# 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?

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