PhD Thesis Proposal - Public Lecture - Samira Soleymani

Title: Essays in Box-Cox Analysis

Abstract: The Box-Cox method has been widely used to improve the estimation accuracy in different fields, especially in econometrics and time series. In this thesis proposal, we initially review Box-Cox transformations (Box and Cox, 1964) and other alternative parametric power transformations. Following, the maximum likelihood method for Box-Cox transformation is presented by discussing the problems of previous approaches in the literature.

This work consists of the exact analysis of Box-Cox transformations based on the truncation effect in the transformed domain. We introduce a new family distribution for Box-Cox transformations in both original and transformed scales. A likelihood analysis of the Box-Cox distribution is presented when truncation is considered. The exact likelihood function is shown to behave poorly, and it is argued that an approximate Box-Cox likelihood approach addresses these difficulties. Numerical problems may arise in prediction and simulation if we generate transformed random variables from a complete normal distribution rather than the truncated distribution. Moreover, we will extend a simulation for the Box-Cox regression model.

A new efficient algorithm will be developed for simulating Box-Cox transformed time series. Box-Cox family distribution in the transformed space, apart from log-normal case, is a truncated normal distribution. We have obtained an exact solution to this simulation problem using matrix factorization methods and this can be used to provide an exact simulation method for Box-Cox transformed ARMA series. Further, we will propose an approach involving a modified Durbin-Levinson algorithm to provide a simulation method for Box-Cox transformed generalized linear time series models.
Later, we will present the optimal forecast for Box-Cox transformed time series. My future research plans will be focused on developing a numerical method to obtain the optimal forecast for any specific loss function.  Finally, Box-Cox transformations will be employed in machine learning and we will discuss Box-Cox analysis for penalized regression and random forest regression.

Keywords: ARMA, Box-Cox transformation, Cholesky decomposition, Cross-Validation, Durbin- Levinson, EM algorithm, Loss function, truncated normal distribution

Supervisor: Ian McLeod