1. Stat310 Estimation
Hadley Wickham
Saturday, 11 April 2009
2. 1. What’s next
2. Recap
3. Example: speed of sound
4. T distribution
5. Comparing estimators
6. Interval estimators
Saturday, 11 April 2009
3. What’s next
Today: ﬁnish of estimation
Thursday & Tuesday: testing
Last class: where next? Other stats
courses and why you should bother
Help session tomorrow.
Saturday, 11 April 2009
4. Recap
What is the distribution of the average of
n iid normal random variables with the
same mean and variance?
How can you form a 95% ci for a random
variable with that distribution?
Saturday, 11 April 2009
5. Example
We want to ﬁgure out what the speed of
sound is. We do this by performing an
experiment with our velocitometer. A
velocitometer can measure the speed of
anything, but has normally distributed error
with standard deviation 10 meters per
second.
How can we decrease this error?
How can we frame this problem statistically?
Saturday, 11 April 2009
6. Your turn
We perform the experiment 10 times and
get the following 10 speeds:
340 333 334 332 333 336 350 348 331
344 (mean: 338, sd: 7.01)
What is our estimate of the speed of
sound? What is the error (sd) of this
estimate? Give an interval that we’re 95%
certain the true speed of sound lies in.
Saturday, 11 April 2009
7. Hat notation
Usually write the estimate of a parameter
with a little hat over it. Subscript
identiﬁes type of estimator used.
ˆ
µM M
ˆ2 σ 2
ˆ σˆ
µ
ˆ
µM L
Saturday, 11 April 2009
11. Example
We want to ﬁgure out what the speed of
sound is. We do this by performing an
experiment with our velocitometer. A
velocitometer can measure the speed of
anything, but has normally distributed
error with standard deviation 10 meters
per second.
Why is this example not realistic?
Saturday, 11 April 2009
12. Some reasons
No such thing as a velocitometer!
Scientiﬁc experiments usually much more
complicated
Don’t normally know the errors are
normally distributed.
Don’t normally know the standard
deviation of the errors.
Saturday, 11 April 2009
13. Resolution
Possible to overcome all of these
problems, but we’re going to focus on
just one.
What happens if we don’t know the
standard deviation, but have to estimate
it?
Saturday, 11 April 2009
14. Your turn
What is an estimate for the standard
deviation of a normal distribution?
When we have to estimate the sd, what do
you think happens to the distribution of
our estimate of the mean? (Would it get
more or less accurate? What will happen
to the conﬁdence interval?)
What about as n gets bigger?
Saturday, 11 April 2009
15. t-distribution
Xi ∼ Normal(µ, σ )2
¯n − µ ¯n − µ
X X
√ ∼Z √ ∼ tn−1
σ/ n s/ n
Parameter called
degrees of freedom
Saturday, 11 April 2009
16. 0.3
df
1
dens
0.2 2
15
Inf
0.1
−3 −2 −1 0 1 2 3
x
Saturday, 11 April 2009
17. Properties of the t-dist
Heavier tails compared to the normal
distribution.
lim tn = Z
n→∞
Practically, if n > 30, the t distribution is
practically equivalent to the normal.
Saturday, 11 April 2009
18. t-tables
Basically the same as the standard
normal. But one table for each value of
degrees of freedom.
Easiest to use calculator or computer:
http://www.stat.tamu.edu/~west/applets/
tdemo.html
(For homework, use this applet, for ﬁnal, I’ll give
you a small table, if necessary)
Saturday, 11 April 2009
19. Example
Back to the example.
340 333 334 332 333 336 350 348 331
344 (mean: 338, sd: 7.01)
If sd is known: (332, 344)
If not known: (333, 342) (2.23)
Saturday, 11 April 2009
20. Constructing interval
¯n − µ
X
√ ∼Z
σ/ n
¯n − µ
X
√ ∼ tn−1
s/ n
Saturday, 11 April 2009
21. Steps
Form conﬁdence interval for
standardised distribution.
Write as probability statement.
Back transform.
Write as interval.
Saturday, 11 April 2009
22. More complicated case
(n − 1)S 2
∼ χ (n − 1)
2
2
σ
Find 95% conﬁdence interval for standard
deviation in previous case
(sd = 7.01, n = 10)
Saturday, 11 April 2009
23. Standard deviation
Find conﬁdence interval for χ2(9).
Generally want the shortest conﬁdence
interval, but hard to ﬁnd when not
symmetric.
Any of the following are correct. The best
has the smallest interval.
Saturday, 11 April 2009
30. Steps
Form conﬁdence interval for
standardised distribution.
Write as probability statement.
Back transform.
Write as interval.
Saturday, 11 April 2009
31. Comparing estimators
We know two ways of estimating the
standard deviation of the normal
distribution: the usual standard deviation
of the data, or the estimate from
maximum likelihood or method of
moments
How can we compare the two?
Saturday, 11 April 2009
32. What are other
estimators of the
standard deviation?
Saturday, 11 April 2009
33. Your turn
How can we compare different point
estimators?
Say I have u1 and u2 which are functions
of the X's, trying to estimate some θ.
Based on what properties could I choose
between u1 and u2? (probability, mean
and variance)
Saturday, 11 April 2009
34. ˆn ) = θ
E(θ Unbiased
ˆ1 ) < V ar(θ2 )
ˆ
V ar(θ
Minimum variance
Common problem is to ﬁnd UMVE (unbiased
minimum variance estimator) across all possible
estimators
Saturday, 11 April 2009
35. Low bias, low variance Low bias, high variance
High bias, low variance High bias, high variance
Saturday, 11 April 2009
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