NYU Derive the Likelihood Function Statistics Worksheet

Let X1,X2,…,Xm and Y1,Y2,…,Yn be independent random samples fromN(µ1,σ21) and N(µ2,σ22), respectively. We wish to test the null hypothesis. Derive Simple vs composite hypothesis
¨
In a hypothesis test,we can have different form of alternative
hypotheses (one-sided and two-sided).
two-ride at
For example:
H0: µ = µ0
vs
H1: µ ≠ µ0
one-sided
H0: µ = µ0
vs
H1: µ > µ0
{
¨
¨
¨
We can also talk about simple and composite hypotheses.
In simple hypotheses, the value of the parameter is completely
specified.
For example, if !! , !” , ⋯ , !# ~% &, ‘ ” (independently and identically
distributed), we could test
÷
($ : & = & %
simple
vs
(& : & = &!
simple
2
Simple vs composite hypothesis
¨
Composite hypotheses are hypotheses that covers a set of values for
the parameter.
simple
composite
For example:
H0: µ = µ0
vs
H1: µ ≠ µ0
H0: µ = µ0
vs
H1: µ > µ0
¨
Hence: one-sided and two-sided alternative hypotheses are always
composite hypotheses.
iid
¨
If !! , !” , ⋯ , !# ~% &, ‘ ” (independently and identically distributed),
what is the✓
\parameter space?
parameters- ( M i o )
p a r a m e t e r space =
E R
µ
112×1131=52
E1Rt¥I¥¥o ÷x÷¥÷÷÷ie
02
3
Neyman Pearson lemma
something
similar
–
t o
a
theorem
Let !! , !” , ⋯ , !# be a sample of iid observations from a distribution with
probability density function (pdf) + !; – .
i
i
i
.
Suppose we want to test
‘s-i-i o’
($ : – = -%
./
(& : – = -!
simple
simple
–
–
s
i
–
–
The likelihood ratio test, that is the test that rejects H0 when
wiggeries
→
liked’ah8dh,
III
∏#'(! + !’ ; -%
ℒ% 1
= #
∏'(! + !’ ; -!
ℒ! 1
.
is less than c where c is chosen so that
Pr Reject ($
($ true) = Pr
ℒ! *
ℒ” *
=
under H o
←
.
f> =
Ha. 0 = 0 0
\
| ($ true = ? = – O J
is the most powerful (has the most power) test of significance level ?.
4
Example:
Generalized likelihood ratio
The generalized likelihood ratio test (GLRT) extends the idea of the
likelihood ratio test to the case of composite hypotheses.
Let !! , !” , ⋯ , !# be a sample of iid observations from a distribution with
probability density function (pdf) + !; – .
Let @ denote the parameter space and let Ω = B$ ∪ B& .
Suppose we want to test
($ : – ∈d-0o ./
(& : – ∈521
-1
–
then, the GLRT is defined as the test that rejects ($ when
sup-.Ω0 ℒ 1; F=
sup-. 0 ℒ 1; is less or equal than I∗ with I∗ chosen so that
–
Pr Reject ($
t f
($ true) = Pr Fo>f M∗ | ($ true = ?
Example:
Generalized likelihood ratio
Let !! , !” , ⋯ , !# be a sample of iid observations from a distribution with
probability density function (pdf) + !; – .
If the pdf + !; – satisfies certain regularity conditions, then as the
number of observations n tends to infinity
−O PQR F
where
sup-.Ω0 ℒ 1; F=
sup-.Ω ℒ 1; –
follows a chi-square (S2 ) distribution with degrees freedom given by
dim B − dim B$ .
Let X1,X2, …,Xm and Y1,Y2, …, Yn be independent random samples from
N(41,0ỉ) and N(u2, ož), respectively. We wish to test the null hypothesis
Ho : 0 = 0
VS.
Hp : 01 702
(a) Derive the form of the generalized likelihood ratio A.
(b) Calling f
(xi-72/(m-1)
£?-1611-5)/(n-1), espress A as a function of f.
(c) Under the null hypothesis Ho, what distribution does
EL (XX)/(m – 1)
2?–1(Y; – )/(n-1)
follow?