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# Statistical Inference | Assignment

The assignment is about the quality of point estimators, specifically focusing on accuracy and precision. It defines various measures such as bias, variance, and mean squared error to evaluate the quality of an estimator.

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Added on  2022-08-29

## Statistical Inference | Assignment

The assignment is about the quality of point estimators, specifically focusing on accuracy and precision. It defines various measures such as bias, variance, and mean squared error to evaluate the quality of an estimator.

Added on 2022-08-29

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Statistical Inference
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Exercise 4.2
Question One
Consider a random sample of size n x1,x2,x3,.......xn from a density function f (x , θ), with
unknown parameter θ . Ones we observe the values of x1,x2,x3,...........xn, the likelihood
function is given by L(θ) =
i=1
n
f ( xi θ)ie the likelihood function is the product of the marginal
distributions.
The score function S(θx1 : n) is derived by taking the differential of the log likelihood fuction
with respect to θ. Ie dl(θ)
. It worthy noting that, the L(θ) and In L(θ) have their maximum at the
same value of θ and therefore it is easier to find the maximum using the logarithms of the
likelihood function.
It follows that, given the pdf of x1,x2,x3......xn which are iid, which is
F ( x ,θ ) = e xθ
1+ exθ x R, then the likelihood function is given by by;
i=1
n
f (xi θ) where i = 1,2,
....n
f ( x , θ ) = d
dx F ( x ,θ ) = exθ
1+ exθ x R d
dx exθ (1+exθ )1 = d
dx = exθ+1
f ( x , θ )=exθ x R,
e x1θ
1+ ex 1θ * ex 2θ
1+ ex 2θ * ex 3θ
1+ ex 3θ *......................* exnθ
1+ exnθ , taking the cumulative function
2 | P a g e
F ( x ,θ ) = e xθ
1+ exθ x R, = f ( x , θ )=exθ (1+ exθ)1 x R, = exθ(1+e x+θ) = exθ+1
f ( x , θ )=exθ +1 x R

i=1
n
f ( xi θ)=
i=1
n
e xiθ
+ 1 x R = e
i=1
n
(xθ)
+1
Log L f (xi θ) =
i=1
n
( xiθ ) log e +log 1 = L f (xi θ) =
i=1
n
( xiθ ) +log1
Question two
S(θx1 : n) = dl(θ)
L f ¿ = dl (θ)

i=1
n
( xiθ ) + log 1 =
i=1
n
(x 1+θ )
L(θx1 : n) = Var(S(θx1 : n)) Var(
i=1
n
(x 1+θ )) = Var(
i=1
n
xi)+Var(n θ) (Smirnov, 2011)
Considering the Var(x1) = π 2
3 it follows that Var(
i=1
n
xi) =
i=1
n
Var ¿ ¿) = 1
n2
i=1
n
var xi
= 1
n2 * n* π 2
3 = 2
3 n
J(θx1 : n) is given by the negative expectation of the second differential of the log likelihood
function. Given as: J(θx1 : n) = -E( d2 l(θ)

i=1
n
( xiθ ) +log 1) (Donnelly et al. 2016)
3 | P a g e

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