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rLlZpnT02ZU
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just because-- well, that's
just we can say a set,
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rLlZpnT02ZU
|
a confidence set.
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rLlZpnT02ZU
|
But people like to
say confidence region.
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rLlZpnT02ZU
|
So an interval is just a
one-dimensional conference
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rLlZpnT02ZU
|
region.
|
rLlZpnT02ZU
|
And it has to be an
interval as well.
|
rLlZpnT02ZU
|
So a confidence interval
of level 1 minus alpha--
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rLlZpnT02ZU
|
so we refer to the quality
of a confidence interval
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rLlZpnT02ZU
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is actually called it's level.
|
rLlZpnT02ZU
|
It takes value 1 minus alpha
for some positive alpha.
|
rLlZpnT02ZU
|
And so the confidence level--
|
rLlZpnT02ZU
|
the level of the confidence
interval is between 0 and 1.
|
rLlZpnT02ZU
|
The closer to 1 it is, the
better the confidence interval.
|
rLlZpnT02ZU
|
The closer to 0,
the worse it is.
|
rLlZpnT02ZU
|
And so for any
random interval-- so
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rLlZpnT02ZU
|
a confidence interval
is a random interval.
|
rLlZpnT02ZU
|
The bounds of this interval
depends on random data.
|
rLlZpnT02ZU
|
Just like we had
X bar plus/minus
|
rLlZpnT02ZU
|
1 over square root of
n, for example, or 2
|
rLlZpnT02ZU
|
over square root
of n, this X bar
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rLlZpnT02ZU
|
was the random thing that would
make fluctuate those guys.
|
rLlZpnT02ZU
|
And so now I have an interval.
|
rLlZpnT02ZU
|
And now I have its boundaries,
but now the boundaries
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rLlZpnT02ZU
|
are not allowed to depend
on my unknown parameter.
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rLlZpnT02ZU
|
Otherwise, it's not a
confidence interval,
|
rLlZpnT02ZU
|
just like an
estimator that depends
|
rLlZpnT02ZU
|
on the unknown parameter
is not an estimator.
|
rLlZpnT02ZU
|
The confidence interval
has to be something
|
rLlZpnT02ZU
|
that I can compute
once I collect data.
|
rLlZpnT02ZU
|
And so what I want is that--
so there's this weird notation.
|
rLlZpnT02ZU
|
The fact that I write theta--
|
rLlZpnT02ZU
|
that's the probability
that I contains theta.
|
rLlZpnT02ZU
|
You're used to seeing
theta belongs to I.
|
rLlZpnT02ZU
|
But here, I really
want to emphasize
|
rLlZpnT02ZU
|
that the randomness is
in I. And so the way
|
rLlZpnT02ZU
|
you actually say
it when you read
|
rLlZpnT02ZU
|
this formula is the probability
that I contains theta
|
rLlZpnT02ZU
|
is at least 1 minus alpha.
|
rLlZpnT02ZU
|
So it better be close to 1.
|
rLlZpnT02ZU
|
You want 1 minus alpha
to be very close to 1,
|
rLlZpnT02ZU
|
because it's really
telling you that whatever
|
rLlZpnT02ZU
|
random variable I'm giving
you, my error bars are actually
|
rLlZpnT02ZU
|
covering the right theta.
|
rLlZpnT02ZU
|
And I want this to be true.
|
rLlZpnT02ZU
|
But I want this--
since I don't know
|
rLlZpnT02ZU
|
what my confidence--
my parameter of theta
|
rLlZpnT02ZU
|
is, I want this to hold
true for all possible values
|
rLlZpnT02ZU
|
of the parameters that nature
may have come up with from.
|
rLlZpnT02ZU
|
So I want this-- so there's
theta that changes here,
|
rLlZpnT02ZU
|
so the distribution
of the interval
|
rLlZpnT02ZU
|
is actually changing
with theta hopefully.
|
rLlZpnT02ZU
|
And theta is changing
with this guy.
|
rLlZpnT02ZU
|
So regardless of the value
of theta that I'm getting,
|
rLlZpnT02ZU
|
I want that the probability
that it contains the theta
|
rLlZpnT02ZU
|
is actually larger
than 1 minus alpha.
|
rLlZpnT02ZU
|
So I'll come back
to it in a second.
|
rLlZpnT02ZU
|
I just want to say
that here, we can
|
rLlZpnT02ZU
|
talk about asymptotic level.
|
rLlZpnT02ZU
|
And that's typically when
you use central limit
|
rLlZpnT02ZU
|
theorem to compute this guy.
|
rLlZpnT02ZU
|
Then you're not guaranteed
that the value is
|
rLlZpnT02ZU
|
at least 1 minus
alpha for every n,
|
rLlZpnT02ZU
|
but it's actually in the limit
larger than 1 minus alpha.
|
rLlZpnT02ZU
|
So maybe for each fixed n
it's going to be not true.
|
rLlZpnT02ZU
|
But for as no goes
to infinity, it's
|
rLlZpnT02ZU
|
actually going to become true.
|
rLlZpnT02ZU
|
If you want this to
hold for every n,
|
rLlZpnT02ZU
|
you actually need to use things
such as Hoeffding's inequality
|
rLlZpnT02ZU
|
that we described at some
point, that hold for every n.
|
rLlZpnT02ZU
|
So as a rule of thumb, if you
use the central limit theorem,
|
rLlZpnT02ZU
|
you're dealing with
a confidence interval
|
rLlZpnT02ZU
|
with asymptotic
level 1 minus alpha.
|
rLlZpnT02ZU
|
And the reason is
because you actually
|
rLlZpnT02ZU
|
want to get the quintiles
of the normal-- the Gaussian
|
rLlZpnT02ZU
|
distribution that comes from
the central limit theorem.
|
rLlZpnT02ZU
|
And if you want to use
Hoeffding's, for example,
|
rLlZpnT02ZU
|
you might actually get away with
a confidence interval that's
|
rLlZpnT02ZU
|
actually true even
non-asymptotically.
|
rLlZpnT02ZU
|
It's just the regular
confidence interval.
|
rLlZpnT02ZU
|
So this is the
formal definition.
|
rLlZpnT02ZU
|
It's a bit of a mouthful.
|
rLlZpnT02ZU
|
But we actually-- the best
way to understand them
|
rLlZpnT02ZU
|
is to build them.
|
rLlZpnT02ZU
|
Now, at some point I said--
|
rLlZpnT02ZU
|
and I think it was
part of the homework--
|
rLlZpnT02ZU
|
so here, I really
say the probability
|
rLlZpnT02ZU
|
the true parameter belongs
to the confidence interval
|
rLlZpnT02ZU
|
is actually 1 minus alpha.
|
rLlZpnT02ZU
|
And so that's because here,
this confidence interval
|
rLlZpnT02ZU
|
is still a random variable.
|
rLlZpnT02ZU
|
Now, if I start plugging
in numbers instead
|
rLlZpnT02ZU
|
of the random
variables X1 to Xn,
|
rLlZpnT02ZU
|
I start putting 1,
0, 0, 1, 0, 0, 1,
|
rLlZpnT02ZU
|
like I did for the kiss
example, then in this case,
|
rLlZpnT02ZU
|
the random interval is actually
going to be 0.42, 0.65.
|
rLlZpnT02ZU
|
And this guy, the probability
that theta belongs to it
|
rLlZpnT02ZU
|
is not 1 minus alpha.
|
rLlZpnT02ZU
|
It's either 0 if
it's not in there
|
rLlZpnT02ZU
|
or it's 1 if it's in there.
|
rLlZpnT02ZU
|
So here is the
example that we had.
|
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