Motivated by the need for effectively summarising, modelling, and forecasting the distributional characteristics of intra-daily returns, as well as the recent work on forecasting histogram-valued time-series in the area of symbolic data analysis, we develop a time-series model for forecasting 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 inference while accounting for parameter uncertainty; an efficient MCMC algorithm is developed for sampling-based posterior inference. Using ten international market indices and approximately 2,000 days of out-of-sample data from each market, the performance of the DQF model compares favourably, in terms of forecasting VaR of intra-daily returns, against the interval-valued and histogram-valued time-series models. Additionally, we demonstrate that the QF-valued forecasts can be used to forecast VaR measures at the daily timescale via a simple quantile regression model on daily returns (QR-DQF). In certain markets, the resulting QR-DQF model is able to provide competitive VaR forecasts for daily returns.
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via …
Estimation in functional linear quantile regression
This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size . Also, we allow the quantile index to vary over a given subset of the open unit interval, so the slope function is a function of two variables: (typically) time a…
Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in $\mathbb{R}^d$
作为分位数和涉及秩和符号的分布函数的单变量概念不规范地扩展到$\mathbb{R}^d, d\geq 2$。它产生了丰富的文学作品。第1章表明，与迄今为止提出的许多定义不同，在Escherzhukov et al.（2017）中引入的基于测量传输的定义具有使单变量分位数和半参数统计推断的成功工具排名的所有属性。因此，我们提出了一个新的中心向外定义的多元分布和分位数函数，以及他们的经验同行，我们得到一个Glivenko-Cantelli结果。我们的方法是几何的，与Monge-Kantorovich的方法相反。（2017），不需要任何力矩假设。由此产生的行列和符号是严格的分布自由，和最大不变的行动下的数据驱动类（保序）的转换生成的家庭绝对连续分布，该属性是半参数的效率保持性能的行列的理论基础。相应的分位数在相同的变换下是等变的。经验提出的分布函数仅在观测值处定义。一个连续的扩展到整个$\mathbb{R}^d$，产生连续的经验分位数轮廓，同时保持单调性和Glivenko-Cantelli功能是可取的。这样的扩展需要解决一个非平凡的问题，光滑插值循环单调性约束下。第二章给出了这个问题的完整解;我们证明了所得分布和分位数函数是Lipschitz的，并给出了Lipschitz常数的一个精确下界。数值研究的经验中心向外分位数的轮廓和他们的一致性进行。
Conditional Quantile Processes based on Series or Many Regressors
Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from t…
This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functiona…
National Technical University of Athens（雅典国立技术大学）；Hellenic Air Force（希腊空军）；World Meteorological Organisation (WMO)（世界气象组织）；University of the West of England（西英格兰大学）
