Flashcards in Chapter 2 Deck (20):

1

## define a parametric family

### a parametric family is a collection of distributions of the same type which differ only in the value of one or more parameters say theta.

2

## theta hat

### out best guess for the unknown parameter theta based on the data set {x1,...,xn}. theta hat depends on the data so it is random

3

## for a simple random sample, the join distribution fX1,...Xn(x1,...xn;theta)=

### product i=1 to n (fX(xi:theta)). Since Xis are IID fXi=FX

4

## kth population moment=

###
expectation(X^k;theta)= integral -inf to inf x^kf(x:theta)dx

for discrete it is the sum x=-inf to inf x^k p(x;theta)

5

## what does the kth population moment mean

### the average value of X^k in the population

6

## kth sample moment=

### mk=(x1^k+x2^k+...+xn^k)/n

7

## what does the kth sample moment mk mean

### the average value of x^k in the sample

8

## sample mean m1=

### (x1+...+xn)/n

9

## method of moments relies on what

### that if the data comes from SIMPLE RANDOM sample then the sample values are representative of population values therefore the expectation (X^k; theta) is roughly equal to the sample moment mk

10

## if the family has one parameter theta then theta hat mom=

### E(X; theta hat mom)=m1

11

## if the family has two parameters alpha and beta then

###
E(X; alpha hat mom, beta hat mom)=m1

E(X^2; alpha hat mom, beta hat mom)=m2

12

## P(Xi<=y)=

### Fx(y; theta hat)

13

## FX^-1(FX(y; theta); theta) =

### y

14

## yk/n=

### FX^-1(k/n : theta hat)

15

## x(k) is roughly equal to

### Fx^-1(k/n+1 ; theta)

16

## define quantiles of the distribution

###
for given value of n and a given distribution FX(x) the quantiles of the distribution are the n values

Fx^-1(k/n+1 ), k=1,...,n

17

## define sample quantiles

### sample quantiles are the n ordered sample values x(1),...x(n) that split the sample into roughly equal parts

18

## dname(x,theta)

### returns value of the density f(x;theta)

19

## pname (x, theta)

### returns value of the probability F(x; theta)= P(X<=x; theta)

20