STATS 730 Lecture 1 1022020 730 lecture 1

STATS 730: Lecture 1 10/2/2020 730 lecture 1 1

Today’s lecture (1) • Sampling distributions – Examples • Describing distributions – Densities – Probability functions – Distribution functions • Moments • Inequalities – Chebychev, Jensen 10/2/2020 730 lecture 1 2

Today’s lecture (2) • Joint distributions – Joint densities/prob functions – Conditional expectation • Convergence Concepts – Convergence in distribution – Convergence in probability 10/2/2020 730 lecture 1 3

Sampling Distributions – basic ideas • Start with a population, described by a distribution. The distribution has parameters that have a subject-matter interpretation • A sample is selected, say independently • A statistic is selected to estimate the parameter • The statistic is used to estimate the parameter. The distribution of the statistic is used to assess the accuracy of the estimate 10/2/2020 730 lecture 1 4

Populations: Two main schemes (1) • IID sample: imagine a “pot” holding a large population of items. Selecting from the pot produces random independent values X 1, X 2, …with probabilities P(a<Xi£b) described by some distribution IID=“independent, identically distributed” 10/2/2020 730 lecture 1 5

Two main schemes (2) • A finite population with values x 1, x 2, x 3, …x. N Select X 1, …, Xn without replacement Pr(Xi=xj) = 1/N • “Finite population sampling” (simple random sampling) 10/2/2020 730 lecture 1 6

Example: Blood pressure • Distribution of blood pressure in the population described by a statistical distribution, say normal N(m, s 2) • Mean Parameter: m interpreted as Average BP in population m estimated by sample mean • Sample mean has distribution N(m, s 2/n): used to construct confidence intervals 10/2/2020 730 lecture 1 7

Sampling distributions basic ideas (2) • 1 st sample: x 1, x 2, …xn® s 1 2 nd sample: x 1, x 2, …xn ® s 2 3 rd sample: x 1, x 2, …xn ® s 3 ……………… • Sampling from pop induces sampling from sampling distribution q: parameter of population • Want q to be close to mean of sampling distribution, which should have small variance 10/2/2020 730 lecture 1 8

Deriving sampling distributions • Given population distribution, and the statistic, the sampling distribution can be determined. • Sometimes by mathematical tricks, or by simulation. • Needed to assess the accuracy of estimates. 10/2/2020 730 lecture 1 9

Two important aspects • Mean of sampling distribution: how does this relate to the parameter? • Bias is difference between the two • Standard deviation of sampling distribution: how variable is the estimate? (Standard error of estimate=std dev of sampling distribution) 10/2/2020 730 lecture 1 10

Example: mean of Poisson eg traffic flows • Traffic flow in 20 fixed length-time periods X 1, …X 20, regarded as samples from Poisson distribution, mean l • Estimate l by mean or median? • Theory: std dev of sample mean is Ö(l/n) eg for n=20, l=5 is 0. 05 10/2/2020 730 lecture 1 11

Example (cont) Median: theory hard, but simulate > theta<-5 > Nsim<-5000 > n<-20 > X<-matrix(rpois(n*Nsim, theta), n, Nsim) > sqrt(var(apply(X, 2, mean))) [1] 0. 4998313 (as we would expect!!) mean(apply(X, 2, median)) [1] 4. 8322 > sqrt(var(apply(X, 2, median))) [1] 0. 6427486 Thus, mean is better!! 10/2/2020 730 lecture 1 12

Describing distributions(1) • Continuous random variables: describe distribution by densities: 10/2/2020 730 lecture 1 13

Examples Normal Gamma t Uniform 10/2/2020 730 lecture 1 14

Describing distributions (2) • For discrete distributions, we use probability functions 10/2/2020 730 lecture 1 15

Examples 10/2/2020 730 lecture 1 16

Describing distributions (3) • For both continuous and discrete we can use the distribution function 10/2/2020 730 lecture 1 17

Expectation 10/2/2020 730 lecture 1 18

Inequalities 10/2/2020 730 lecture 1 19

Chebychev Area £ s 2/e 2 m-e m m+e 10/2/2020 730 lecture 1 20

Jensen g(x) g(m) + g’(m)(x-m) m 10/2/2020 730 lecture 1 21

Relationship of F to f and p 10/2/2020 730 lecture 1 22

Marginal and conditional densities 10/2/2020 730 lecture 1 23

Conditional expectation Expectation calculated using the conditional distribution Useful results: 10/2/2020 730 lecture 1 24

Convergence in distribution: Main use: approximation of distributions 10/2/2020 730 lecture 1 25

Examples • tn converges to normal • Binomial converges to Poisson hus • Gamma converges to normal 10/2/2020 730 lecture 1 26

Central limit Theorem (CLT) 10/2/2020 730 lecture 1 27

Convergence (2) Convergence in probability Eg proportion of heads in n tosses® 0. 5 10/2/2020 730 lecture 1 28

Taylor Series 10/2/2020 730 lecture 1 29

Taylor series(2) Alternative form 10/2/2020 730 lecture 1 30

Taylor series (3) Generalization 10/2/2020 730 lecture 1 31

Taylor series(4) Stochastic version 10/2/2020 730 lecture 1 32

Order Statistics 10/2/2020 730 lecture 1 33