Review of Mathematical Statistics
Yulin Hou
Department of Economics
Florida International University
Spring 2017
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Random Sampling
If Y1 , Y2 ,..., Yn are independent random variables with a common
probability density function f (y ), then Y1 , Y2 ,..., Yn is said to be a
random sample from f (y ).
A random sample is a set of independent identically distributed
(i.i.d) random variables.
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Estimators and Estimates
Typically, we can not observe the full population, so we must
inference based on estimates from a random sample.
An estimator is just a mathematical formular for estimating a
population parameter from a sample data.
An estimate is the actual number the formular produces from the
sample data.
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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What make a good Estimator?
Unbiasedness
An estimator, W of θ, is an unbiased estimator if E(W)=θ
(e.g), sample average is unbiased estimator of the population
mean.
Efficiency
If W1 and W2 are two unbiased estimator of θ, W1 is efficient
relative to W2 when Var(W1 ) < Var(W2 ).
(e.g.)
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Consistency
Law of large number refers to the consistency of sample averages
as estimator for θ, that is, to the fact that: plim(W )=θ
An unbiased estimator is not necessarily consistent.
An biased estimator is not necessarily inconsistent.
If W is an unbiased estimator of θ and Var(W) →0 as n → ∞, then
plim(W )=θ. (e.g. sample average is a consistent estimator of the
population mean )
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Estimators as Random Variables
Each of our sample statistics (e.g. the sample mean, sample
variance, etc.) is a random variable.
Each time we pull a random sample, we will get different sample
statistics.
If we pull lots and lots of samples, we will get a distribution of
sample statistics.
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Central limit theorem
The central limit theorem states that a random sample for any
population (with finite variance) has an asymptotic normal
distribution
Y n ∼ N (µ, σ2 )
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Econometrics
Spring 2017
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Hypothesis testing
A premise or claim that we want to test
H0 and H1
Null Hypothesis–H0
Alternative Hypothesis–H1
eg.It is believed that a candy machine makes chocolate bars that
are on average 5g. A worker claims that the machine after
maintenance on longer makes 5g bars. Write H0 and H1 .
H0 : µ = 5g
H1 : µ 6= 5g.
Yulin Hou, FIU Eco
Econometrics
Spring 2017
8 / 13
Examples
A company has stated that their straw machine makes straws that
are 4 mm diameter. A worker believes the machine no longer
makes straws of this size. Write H0 and H1 .
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Examples
Doctors believe that the average teen sleeps on average no longer
than 10 hours per day. A researcher believes that teens on
average sleep longer. Write H0 and H1 .
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Examples
The school board claims that at least 60 percent of students bring
a phone to school. A teacher believes that this number is too high.
Write H0 and H1 .
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Examples
The school board claims that at least 60 percent of students bring
a phone to school. A teacher believes that this number is too high.
Write H0 and H1 .
Yulin Hou, FIU Eco
Econometrics
Spring 2017
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Continue
Possible outcomes of hypothesis testing
Reject null hypothesis
Fail to reject null hypothesis
Level of confidence: how confident are we in our decision?
C=90%; 95%; 99%
confidence interval
Levels of significance
α=0.10; 0.05; 0.01
Yulin Hou, FIU Eco
Econometrics
Spring 2017
13 / 13