Crash Course - Statistics

Summary

General notations

• $\mathrm{\Omega }$ abstract set representing the sample space of a random experiment. The elements in $\omega \in \mathrm{\Omega }$ are the possible outcomes of the experiment.
• $\mathcal{P}(\mathrm{\Omega })$: power set of $\mathrm{\Omega }$, the set of all possible subsets of $\mathrm{\Omega }$.
  
• Most of subsets $A,B,\dots$ will be thought as events.
• Collection of subsets $\mathcal{A},\mathcal{B},\dots$.
• The empty set $\mathrm{\varnothing }$.
• $\mathcal{B}$ is a $\sigma$-field, usually connected with the sample space $\mathrm{\Omega }$.
• $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-field of $\mathbb{R}$.
• $\mathbb{P}$ is a probability measure function $\mathbb{P}:\mathcal{B}\to [0,1]$.
• $(\mathrm{\Omega },\mathcal{B},\mathbb{P})$ is a probability space.
• $\bigsqcup$ is a shortcut to denote a disjoint union, for example writing $A\bigsqcup B$ means that the sets $A$ and $B$ are disjoint, while writing $A\cup B$ means that the sets $A$ and $B$ are not disjoint.
• $X$ denotes an univariate random variable.
• Bold letters, ${\mathbf{X}}_n=(X_1,X_2,\dots ,X_i,\dots ,X_n)$ denotes a random vector with $n$ elements, where each $X_i$ is a random variable. Instead, ${\mathbf{x}}_n=(x_1,x_2,\dots ,x_i,\dots ,x_n)$ denotes a vector, where each $x_i$ is a scalar.
• $\mathbf{X}$ denotes a random matrix. More precisely, $\mathbf{X}=({\mathbf{X}}_1,{\mathbf{X}}_2,\dots ,{\mathbf{X}}_j,\dots ,{\mathbf{X}}_k)$, where ${\mathbf{X}}_j$ is a random vector related to the column $j$ with $n$-elements. In matrix context, the index $i$ denotes a generic row, while $j$ a generic column.

Syllabus

• Part 1. Probability
  1. Set and events
  2. Probability measure
  3. Measurable maps
  4. Independence
  5. Expectation
  6. Conditional expectation
  7. Moment generating functions
  8. Convergence concepts
• Part 2. Statistics:
  1. Introduction
  2. Moments
  3. Likelihood
  4. Hypothesis tests
  5. Multivariate data
• Part 3. Linear models:
  1. Linear models
  2. Classic least squares
  3. Generalized least squares
  4. Restricted least squares
• Part 4. Time Series:
  1. Time series
  2. AR and MA processes
  3. ARMA processes
  4. GARCH processes
  5. GARCH(1,1) Moments
• Part 5. Diagnostic tests:
  1. Stationarity tests
  2. Autocorrelation tests
  3. Normality tests
  4. Value at Risk tests
• Part 6. Notable Distributions:
  1. Bernoulli
  2. Exponential