Before revision:  Specifically, we consider a quantile function as an observation, and develop a new class of dynamic models for quantile-function-valued (QF-valued) time series. In order to make statistical inferences and account for parameter uncertainty , we propose a method whereby a likelihood function can be constructed for QF-valued data, and develop an adaptive MCMC sampling algorithm for simulating from the posterior distribution.
After revision:  Specifically, we consider a quantile function as an observation, and develop a new class of dynamic models for quantile-function-valued (QF-valued) daily summaries for intra-daily returns. We call this model the dynamic quantile function (DQF) model. Instead of a histogram, we propose to use a g-and-h quantile function to summarise the distribution of intra-daily returns. We work with a Bayesian formulation of the DQF model in order to make statistical inferences and account for parameter uncertainty , we propose a method whereby a likelihood function can be constructed for QF-valued data, and develop an adaptive MCMC sampling algorithm for simulating from the posterior distribution.