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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

qname(x, theta)

returns value of the quantile F^-1(x, theta)