CS计算机代考程序代写 STAT314. Poisson MLE. Derivation.

STAT314. Poisson MLE. Derivation.

STAT314. Poisson MLE. Derivation.

Elena Moltchanova

2021S2

Let xi|µ ∼ Pois(µ) for i = 1, …, n. Then the joint likelihood (i.e., the product of individual likelihoods):

L = f(x|µ) = f(x1, …, xn|µ) =

i

f(xi|µ) =

i

µxie−µ

xi!
=
µ

i
xie−nµ∏
i xi!

.

for µ > 0.

First, let’s log the likelihood to make things easier:

logL =

i

xi logµ− nµ− log
(∏

i

xi!
)
.

Now, let’s differentiate with respect (w.r.t.) to µ:

∂ logL
∂µ

=

i

xi
1
µ
− n.

Set it to 0, and solve for µ:


i

xi
1
µ
− n = 0 ⇐⇒ µ̂ =


i xi
n

= x̄.

Remember, that that is not yet a point where the function reaches maximum. It may be a minimum or an
inflection point. Always check for the second order condition:

∂2 logL
∂µ2

=

i

xi

(

1
µ2

)
.

You can substitute specific µ̂ to find that


i

xi

(

1
µ̂2

)
= −


i xi
x̄2

= −
nx̄

x̄2
= −

n


< 0 ∀x̄ > 0.

Or you can use the fact that µ2 > 0 for all µ > 0 to arrive at the same conclusion. Thus m̂u = x̄ is the
maximum likelihood estimate (MLE) of the Poisson intensity parameter µ.

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