMPONENTS IN RANDOM EFFECTS MODEL

Bayes decision rule of variance components for one-way random effects model is derived and empirical Bayes (EB) decision rules are constructed by kernel estimation method. Under suitable conditions, it is shown that the proposed EB decision rules are asymptotically optimal with convergence rates near O(n-1/2). Finally, an example concerning the main result is given.

Impact of sampling interval on variance components of epoch-wise residual error in relative GPS positioning: A case study of a 40-km-long baseline

The study presents sampling interval impacts on variance components of the epoch-wise residual errors in relative GPS positioning. In the variance components estimation process, the 2-way nested ANOVA method was used. For that purpose, GPS observation data during four months at two permanent GPS stations, establishing a 40-km-long baseline as a part of the Montenegrin permanent network(Monte Pos), were used. The study results showed that there is no statistically significant impact of sampling interval changes on epoch-wise variance components related to the residual tropospheric and ionospheric delays(effect a) when it comes to such a baseline. However, it is not the case with epoch-wise variance components related to the interstation-distance-independent residual ‘far-field’ multipath effect(effect b). It turned out that the absolute values of relative differences of standard deviations of the effect a on the relative GPS coordinates(e, n and u) had maximum values 11.1%, 10.2% and 8.9%,respectively. Keeping the same order of presentation for the effect b, the values of 5.9%, 9.9% and 12.5%were obtained. In addition, absolute values of relative differences of standard deviations of horizontal and vertical position had maximum values of 3.8% and 7.7%, respectively.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

Comparison of REML and MINQUE for Estimated Variance Components and Predicted Random Effects

Linear mixed model (LMM) approaches have been widely applied in many areas of research data analysis because they offer great flexibility for different data structures and linear model systems. In this study, emphasis is placed on comparing the properties of two LMM approaches: restricted maximum likelihood (REML) and minimum norm quadratic unbiased estimation (MINQUE) with and without resampling techniques being included. Bias, testing power, Type I error, and computing time were compared between REML and MINQUE approaches with and without Jackknife technique based on 500 simulated data sets. Results showed that MINQUE and REML methods performed equally regarding bias, Type I error, and power. Jackknife-based MINQUE and REML greatly improved power compared to non-Jackknife based linear mixed model approaches. Results also showed that MINQUE is more time-saving compared to REML, especially with the use of resampling techniques and large data set analysis. Results from the actual cotton data analysis were in agreement with our simulated results. Therefore, Jackknife-based MINQUE approaches could be recommended to achieve desirable power with reduced time for a large data analysis and model simulations.

Nonlinear total least-squares variance component estimation for GM(1,1)model

The solution of the grey model(GM(1,1)model)generally involves equal-precision observations,and the(co)variance matrix is established from the prior information.However,the data are generally available with unequal-precision measurements in reality.To deal with the errors of all observations for GM(1,1)model with errors-in-variables(EIV)structure,we exploit the total least-squares(TLS)algorithm to estimate the parameters of GM(1,1)model in this paper.Ignoring that the effect of the improper prior stochastic model and the homologous observations may degrade the accuracy of parameter estimation,we further present a nonlinear total least-squares variance component estimation approach for GM(1,1)model,which resorts to the minimum norm quadratic unbiased estimation(MINQUE).The practical and simulative experiments indicate that the presented approach has significant merits in improving the predictive accuracy in comparison with control methods.

Analysis of stable components in the extended-range forecast for the coming 10-30 days in winter 2010 and 2011

In this paper we try to extract stable components in the extended-range forecast for the coming 10–30 days by using empirical orthogonal function （EOF） analysis, similarity coefficient, and some other methods based on the National Center for Environmental Prediction （NCEP）/National Center for Atmospheric Research （NCAR） reanalysis daily data. The comparisons of the coefficient of variance of climatological background field and truth data in winter between 2010 and 2011 are made. The method of extracting stable components and climatological background field can be helpful to increase forecasting skill. The forecasting skill improvement of air temperature is better than geopotential height at 500 hPa. Moreover, this method improves the predictability better in the Pacific Ocean. In China, the forecast in winter in Northeast China is more uncertain than in the other parts.

Dependence of epoch-wise two-way nested ANOVA estimates of variances of unmodeled effects present in relative GPS positioning on satellite elevation cutoff angle and PDOP mask

Impact of satellite elevation cutoff angle and position dilution of precision(PDOP)mask change on epoch-wise variance components of unmodeled effects that accompany relative Global Positioning System(GPS)positioning is presented herein.Data used for this study refer to the winter and summer periods of the years with minimal(2008)and maximal(2013)solar activity.These data were collected every 30 s in static mode,at two permanent GPS stations located in Montenegro,establishing a mediumdistance(116-km-long)baseline with a height difference of approximately 760 m between its endpoints.The study showed that changing satellite elevation cutoff angle,with a fixed PDOP mask,affects epochwise two-way nested ANOVA estimates of variances related to the‘far-field’multipath(considered as the nested factor herein)and the combined unmodeled effect of tropospheric and ionospheric refraction(considered as the nesting factor herein).However,changing of PDOP mask,with a fixed satellite elevation cutoff angle,doesn’t affect epoch-wise two-way nested ANOVA estimate of variance of the combined unmodeled effect of tropospheric and ionospheric refraction,but,generally,affects the estimate of variance of the‘far-field’multipath(possibly mixed with a part of a‘shorter-term’ionospheric refraction),which is especially pronounced for the summer period.It should also be noted that there is a significant influence of satellite elevation cutoff angle change on both epoch-wise horizontal and vertical position accuracy,only for the summer period,especially in the presence of maximal solar activity,while there is no significant impact of PDOP mask change on epoch-wise positional accuracy.

Recent Advances in the Geodesy Data Processing

Geodetic functional models,stochastic models,and model parameter estimation theory are fundamental for geodetic data processing.In the past five years,through the unremitting efforts of Chinese scholars in the field of geodetic data processing,according to the application and practice of geodesy,they have made significant contributions in the fields of hypothesis testing theory,un-modeled error,outlier detection,and robust estimation,variance component estimation,complex least squares,and ill-posed problems treatment.Many functional models such as the nonlinear adjustment model,EIV model,and mixed additive and multiplicative random error model are also constructed and improved.Geodetic data inversion is an important part of geodetic data processing,and Chinese scholars have done a lot of work in geodetic data inversion in the past five years,such as seismic slide distribution inversion,intelligent inversion algorithm,multi-source data joint inversion,water reserve change and satellite gravity inversion.This paper introduces the achievements of Chinese scholars in the field of geodetic data processing in the past five years,analyzes the methods used by scholars and the problems solved,and looks forward to the unsolved problems in geodetic data processing and the direction that needs further research in the future.

The mathematical weighting of GNSS observations based on different types of receivers/antennas and environmental conditions

Stochastic models play an important role in achieving high accuracy in positioning,the ideal estimator in the least-squares(LS)can be obtained only by using the suitable stochastic model.This study investigates the role of variance component estimation(VCE)in the LS method for Precise Point Positioning(PPP).This estimation is performed by considering the ionospheric-free(IF)functional model for code and the phase observation of Global Positioning System(GPS).The strategy for estimating the accuracy of these observations was evaluated to check the effect of the stochastic model in four modes:a)antenna type,b)receiver type,c)the tropospheric effect,and d)the ionosphere effect.The results show that using empirical variance for code and phase observations in some cases caused erroneous estimation of unknown components in the PPP model.This is because a constant empirical variance may not be suitable for various receivers and antennas under different conditions.Coordinates were compared in two cases using the stochastic model of nominal weight and weight estimated by LS-VCE.The position error difference for the east-west,north-south,and height components was 1.5 cm,4 mm,and 1.8 cm,respectively.Therefore,weight estimation with LS-VCE can provide more appropriate results.Eventually,the convergence time based on four elevation-dependent models was evaluated using nominal weight and LS-VCE weight.According to the results,the LS-VCE has a higher convergence rate than the nominal weight.The weight estimation using LS-VCE improves the convergence time in four elevation-dependent models by 11,13,12,and 9 min,respectively.

Estimation of the Genetic Parameters on Reproduction Performances of the China Yorkshire Line SⅡ_1

[Objective] This study was to investigate the genetic parameters on reproduction performances of China Yorkshire line SII,. [Method] Using China Yorkshire line SII, as experimental animal, its genetic parameters on reproduction performance, such as total number of born （TNB）, the alive number of born （ANB）, the litter weight of born （LWB） and the litter weight at age of 21 days （/W21 ） , were estimated by using the DFREML method of multi-traits animal model. [Result] The heritability estimates of TNB, ANB, LWB and LW21 were 0.11,0.19, 0.14 and 0.05, respectively; and the genetic correlations between the traits as TNB/ANB, TNB/LWB, TNB/LW21, ANB/LWB, ANB/LW21, LWB/LW21 were 0.91, 0.76, 0.65, 0.85, 0.59 and 0.38, respectively. [ Conclusion] Main genetic parameters of reproduction traits in China Yorkshire line all, have all reached or even exceeded the requirements of breeding.

THE RELATIVE EFFICIENCIES OF LEAST SQUARES IN LINEAR MODELS

The present paper daisses the relative efficiencies of the least square estimates in linear models. For Gauss-Markoff model: Y=Xe + e E(e)= 0, Cov(e)=V, an new efficiencyo f least square estimate for linearly estimable function c'r is proposed and its lower bound is giv-en. For variance component model: Y=X + e, E(e)=0, Cov(e)=, an new efficiency of least square estimate for linearly estimable function C'r is introduced for the first timeand its lower bound, which is independent of unknown parameters, is also obtained.

Longitudinal genetic analysis of growth-related traits in red swamp crayfish Procambarus clarkii (Girard)

The red swamp crayfish,Procambarus clarkii,is an economically important species especially in China.Their exoskeleton places serious constraints on growth and culture management.Their growth is achieved through intermittent molting/ecdysis.The longitudinal genetic dynamics for growth-related traits at different ecdysial points in P.clarkii has been unclear to date.In this study,conditional genetic analysis was carried out for growth-related traits(body weight,body length,chela length,and cephalothorax length)based upon a mixed genetic model with conditional additive,dominance,and genotype by environment effects in P.clarkii.A complete diallel cross was made among three geographic populations of P.clarkii for the genetic mating design.Results of the conditional genetic analysis showed that from 4 th molt to 9 th molt the conditional additive variations were increased significantly whereas the conditional non-additive genetic variations(dominance and genotype by environment interaction)were decreased significantly for these growth-related traits.This indicated that lots of new expression of additive effect genes for body weight,body length,chela length,and cephalothorax length occurred during ontogeny,and environment played a signifi cant role in the expression of genes affecting these growth-related traits.Growth of the four traits was mainly affected by non-additive genetic effects in early developmental stage(prior to 4 th molt).The cumulative conditional additive variation for the growth-related traits from 4 th molt to 9 th molt accounted for a large majority of the total conditional additive variations from 2 nd molt to 9 th molt,indicating that this period was very important for the growth of this species.Using the conditional analysis method,dynamics of growth-related traits during an important ontogenetic phase of red swamp crayfish was uncovered.Our results provide valuable insights into refining production of this species.

Bootstrap inference of the skew-normal two-way classification random effects model with interaction

In this paper,we consider the statistical inference problems for the fixed effect and variance component functions in the two-way classification random effects model with skewnormal errors.Firstly,the exact test statistic for the fixed effect is constructed.Secondly,using the Bootstrap approach and generalized approach,the one-sided hypothesis testing and interval estimation problems for the single variance component,the sum and ratio of variance components are discussed respectively.Further,the Monte Carlo simulation results indicate that the exact test statistic performs well in the one-sided hypothesis testing problem for the fixed effect.And the Bootstrap approach is better than the generalized approach in the one-sided hypothesis testing problems for variance component functions in most cases.Finally,the above approaches are applied to the real data examples of the consumer price index and value-added index of three industries to verify their rationality and effectiveness.

A General Theory of Estimation of Variance-Covariance Components

Starting from the more general functional model and being based on their work of K. R. Koch (1986) and Ou Ziqiang (1989), marginal likelihood function of variance components is derived and is identical to the orthogonal complement likelihood function in this paper. Minimum norm quadratic unibiased estimator (MINQUE) is developed, which expands the formula by C. R. Rao (1973). It is proved that Helmert type estimation, MINQUE, BQUE and maximum likelihood estimation are identical to one another. Besides, a universal formula for accuracy evalution is presented. Through these work, the paper establishes a universal theory of variance covariance components.

Robust Estimation of Variance Components Model

The classical least-squares methods may only solve LS β when the variance-covariance (matrix ∑(σ2 ∑)) is known (σ2 is unknown and ∑ is known) in linear model. The author thinks that maximum likelihood type estimation (M-estimation) should replace LS estimation. The paper discusses robust estimations of parameter vector and variance components for corresponding error model based on the principle of maximum likelihood type estimations (M-estimations). The influence functions are given respectively.

A compressed variance component mixed model for detecting QTNs and QTN-by-environment and QTN-by-QTN interactions in genome-wide association studies

Although genome-wide association studies are widely used to mine genes for quantitative traits,the effects to be estimated are confounded,and the methodologies for detecting interactions are imperfect.To address these issues,the mixed model proposed here first estimates the genotypic effects for AA,Aa,and aa,and the genotypic polygenic background replaces additive and dominance polygenic backgrounds.Then,the estimated genotypic effects are partitioned into additive and dominance effects using a one-way analysis of variance model.This strategy was further expanded to cover QTN-by-environment interactions(QEIs)and QTN-by-QTN interactions(QQIs)using the same mixed-model framework.Thus,a three-variance-component mixed model was integrated with our multi-locus random-SNP-effect mixed linear model(mrMLM)method to establish a new methodological framework,3VmrMLM,that detects all types of loci and estimates their effects.In Monte Carlo studies,3VmrMLM correctly detected all types of loci and almost unbiasedly estimated their effects,with high powers and accuracies and a low false positive rate.In re-analyses of 10 traits in 1439 rice hybrids,detection of 269 known genes,45 known gene-by-environment interactions,and 20 known gene-by-gene interactions strongly validated 3VmrMLM.Further analyses of known genes showed more small(67.49%),minor-allele-frequency(35.52%),and pleiotropic(30.54%)genes,with higher repeatability across datasets(54.36%)and more dominance loci.In addition,a heteroscedasticity mixed model in multiple environments and dimension reduction methods in quite a number of environments were developed to detect QEIs,and variable selection under a polygenic background was proposed for QQI detection.This study provides a new approach for revealing the genetic architecture of quantitative traits.

Bootstrap Inference on the Variance Component Functions in the Two-Way Random Effects Model with Interaction

In this paper,using the Bootstrap approach and generalized approach,the authors consider the one-sided hypothesis testing problems for variance component functions in the two-way random effects model.Firstly,the test statistics and confidence intervals for the sum of variance components are constructed.Next,the one-sided hypothesis testing problems for the ratio of variance components are also discussed.The Monte Carlo simulation results indicate that the Bootstrap approach is better than the generalized approach in most cases.Finally,the above approaches are applied to the real data examples of mice blood p H and molded plastic part’s dimensions.

THE INEFFICIENCY OF LEASTSQUARES IN GAUSS-MARKOV ANDVARIANCE COMPONENT MODELS

The present paper deals with the inefficiency of the least square estimates in linear models.FOr Gauss-Markov model, a new efficiency is proposed and its lower bound is given. FOr variancecomponent model, an efficiency is introduced and its lower bound, which is independent ofunknown parameters, is obtained.

Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions

The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation（VCE） and minimum standard deviation（MSTD） approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.

SMALL SAMPLE PROPERTIES OF THEPOWER FUNCTION OF F TESTS IN TWO-WAYERROR COMPONENT REGRESSION

In this paper we investigate properties of the power function of the generalized least squaresF ted for linear hypotheses under regression models with two-way error component model. Thecovariance structure of the model depends on the correlation coefficients ρ1 and ρ2 correspondingto the random effects. This model has been frequently applied to the analysis of panel data.In general, we show that the power is a monotonically increasing function of ρ1 (ρ2) in a regionwhich is ciO6e to the ρ1 (ρ2) axis, and a monotonically decreasing function of ρ1 (ρ2) in a regionclose to the ρ2 (ρ1) axis.