Sequential quantiles via Hermite series density estimation (Q89436)

Let \(x_1,x_2,\dots,x_n\) be iid random variables with density function \(f(x)\) and distribution function \(F(x)\). The empirical distribution function is defined as \(\widehat{F}(x)=\frac{1}{n}\sum_{i=1}^{n}I\{x_i \leq x\}\), where \(I\) is the usual indicator function. Clearly, \(\widehat{F}(x)\) can also be written in terms of order statistics \(x_{(i)}\), \(i\geq 1\). The kernel distribution function estimator is defined as \(\widehat{F_1}(x)=\frac{1}{n}\sum_{i=1}^{n}\int_{-\infty}^{\infty} \frac{1}{h_n} K\Big(\frac{x_i-x}{h_n}\Big) dx\), where the kernel function \(K\) is a non-negative function integrating to one with mean zero and bandwidth \(h_n >0\) as a smoothing parameter. The quantile function is defined as: \(q(p)=\text{inf}\{x:F(x)\geq p \}\). The \(p\)-th quantile can be estimated from the order statistics. For example, if \(p\in \Big(\frac{i-1}{n},\frac{i}{n}\Big]\), then \(\hat{q}(p)=x_{(i)}\). For certain values of \(p\), \(\hat{q}(p)\) is interpolated between \(x_{(i)}\) and \(x_{(i+1)}\). To improve the efficiency of such estimators, certain weighted average of several order statistics with suitable weights (known as L-estimators) is used as an alternative estimator. Perhaps the most popular class of such estimators uses the kernel as the weight function. Thus the kernel quantile estimator is defined as: \(\hat{q}(p)=\sum_{i=1}^{n} \int_{\frac{i-1}{n}}^{\frac{i}{n}} \frac{1}{h_n}K\Big(\frac{s-p}{h_n}\Big)\, x_{(i)}\, ds\). After providing some background, the authors use Hermite polynomials to develop the Gauss-Hermite expansion of the density function \(f(x)\), which leads to a Gauss-Hermite based estimator of the cumulative distribution function (CDF). They apply these methods to a obtain sequential quantile estimation in the settings of static and dynamic quantile estimation. The treatment of quantile estimation is made concrete by considering cases of independent and identically distributed and non-identically distributed data streams. Observations are continuous random variables observed sequentially one at a time. The quality of the Gauss-Hermite based CDF estimator and the quantile estimation is investigated. The proposed techniques are compared with existing leading algorithms for both simulated data and real data, and suitable conclusions are made.