Univariate Density Estimation
Parametric Desity Estimation
Draw from a normal distribution
Given \({{X}_{1}},{{X}_{2}},\ldots ,{{X}_{n}}\) \(i.i.d\) draw from a normal distribution with mean of \(\mu\) and variance of \({{\sigma }^{2}}\) the joint \(PDF\) can be expressed as:
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\prod\limits_{i=1}^{n}{\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{i}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}}\]
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{1}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{2}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \cdots \times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{n}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\]
The term \(\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}\) is a constant multiplying this term for \(n\) times gives \({{\left( \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}} \right)}^{n}}=\frac{1}{{{\left( 2\pi \sigma \right)}^{\frac{n}{2}}}}\).
Replication can have a different meaning to the different discipline. Replication in studies that uses public data sources involve sharing codes to ensure consistency in the results. Here, I share some chunk of the codes in five steps using RStudio, that can ensure ease in shareability. These 5 steps are the preambles prior I proceed toward data management.
For this a user will require to install R here and R-Studio here.
Theorem 1: Gaussian Tail Inequality
Given \({{x}_{1}},\cdots ,{{x}_{n}}\sim N\left( 0,1 \right)\) then, \(P\left( \left| X \right|>\varepsilon \right)\le \frac{2{{e}^{-{{{\varepsilon }^{2}}}/{2}\;}}}{\varepsilon }\) and \(P\left( \left| {{{\bar{X}}}_{n}} \right|>\varepsilon \right)\le \frac{2}{\sqrt{n}\varepsilon }{{e}^{-{n{{\varepsilon }^{2}}}/{2}\;}}\overset{l\arg e\ n}{\mathop{\le }}\,{{e}^{-{n{{\varepsilon }^{2}}}/{2}\;}}\).
Proof of Gaussian Tail Inequality
Consider a univariate \({{x}_{1}},\cdots ,{{x}_{n}}\sim N\left( 0,1 \right)\), then the probability density function is given as \(\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}}\).
Let’s take the derivative w.r.t \(x\) we get:
\[\frac{d\phi \left( x \right)}{dx}={\phi }'\left( x \right)=\frac{d\left( \frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{dx}=\frac{1}{\sqrt{2\pi }}\frac{d\left( \,{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{dx}=\frac{1}{\sqrt{2\pi }}\frac{d\left( \,{{e}^{-\frac{{{x}^{2}}}{2}}} \right)}{d\left( -\frac{{{x}^{2}}}{2} \right)}\frac{d\left( -\frac{{{x}^{2}}}{2} \right)}{dx}=\frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{x}^{2}}}{2}}}\left( -x \right)=-x\phi \left( x \right)\]
Let’s define the gaussian tail inequality.