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, while $X_n=(x_1,x_2,\dots ,x_i,\dots ,x_n)$ a generic collection of $n$ realization of the random variable. Where small letter, e.g $x_i$ denotes a scalar.
• Bold and capital letter stands for a matrix, i.e. ${\mathbf{X}}_{n,k}$ denotes a matrix of $k$ random variables each one with $n$ observations, where bold with small letter denotes a vector ${\mathbf{x}}_j=(x_1,x_2,\dots ,x_i,\dots ,x_n{)}^{\mathrm{\top }}$ for $j=1,2,\dots k$. In matrix context, the index $i$ denotes a generic row, while $j$ a generic column.

Syllabus

• Part 1. Probability
  1. Set and events
     • Set operations
     • Indicator function
     • Limits of sets
     • Fields and $\sigma$-fields
  2. Probability measure
     • Consequences of the axioms
     • Maps and inverse map
     • Induced distribution function
     • Independence
  3. Random variables
  4. Expectation
     • Review of inequalities
  5. Conditional expectation
     • Properties of conditional expectation
     • Conditional probability
  6. Characteristic functions
  7. Convergence concepts
     • Types of convergence
     • Laws of large numbers
     • Central limit theorem
• Part 2. Statistics:
  1. Introduction
  2. Sample moments
     • Expectation
     • Variance and covariance
     • Skewness
     • Kurtosis
  3. Maximum likelihood
  4. Hypothesis tests
  5. Multivariate data
• Part 3. Linear models:
  1. Linear models
     • The matrix of data
     • Joint, conditional and marginals
     • Conditional expectation model
  2. Classic least squares
  3. Generalized least squares
  4. Restricted least squares
  5. Multiequationals linear models
• Part 4. Time Series:
  1. Time series
     • Stationarity
     • Lag operator
     • Notable processes
  2. AR and MA processes
  3. ARMA processes
  4. GARCH processes
  5. GARCH(1,1) Moments
• Part 5. Diagnostic tests:
  1. Normality tests
  2. Stationarity tests
  3. Autocorrelation tests
  4. Value at Risk tests
• Part 6. Robustness:
  1. Tukey Functions
• Part 7. Distributiobs:
  1. Gaussian Mixture