GENERALIZED PIVOTS GENERALIZED PIVOTS FOR VARIANCE COMPONENTS FOR - - PowerPoint PPT Presentation

▶

Dec 17, 2022 110 likes •256 views

GENERALIZED PIVOTS GENERALIZED PIVOTS FOR VARIANCE COMPONENTS FOR VARIANCE COMPONENTS IN MIXED LINEAR MODELS IN MIXED LINEAR MODELS Barbora Arendack THE MODEL: Y = n-dimensional vector of observations X, A 1 = known matrices 2 I , ~

SLIDE 1

GENERALIZED PIVOTS GENERALIZED PIVOTS FOR VARIANCE COMPONENTS FOR VARIANCE COMPONENTS IN MIXED LINEAR MODELS IN MIXED LINEAR MODELS

Barbora Arendacká

SLIDE 2

THE MODEL:

Y = n-dimensional vector of observations X, A1 = known matrices

u~N q0,  1

2 I  ,~N n0,  2 I 

covu,=0

 , 1

2, 2 = unknown parameters

1

2≥0,  2 0

W 1 = A1 A1

THE TASK: construction of confidence intervals on 1

R A1⊈R X 

Further we suppose

SLIDE 3

Invariance with respect to the group of translations in mean: Minimal sufficient statistics: maximal invariant

BX

T BX=I n−rank X 

BX BX

T =M X =I n−X  X T X  − X T

spectral decomposition of V1

V 1=∑i=1

iE i

12...r≥0 1,.... ,r

Olsen, Seely, Birkes (1976)

V 1=BX

T W 1 BX

SLIDE 4

Balanced one-way random effects model

Y ij=iij ,i=1,... , I , j=1,... ,J

i~N 0,1

2 ,ij~N 0, 2 , mutually independent

U 1=S B

2~J 1 2 2I−1 2

between-groups sum of squares

U 2=SW

2 ~ 2I J−1 2

within-group sum of squares

1=J ,2=0

General case

U 1,... ,U r

U r~

2r 2

is not too restrictive supposing

SLIDE 5

Pivot

RU , 1

2

distribution independent of all parameters

(1-α )100% confidence set find q1 , q2 such that Construction of confidence intervals on 1

U 1 11

2 2

~1

2 ,......,

U r−1 r−11

2 2

~r−1

, U r 

2 ~r 2

U=U 1,...,U r

Pivot

SLIDE 6

Difficulties caused by nuisance parameters generalized inference Tsui, Weerahandi (1989), Weerahandi (1993) Extension of “classical” methods for testing and constructing confidence intervals in such a way that the quantities these methods are based on (pivots and test statistics) are let to be FUNCTIONS not only

f the RESPECTIVE RANDOM VECTOR

and the parameter of interest but of the OBSERVED DATA and ALL THE PARAMETERS as well The only requirement:

for each fixed data the generalized quantities behave

like their “classical” counterparts

SLIDE 7

GENERALIZED PIVOTS

U =U 1 ,...,U r u=u1 ,... ,ur

Generalized pivot: a function RU ,u , 1

2 , 2 with properties:

can find q1 ,q2 (1-α )100% confidence set For each observed u

SLIDE 8

A drawback The confidence set is constructed for fixed u, the distribution of usually depends on u, and so the set preserves the frequency properties of an (1−α )100% confidence set only conditionally, for given u, and its ACTUAL CONFIDENCE LEVEL must be CHECKED BY SIMULATIONS

RU ,u , 1

2 , 2

On the other hand Inclusion of observed data makes construction of expressions with distribution independent of nuisance parameters much easier

SLIDE 9

An example of a generalized pivot

U i i1

2 2

=Qi~i

U r 

2 =V~r 2

ui i1

2 2

bserved

value

bserved

value

ur 

1

∑i=1

r−1

ui− ui i 1

2 2



ur ur

∑i=1

r−1

iui i 1

2 2

R=∑i=1

r−1

ui−Qiur V 

∑i=1

r−1

iQi

robs= 1

SLIDE 10

Computing the bounds of the generalized interval R=∑i=1

r−1

ui−Qiur V 

∑i=1

r−1

iQi For fixed u

P q/2≤R≤q1−/2=1−

Monte Carlo methods Numerical integration + numerical solving of the equation

Cu={1

2;q/2≤Ru ,u , 1 2≤q1−/2}={ 1 2; q/2≤ 1 2≤q1−/2}

Simulate the distribution of R by generating a large number of Q1 ,...., Qr-1 ,V – independent, chi squared distributed random variables

1−/ 2=PR≤q1−/2=P∑i=1

r−1

Q iiq1−/2ur V ≥∑i=1

r−1

ui

Negative bounds are put equal to zero

SLIDE 11

Unless r =2, R is far from being unique Zhou, Mathew (1994)

R=

∑i=1

r−1

ciui−Qiur V 

∑i=1

r−1

i ciQi

nonnegative constants ci My suggestions for the choice of ci inspired by test statistics for testing the nullity of :

ci=1

ci=1/i ci=i

1

SLIDE 12

Another and not the only other generalized pivot Park, Burdick (2003)

∑i=1

r−1

ui i 1

2ur

V =∑i=1

r−1

ui i1

2ur 2

U r

∑i=1

r−1

U i i 1

2 2=∑i=1 r−1

Q i

the observed values are equal R defined as a solution for of the non-linear equation

1

observed value is 1

SLIDE 13

Again, constants can be added However, with the computational attractiveness is lost Computation of bounds of the interval: Monte Carlo methods or numerical approach

ci≠1

∑i=1

r−1

Q i~

=∑i=1

r−1

 i

SLIDE 14