Estimating and forecasting a time series of densities using a functional data approach

Functional data analysis is a branch of statistics that uses a collection of statistical methods to analyse data in the form of functions or curves. Density functions are a specific case of functional data. In recent years, forecasting functional data has received much attention; however, forecasting functional data when functions are density functions has not been extensively addressed in the literature. This thesis has demonstrated that a time series of densities can be effectively forecast using a functional data approach. The first contribution of this thesis is to propose an estimation method to nonparametrically estimate a time series of densities, taking account of the time ordering of densities. A data set comprising many observations is recorded at each time period, and the associated probability density function is to be estimated for each time period. A logspline approach is applied to each data set separately where each estimated density has common knots but different coefficients. These estimated densities form a 'functional time series'. The second contribution of this thesis is to propose three conditional density estimation methods with time as the discrete conditioning variable: (1) a conditional logspline approach accounting for the time variation by flexibly adjusting the smoothness of the conditional density functions; (2) a conditional logspline approach applied to all data simultaneously with common knots and kernel weights to account for the time variation; (3) a conditional kernel approach with two new bandwidth selection methods where the kernel weights account for the time variation. The proposed conditional density estimation methods are compared with conditional kernel estimation using two existing bandwidth selection methods: the uniform reference rule and the bootstrap method. The third contribution of this thesis is to propose a decomposition and forecasting algorithm to forecast future densities by decomposing the functional time series into orthonormal basis functions and their uncorrelated coefficients. Different sets of future densities are forecast using univariate time series models applied to the coefficients obtained from various decomposition methods: the functional principal component analysis with log-transformation, three multidimensional scaling methods and eight projection pursuit methods. The prediction intervals of the forecast density functions are constructed. The fourth contribution of this thesis is to propose a new method to evaluate the reliability of density forecasts assessing calibration using probability integral transforms, considering thousands of observations for each future time period. We apply our methods to two simulated data sets (unimodal and bimodal) and four real data sets. The four real data sets comprise United Kingdom and Australian income and age data over many years with thousands of observations per year. Proper scoring rules are then used to evaluate the relative accuracy of the density forecasts by assessing both calibration and sharpness simultaneously.