Efficient Estimation Of Multivariate Parameters Under Several Partial Envelope Models

An envelope is a relatively new dimension reduction methodology for decreasing estimative and predictive variation relative to standard methods in multivariate statistics,sometimes by amounts equivalent to increasing the sample size many times over.In essence,an envelope is a form of targeted dimension reduction which is descendent from sufficient dimension reduction,and inherits its underlying philosophy from Fisher’s notion of sufficient statistics.Envelope model uses dimension reduction techniques to remove immaterial variation in the data and has the potential to gain efficiency in estimation and improve prediction.When the envelope subspace is equal to the full space,the envelope model reduces to the standard model,but as long as the dimension of predictor variables of special interest is less than the dimension of response variables,the partial envelope model is still applicable,so the partial envelope model is more flexible than the envelope model.We adopt dimension reduction ideas of the partial envelope model to centre on some predictors of special interest and set more relaxed constraints,which can further improve parameter estimation efficiency and effectively reduce dimension.This paper focuses on the efficient estimation of multivariate parameters and the dimension reduction of response variables under the reduced-rank partial envelope model,groupwise partial envelope model,scaled partial envelope model,and simultaneous partial envelope model.The main work of this dissertation includes the following contents:(1)The efficient estimation of reduced-rank partial envelope model in multivariate linear regression is investigated.Firstly,in order to further improve the efficiency of parameter estimation and reduce the number of estimated parameters,we combine partial envelopes with reduced-rank regression to form reduced-rank partial envelope model which can efficiently perform parameter estimation and dimension reduction.This method has the potential to perform better than both.Further,we demonstrate maximum likelihood estimators for the reduced-rank partial envelope model parameters,and exhibit asymptotic distribution and theoretical properties under normality.Meanwhile,we show selections of rank and partial envelope dimension.Lastly,under the normal and non-normal error distributions,simulation studies are carried out to compare our proposed reduced-rank partial envelope model with the ordinary least squares,reduced-rank regression model,partial envelope model,and reduced-rank envelope model.A real-data analysis is also given to support the theoretic claims.After extensive simulation studies and a real-data analysis,the reduced-rank partial envelope estimators have shown promising performance.(2)The efficient estimation for groupwise partial envelope model in multivariate linear regression is studied.Firstly,in order to incorporate group information for different groups,we extend the partial envelopes to the groupwise partial envelopes which can improve the efficiency of parameter estimation and enlarge the scope of partial envelope model.It maintains the potential of the original partial envelope methods to increase efficiency and allows for both different regression coefficients and different error structures for diverse groups.Further,we demonstrate the maximum likelihood estimation under the groupwise partial envelope model.Meanwhile,we give asymptotic distribution and theoretical properties.Lastly,simulation studies are carried out to compare our proposed groupwise partial envelope model with the standard model,partial envelope model,and separate partial envelope model.From the simulation results and a real-data analysis,we can see that the performance of the groupwise partial envelope estimators is much better than that of the standard model estimators,partial envelope estimators,and separate partial envelope estimators.(3)The scaled partial envelope model in multivariate linear regression is investigated.Inference based on the partial envelope model is variational or non-equivariant under rescaling of the responses,and tends to restrict its use to responses measured in identical or analogous units.The efficiency acquisitions promised by partial envelopes frequently cannot be accomplished when the responses are measured in diverse scales.Here,we extend the partial envelope model to a scaled partial envelope model that overcomes the aforementioned disadvantage and enlarges the scope of partial envelopes.The proposed model maintains the potential of the partial envelope model in terms of efficiency and is invariable to scale changes.Further,we demonstrate the maximum likelihood estimators and their properties.Lastly,simulation studies and a real-data example demonstrate the advantages of the scaled partial envelope estimators,including a comparison with the standard model estimators,partial envelope estimators,and scaled envelope estimators.(4)Efficient simultaneous partial envelope model in multivariate linear regression is studied.Our goal is to associate their advantages by simultaneously reducing the predictors of special interest X1 and the responses Y to decrease both predictive and estimative variation.Firstly,we propose the simultaneous partial envelope model which can dramatically improve the efficiency of parameter estimation and effectively reduce dimension.Further,we show the maximum likelihood estimators for simultaneous partial envelope model parameters.Meanwhile,we give the asymptotic distribution and theoretical properties,selections of rank and envelopes dimension.Lastly,the simulation results and a real-data analysis demonstrate that the performance of the simultaneous partial envelope estimators is much better than that of the ordinary least squares estimators,X1-envelope estimators,Y-envelope estimators,and simultaneous envelope estimators.