Statistical Theory

Overview

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Explore advanced statistical theory concepts through a comprehensive 5-hour video series. Delve into topics such as sum of squared normal variables, best predictors using functions, alternative expected value formulas, and incomplete beta functions. Learn about confidence intervals for population medians, discrete order statistics, and Poisson probabilities. Discover proofs for the median minimizing absolute loss, Markov and Chebyshev inequalities, and the weak law of large numbers. Study the binomial theorem, big O notation, and correction methods like Holm-Bonferroni and Simes. Examine matrix properties, gamma function characteristics, and distribution approximations. Investigate confidence intervals for population means, moments of sample means, and noncentral chi-square distributions. Gain practical skills using R for bivariate normal probability calculations and generating double exponential data. Explore statistical concepts such as rotational invariance, Kruskal's proof, and deriving cumulative distribution functions for inverse gamma distributions.

Syllabus

Statistical Theory: Sum of Squared Normal mean=mu var=1 variables.
"Best" predictors of Y using a function of X..
Alternative Formula for the Expected Value.
Incomplete Beta Function as the Sum of Binomial Probabilities.
CI for Population Median using Order Statistics.
Discrete Order Statistics with Illustration using R.
Sum of Poisson Probabilities equal a Chi-square Probability.
Using R to Find an Exact CI for a Poisson Parameter.
The Median Minimizes Absolute Loss. 3 proofs when X is continuous..
Markov Inequality. Chebyshev Inequality. Weak Law of Large Numbers..
Proof of Binomial Theorem with specific cases of the General Binomial Theorem.
Big O, Little o Notation. Examples with Cumulant and Moment Generating Functions.
Proof of Holm Bonferroni Correction Method.
Proof of Simes Correction Method.
2 formulas between the determinant, trace and eigen values of a matrix.
Properties of the Gamma Function (part 1 of 2).
Properties of the Gamma Function (part 2 of 2).
Chi square approximation to an F Distribution.
Asymptotic C I for the Difference of 2 Independent Population Means.
Exact C I for the difference of 2 independent normal population means.
1st 4 moments of the sample mean when x is a Bernoulli random variable.
A df=1 noncentral chi sq distribution as a Poisson weighted mixture of central chi sq distributions.
Using R: Calculating Probability for a Bivariate Normal Random Variable.
Statistical Distance.
Extended Cauchy-Schwarz Inequality.
Rotational Invariance.
Generating Double Exponential Data from Scratch.
Kruskal's Proof of the Joint Distribution of the Sample Mean and Variance.
Derive the CDF of an Inverse Gamma Distribution.