Mathematical Statistics

Population

r.v.

distribution

distribution family

exponential family

Sample

sample space

sample size

r.v. | observation

sampling

simple random sample

i.i.d.

Statistics

sampling distribution

general

1

sample mean

sample variation / sd

sample moment

sample covariance / relavant

2

ordered statistics

sample median

extreme of sample

sample p-fractile / p-quantile

3

sample coefficient of variation

sample skewness

sample kurtosis

4

empirical distribution function

sufficient statistic

# ordered is sufficient

Distributions

discrete

0-1

binomial

poisson

0-1-...-k

multinomial

continuous

uniform

exponential

normal

x2

t

F

cauchy

gamma

beta

Z

multivariate normal

Method

moment estimator

maximum likelyhood estimator | MLE

Assessment

MSEM

MESM = Var + bias^2

MESM = Var

相对效率 = MSEM / MSEM

bias

unbiased | UE

estimable

biased

一致最优估计

UMVUE

CR inequation

general

smooth condition

regular condition

Fisher 信息量

UE and another condition

conclusion

iid

充分完备统计量

parameter estimation

point

interval

hypothesis test