AD-R123 844 7UNCLASSIFIED
CONFIDENCE INTERVALS FOR FUNCTIONS OF VARIANCE COMPONENTS(U) COLORADO STATE UNIV FORT COLLINS DEPT OF STATISTICS F A GRAYBILL 14 OCT 82 N88914-78-C-8463 F/G 12/1
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Source: Colorado State University, Fort Collins, Colorado
Final Report on "Confidence Intervals for Functions of Contract Number N00014-78-C-0463
Variance Components."
by Franklin A. Graybill In this research__roject confidence intervals were determined for the functions included one-sided
of variance components for the situations listed below. -These and two-sided confidence intervals.
Since no method is available for exact (exact confidence coefficients) for Sconfidence intervals on the functions of variance components considered, confidence In each case large
intervals with approximate confidence coefficients were examined.
simulation studies were conducted to determine how.good the confidence intervals are. In cases where alternative methods are available they were compared to the procedures in this project
Ix
Confidence
uteals were examined for the cases listed below. AO~u1.
Model
S
+Bi i +
+ Ci CC j
DTIC TAB Unanmounced
0ustlfoatlon
k
@ VarEA]
-
2; var [B
] = o.
Var[C i
'
'
TIS ORwI
1. Two-fold nested model S+ A
l
3 M 02
Oslo
ii
so"
All random
0
_Dietributin/ Availabili-tyCodes Avail - and/or Dist Special
variables Ai, Bij, Cij k independent normal.
.are i
1,
Also
... , I; j - 1, ... , J; k
1,
D T IC
..., K.
Confidence intervals on the following were determined A
B
02_+_02_+_a2___a2_2.JAN
C
2/2; 02/2; A T' B T' B/° A
A
Total
ELECTE
f
27
3
/T' !STRIBUI ON STATEMLET
2 /02
I
C/T'
82
rApoved for public .mlea Distribution Unlimited
10 25
098
I
D
October 14, 1982
W
DA
-2-
2. Three-way cross classification model with interactions =
YJk J[A
+ Ai + Bj + Fij + C + G +H +P ijll I j j k 1k jk +uk
=t[B] -
I
2 vaUBY A' ; ja[
Var[A i Var[G
I ij
[Ck]
e[Fij] =
2
':' The random variables Ai, B
3.
+
-I+A
f[A
*
2
aC;
ijk
F1 j , Ck, Gk
I; j - 1, ..., J; k 2
2
-
,
Hjk
Pijk are independent normal.
1, ..., K.
a2 were determined.
C
Two-way cross classification model without interaction Y
-.
0.
" a2
A, aB
.°
ijk
F var[C k
. var[P
Confidence intervals on
(Hjk]
-
2
]a
r[ B;vrFii
jk I-aH'
G vr[
Also i = 1, ...,
t[GikI
2;
var[H] =
og
-
-
J I[Ba]
var[Eij]
".
=
1,
B
+ E
C[EiJ
-
12.
-
0; var[Ai]
-
1; var[B]
-a
The random variables Ai, Bi. Eij are independent normal.
... , I; j "
Also
1, ... , J.
Confidence intervals were determined on the following.
02 0
T
4.
+ a2 + a2 ; a2 /a 2 . U2 /0 2 B
E'AT'
a2 /a 2 ; a2 /a2 ; a2 1/(a
BT'ET'A
B
A
2 A
+ a2 ); 0 2 /(a 2 + a2 ). B
B
A
B
Two-way cross classification model with interaction
+ A
+
Bj + Gij + Ek
EA~I[AC[Bj]
-
t[GEik]
Yijk
*
A
- 0; varCAi]
-
aF; var[B]
""
var[Eik] = a1. The random variables A., Bi, GI
4
Also
*i
Confidence intervals were determined for the following.
1 - 1, ..., I; j
-
,
-
a2; var[G
Eujk are independent normal.
1, ... , J; k - 1, ..., K.
.G;
-3-
2 + 02 + 02 + 02; a2/ay2. U2/02; 02/r2. U2 /ay
02-F
T
5.
Let U
A
B
nS
2
G
/e
£ i i
freedom for i
-
E
A T
B T
GT'
E T
be independent chi-square random variables with n degrees 2 2 1, 2, where S are observable. Confidence intervals procedures
i
£
were determined for Cle 1 + C2
2
for C1 > 0, C2
.O.
These procedures were compared
with standard procedures developed by Welch and Satterthwaite and found to be significantly better.
6.
For the model considered in 5, a procedure was developed to determine "exact" 1 - a confidence coefficients on C10 1
+
C2 02 for C1
1 0, C2
> 0.
The vord "exact"
means that for any c > o the confidence coefficient was within e of the specified confidence coefficient 1 - a.
These confidence intervals were compared with
those in 5 and found to be better but not significantly better.
7.
Let U,
n S2 /e
be independent chi-square random variables with ni degrees
of freedom for i - 1, 2, 3 where the S2 are observable. Intervals were determined and evaluated for CIe
Approximate confidence
+ C202 + C3e3 for C,
1 0. They
were found to be better than the conventional procedures developed by Welch and Satterthwaite.
41
-4-
-~
. 8.
= n S 2 /0
Let U
i
i i I
freedom for i
be independent chi-square random variables with n1 degrees of
1, 2, 3, 4 where the 52 are observable.
=
Approximate confidence
intervals on 9 were determined and evaluated where C18 + C28 11 22
C3e 44 3 3 + C4e where C
> 0.
It was found that a procedure using Satterthwaite's method was
quite adequate.
9.
LetU
= nS
2
be independent chi-square rnadom variables with n1 degrees of
/0
1, 2, where S2 are observable. i
freedom for i
were obtained and evaluated for C18
Approximate confidence intervals
- C2a2 where C
>0. These confidence
intervals were quite good.
10.
Let U
be independent chi-square random variables for I
= n S2/0
i
i i
where S2 are observable.
i
(a)
+ e2We
1+O2
(d) (61
4
The procedures are in various stages of completion
81 + 82 - 83
(b) (e .(c)
Approximate confidence intervals were considered for the
listed below.
functions of 0
1, 2, 3, 4 1
-
_" - . _ ." - . '- . . ". .
83 -
2)/e
4
3.
.: ._
" .. . i.
T , • _ :, . .
.
.-
-,
.... . .. =
:
.
- :
/
.
./ -
.
-
I
.-;
-5-
11.
One-factor nested model with unequal numbers Y
ij
ij
ICA I
+ A
i
+ E
i
J[Ei I -0;
var[A 3
are independent normal.
02; var[E1iJ
Also j
1, 1
...,
2.
n; i
The random variables Ai, E 1,
...
,
I.
Approximate confidence intervals were derived for u 2+ 02. yet completed.
4
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O
3.
RECIPIENT'S CATALOG NUMBER
S.
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Final Report "Confidence Compoinents"Fia Variance of on ~Functions Intervals forFinal
-
S. PERFORMING ORG. REPORT NUMIER
7.
CONTRACT OR GRANT NUMBER(s)
S.
AUTHOR(#)
NR 042-402 N00014-78-C-0463
Franklin A. Graybill S.
PROGRAM ELEMENT. PROJECT. TASK AREA & WORK UNIT NUMBERS
10.
PERFORMING ORGANIZATION NAME AND ADDRESS
Colorado State University Department of Statistics Fort Collins, Colorado 80523 II.
.
CONTROLLING
12.
FFICE NAME AND ADDRESS
"
REPORT DATE
14 October 1982
Office of Naval Research Statistics and Probability Program 22217 Arlington, VA 14. MONITORING AGENCY NAME & ADDRESS(Il dlilerent from Controlling Office)
13.
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DD
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