Neeraj Nagarkatti wrote:
>
> Question:
>
> Let X_1,...,X_n be a random sample from the Uniform U[0,t] distribution,
> i.e. with pdf:
>
> f(x|t) = 1/t (0 < x < t).
>
> Obtain the maximum likelihood estimator of t.
>
> Now the model solution says:
>
> Because it's a non-regular case, you can't differentiate it, so the
> likelihood is:
>
> L(t) = 1/t^n, 0 < x_i < t, (i=1...n)
>
> which is the same as:
>
> L(t) = 1/t^n, 0 < max(X_i) < t.
>
> And clearly:
>
> ^
> t = max(X_i).
>
> Now I don't quite understand why the mle has to be the sample maximum.
> Can any1 shed any light as to why this is the case?
The MLE method relates to C. Peirce's concept of "abductive reasoning",
which he proposed to complete the pattern started by deductive and
inductive reasoning; it resembles inductive reasoning in being heuristic
rather than rigorous.
Deductive reasoning: assume A; assume A=>B; conclude B
Inductive reasoning: assume A; assume B; conjecture A => B
Abductive reasoning: assume B; assume A=>B; conjecture A.
In this case, small but consistent values of t explain the data better
than larger values do, in that they make the joint probability density
of (X_1,...,X_n) higher.
Another related philosophical concept is "Occam's razor", a heuristic
telling us to prefer the simplest model that explains the facts.
"Simplest" in this case is interpreted as "making the data most probable
subject to the model", thus requiring less additional assumption of
random chance to complete the full explanation of what was observed.
Try thinking about the discrete version of the problems; it is easier
to think about because it lets one think about probability rather than
probability density. Suppose you know a bag contains counters numbered
from 1 to n and you draw a sample of size 1, and that counter is
numbered 2. Under which hypothesis is this best explained by what you
already knew: n=1000, n=10, n=10, or n=2?
(Another interesting but unrelated fact: most "obvious" estimators in
this problem can give estimates inconsistent with the data! Examples: 2x
the sample mean or 2x the sample median (for sample size >2 in each
case). In either case t-hat has a nonzero probability of being les than
X_max.
-Robert Dawson
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