# Statistics

## Neyman-Pearson and some other Uniformly Most Powerful Tests

Introduction Suppose data consisting of i.i.d. observations $$X^n=(X_1,X_2,\cdots,X_n)$$ are available from a distribution $$F(x,\theta),\,\theta\in\Theta\subset\mathbf{R}.$$ The exact value $$\theta$$ corresponding to the distribution that generated the observations is unknown. The problem is, using the available data $$X^n,$$ construct tests for making decisions on the possible value of unknown parameter $$\theta$$.

## Bayesian Estimation

CEO Salary Estimation Problem Consider the following problem. An investigative reporter wants to figure out how much salary makes the CEO of an investment bank X. For this he conducts interviews with some of the employees of that bank and writes down their salaries, which forms the following sample

## Win Odds Confidence Intervals in R

Continuous distributions Mann-Whitney estimate for the win probability Consider two independent, continuous RVs (random variables) $$\xi$$ and $$\eta$$. The following probability is called the win probability of RV $$\eta$$ against the RV $$\xi$$

## Exponential Distribution

Introduction This is a short reminder of some simple properties of exponential distributions. The continuous random variable (RV) $$\xi$$ has an exponential distribution with the rate $$\lambda>0$$ if its CMD (cumulative distribution function) has the following form

## Optimality of Estimators in Regular Models

Consistent Estimators In the estimation problem of a one-dimensional parameter $$\theta$$ from an i.i.d. sample $X^n=(X_1,\cdots,X_n),\ \ X_i\sim F(x,\theta),\,\theta\in\Theta\subset R$ we introduced the notion of consistency of an estimator.