The Linear Regression Model
A normal linear regression model assumes an output variable \(Y\), a vector of covariates \({\bf x}=(x_1,\cdots ,x_p)\), and \(Y\) is linear deterministic function \(\bf \beta^Tx\) with additive Gaussian noise.
This series is a note of Y. P. Mack and M. Rosenblatt's work in 1979.
Introduction
Suppose we have a bounded, twice differentiable density function \(f\) on \(\mathbb R^p\). Let \(X_1,X_2,\cdots,X_n\) be \(n\) i.i.d. samples of \(f\). Let \(x\in\mathbb R^p\) be a fixed point so that \(f(x)>0\). We want to estimate \(f(x)\) based on \(n\) samples. The KNN estimate is given by \[ f_n(x)=\frac{1}{n(Z_n)^p}\sum_{j=1}^n\omega(\frac{x-X_j}{Z_n}), \] where \(Z_n\) is the distance between \(x\) and its \(k\)th nearest neighbor, and \(\omega\) is a bounded integrable weight function with \(\int\omega(u){\rm d}u=1\).
Stories
The idea about building my own blog was originated from a long time ago when I first read Terry Tao's homepage. In 2019, when I became a graduate student, I realized it is necessary to write down the mathematics to keep track of my research. Honestly speaking, repetitions and failures really make one suffer. It is those tiny progresses that encourage me to hold on.
