Updated: 21/10/25 Ref: “Introduction to Mathematical Statistics” by Hogg, McKean, Craig. “Statistics for Mathematicians” by Panaretos. [Hypothesis Testing] Consider a random variable ${ X }$ with density ${ f(x; \theta) , }$ ${ \theta \in \Omega . }$ Suppose we think ${ \theta \in \Omega _0 }$ or ${ \theta \in \Omega _1 , }$ where ${ \... Read more 20 Oct 2025 - 6 minute read
Consider the plane ${ \mathbb{R} ^2 . }$ Consider the origin ${ (0, 0) . }$ Consider the natural error generating process: The random error vector is ${ \mathcal{E} = (\varepsilon _1, \varepsilon _2) . }$ The components ${ \varepsilon _1, \varepsilon _2 }$ are independent and identically distributed. The distribution of ${ \mathcal{E} }$... Read more 08 Oct 2025 - 1 minute read
Ref: “Data Analysis for Social Scientists” by Duflo, Ellison. Lec-5. Link to the lecture: Link. Scikit-learn’s Density estimation documentation. Link to the page: Link. “Introduction to Mathematical Statistics” by Hogg, McKean, Craig. “Introduction to Nonparametric Estimation” by Tsybakov. “Smoothing methods in Statistics” by Simonoff... Read more 04 Oct 2025 - 2 minute read
Ref: “On the similarity of the Entropy Power Inequality and the Brunn Minkowski inequality” by Costa, Cover. “The Convolution Inequality for Entropy Powers” by Blachman. “Simple Proof of the Concavity of the Entropy Power with respect to added Gaussian Noise” by Dembo. “Stochastic Models, Information Theory, and Lie Groups, Vol 1” by Ch... Read more 24 Jul 2025 - 13 minute read
Ref: Marc van Leeuwen’s answer here: Link. Let ${ E }$ be an ${ n }$ dimensional Euclidean space. What are the origin and distance preserving maps ${ f : E \longrightarrow E }$? Obs: Let ${ E }$ be an ${ n }$ dimensional Euclidean space. Let ${ f : E \longrightarrow E . }$ Then the following are equivalent: ${ \lVert f(x) - f(y) \rVe... Read more 16 Jun 2025 - 7 minute read