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  3. University Of Nottingham Malaysia
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  5. Statistical Inference

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Statistical Inference Summary
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  • Grado
  • Statistical Inference
  • Institución
  • University Of Nottingham Malaysia

Overview on: - Score Statistics, Maximum Likelihood Estimation (MLE), Fisher Information - Sufficiency and Completeness with Neyman's Factorisation Theorem and Exponential Family - Parameter Estimation: Desirable Properties of Estimators, Method of Moments, MLE and its properties - Cramer-Rao L...

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Información del documento

  • 8 de junio de 2024
  • 24
  • 2023/2024
  • Resumen

Temas

  • properties of e
  • hypothesis testing
  • confidence intervals
  • cramer rao lower bound
  • delta method
  • dirichlet distri
  • sufficiency and completeness
  • prior and posterior distributions
  • bayesian prior distributions

Escuela, estudio y materia

  • University Of Nottingham Malaysia
  • Statistical Inference
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nghueyern

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we want to infor the parameters ! In this case ,
we are parametric :
models


Chapter 1 :
a) Likelihood are functions of an unknown parameter .
O


110(U) = Px(NIO) < for a
single case


IID obs , due to independence

(1014) = Px(ip)
,
roan

↳ can
use (101) =
log((10ln)) .


Why ? log transformation

log is an
increasing
is one-to-one
,

function
, >N




Why MLE ? Likelihood
says how likely a value of the parameter is
given the data


to inter from the data :
maximise the likelihood

↳ find of the the data
mostly likely value parameter given .




b) Score Statistic , VIX

vix)


0 ETu(x)]
=
l'10m)

0
=
vologPx(n10) - derivative of
log-likelihood !
Fisher information :↑210) ; distribution changes
·
quickly
=
when
>
-
② var(v(x)) = -
Ele"(Olns] =
210) O departs from 00 ; Oo
estimate well

&
210) =
nicd


c) Sufficiency :




Def :
A partition A of sample space - is called sufficient for O if for all AjzA ,
Px/MO , MEAj)

is independent of 0
.


sufficient stat of without .
0
>
knowing a we can find the probablity an event the need to know
-

,



>
-
E at least I sufficient partition ,
ie.
knowing all the n Idatal outcome




↳ minimal Sufficient

Def If sufficient partition A sufficient paration B set
:
is
a such that
given any other , any
element of B is contained in a set element of A then A is said to be minimal sufficient
.
,




>
If T isAncient
for O and Amplete then T is minimal sufficient (Bahadur Incorem (
-
.


,

, TO FIND SUFFICIENT STATISTIC !!

① Meyman's Factorisation Theorem to this !
- My show


A statistic T is sufficient for 0 Px(u10) =
g(0 , T(x)) n(X) ·




② Exponential Family

If X, Xn are IID from a dist of the
exp family


·
·

,
.
...




Px(n10) =
expLACOBIn) + <10) + D(u)} =
try to show this

lif you start with I obe,
Then T = ZBIXi) is sufficient for 0 to because the [BIxi)
state
.



need
clearly
comes from likelihood :
TTPx (n/0)

from def .




>
-
⑤ T is sufficient for O If

(i) for all n and a such that TIu) = a
,

Px(n10 , T(n) =
a) is independent of .
0


(ii) for all 2 and I',


T(u) E PxIMIO) is of O
=
T(u'l
independent
↑x (n'10)


AND T is minimal sufficient if(you show the other direction ,
is ,
E)

Px(n(0) => T() =
TIu'l

↑ x In '10)




d) completeness

family [PX (410) 083

&
Def :
A :
of distributions on - is called complete

if E[hix)] = 0 for all 00 >
- P(nIX) =
010) = 1 for all .
OE h(X) is a zero function


any statistic n(x)
for such that the above expectation makes sense .


>
- A statistic T is said to be
complete if its
family of distributions &P + CtIO) 08] :
is complete



① Exponential
family

Suppose (X , Xn)
X is IID sample from the probability model
=
..., an


Px(n(0) =
expCAIOBIn) + col + DinI] ,
Do


and let T = [B(Xi) denote the corresponding sufficient statistic.


If & contains an open interval ,
then T is .
complete
↓ (R 1 - 0 , 3)
Eg :
,

space of O

, *
completeness of X =
completeness of the
family of distributions of .
X




Chapter 2 :
Goal :
estimate gloy ,
a function of 0
.


Setting : Let (x) be our estimate of g(0) when we observe X =
.
a



An estimator is a function of the r v
.
.
XI, ...,
Xn
.
-



L 1) Estimator not
should take values outside the parameter space .




I
2) Unbiasedness


E[(x)) =
g(0) +
bg(0)

want bg10) = 0 g(x)] =
g(0) VOzO


desirable 3) small volatility
of Squared Error IMSE) > 0
properties mean -




estimators Def :
McE =
ESigIX)-glOT"] =...



=
var((x)) + (b(0))
4)
Consistency
g(x) +
g(0)asn + 0
,

it .
for every 320 , PlIIX)-gl01K) < 10) + 0 as n+



>
-

E(g(X)] + g(0) and var(g(x)) + 0 as n + & = g(X) is consistent for glo)

construct estimators :




a) method of moments


for the r-th moment, E(x] + x
>
n+ 0
-
,




Exr] = Xi [ law of
large numbers]

[theoretical] =
[data]
Exi
Eg
: Eix] =
N

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