11. SERIES EXPANS1ONS I). l.. [amircz.rmd 1). 11.Jensen I)cptrtmcnt of 131hem2tics IIcpartm

’12distributions, extending the bounds of Kotz
ei {IL( 1967). Some nutncrical studies arc rcporlcd in Scclion 4. 97
Copyrighl (3
1{~’~1 iltarcrl l)c~~cr. Inc. by
In Part 1 of this arc idcnt ificd vectors
;md Rmnircz ( 199 I )), t’,vo types of model misspeciftcation
\lisspcciIicd
Iocation
rnodrls
arise
~vhcn
obscmation
ptlft.fiorlf, lI)CCII CZrtd~lt)[ItIl (’t-rUr [J(JtirlCi.f
series
O. )\llS’1’R.\C”l I’rt)pcrtics of nonstandard
l!t~tclling’s
(1931) 72 distributiol]s
are stucliccl under errors 3rc gi\en
models having mis<pccificd scale. Local and global bounds un truncation nonsl:tnd:trd ‘ITJislrihulions. 1. 1>”1’Rol)[”(;”I’I()\ 1lotclliny’s ( [q.! [) study (cf. Jensen and illvcs[ig:ltcd.
MI SSPECIFIED
T’
TESTS.
11
99
l–G(cz/y*;
p,v)
s
a’
<
I–G(CJY};
P!V)
(2.3)
Ivhcrc Y* is tl)c geometric mean of {)1, ... ,yP}. 1/1; S1;1.”1’ [f cmis the critical value for a 71tcst at nominal ICVCI and actua! level a+, 6. a then the potvcr is at Icast a+ at each alternative O # O ill Rf’,and thus the test is unbiased. 3. ERROR lWIJXWi Result 1 of Section 2 Icads us to consider tvcighted scnes of f-distributions. u/. ( 1967), continuing indcpcndcnt wtighted chi-squarccl distributions. Kotz et of
COMNUN. STATIST.
-SINUL.4.,
20(1),
97-108
(1991)
NIIsSPECIFIKDP
11. SERIES
TESTS.
EXPANS1ONS
I). l;.. [<amircz.rmd 1). 11.Jensen
I)cp:trtmcnt
of \131hem2tics [“llir~rsi[! of\-irgillifi
98
RAMIREZ
AND JENSEN
2. PRELIfilINAIUl~S
2.1 Notation. WC continue the usage from Part I of this study, Jvhcrc c(f~aml p<//_ rcfcr to cumulative distribution and probability (p, Z); the chi-squared distribution J; the central JVishart distribution density functions. Special distributions include the Gaussi,an law NP(P,X) on the Euclidean space 0+’ having location-scale pmxrnctcrs
Y/\ 22W3
tmdc[sp
(:h:lrk)ttcsYllk!,
IIcpartmcnt of Statistics Virginia I’c)l>tecllnic lnstitutc lll:lCks[~Urg, J-i\ 2406 1
t7fodci tt1i.rsb~ec[ficulion,
arrtl(lti.r,ห้องสมุดไป่ตู้lie! l~”ords. ,\/tilllr,itidlf Ioralion-scd[e
SUCII
‘1IIc present ppcr
’12 distributions
tllcmsclvcs.
nlisspcc”ificd scale,
distributions
can bc cxpal~dcd as ~rcightccl series of of tl)c paper follows.
1“-distributions, cxtcmiing the tvork t)f Kotz cl (1/. ( 1967). /\n outline
-Y((YIZF+ -“ + yPZ~)/P’)such that the following conditions hold: (i) {21, ... . Z,, ~ arc mutually independent; (ii) S(2,) = A’i(6j, 1), 1< i <p; (iii) Y(V) = )$(V–p+ 1).
o 111.UIIIIICSccl~r in the uw of I lutrlling’s ( I’W7) Iz charts in statistical
and
in prrw~nncl selection rules b~scd on “12. I’:u1 I of this study is comxrncd ordcrinu
(?)
[Yl, ... , yp]’ and o = [01, ... . 0,]’ such that O = Z- 1/2p. Let gp({; V, y, 0=0.
and Gp(f; v, y, O) bc
the corresponding pd’’and cd~of 72, with gP(t;v, y) and GP(I;v, y) for the ccntm.1 case ~vhcn The principal results for 72 under rnisspecificd scale arc the following. RESULT 1. The statistic 72 admits a stochastic representation in which ~(72/v) =
2.3 Summary from Part I. The following results cm-y over from Part 1 of this study. Parameters of the distribution &’(7z) under misspccificd scale arc (v, y, 0), ~vith y =
... , YP, d) k
and
RESULT 2. The cdf GP((;V, yl,
the remaining y’s held fixed.
a decreasing function of each y, for 1, V,
0, and
RESULT 3. The stochastic bounds G(I/yl; p, v, ~) < GP((; v, y], ... . yP, ~) S G(t/YP; Pl ‘, ~.) with 1 = 0’0. (2.1)
Section 2 summarizes the principal findings from Part I as they pertain to dcvcloprncn[s here. Section 3 clcvclops local and global bounds on truncation of J--distributions arising from nonstandard crmrs for Ivcightcd series
‘[’his by
formally describes our model under misspecification
Z #Q.
tlmotc
{Y,>... ~ y,> 0) the ordered roots of the deterntirrantal eqwrtion
]Z – yQ I = O.
X2(V, ) having v dcgrccs of freedom and nonccntrality 2 M;(v, Z) having
v
degrees of freedom and srxdc pararficof order
ters Z; and the pa’’g((; p, v, J) and cdf C([; p, v, 2) of the standard ‘F distribution p and argument t, having v degrees of freedom and nonccntrality of distribution of Z. 2.2 illisspccilid
{X 1,..., X.) lMVChctcrogcncous
rnc:ms. On the otlwr hand, if 72 may bc rcprcscntccf as of these distributions ffiil to coincide,
process control.
7-2 = r }“‘ \\- 1Y in terms of’ a G;lussi;m vector Y :wl a \Vishart matrix \\”, then n~isspccificd sc:ilc modcts :Uisc \vllcn tl]c sc:~lcpwunctcrs
Scale.
J. @(Z)
signifies the law
Consider
the representation
P
=
v
Y‘ IV’1 Y in Ivhich =
II)(v, Q).
(Y, W) are independent
such that -$?(Y) = AL(P, X) and Y’(JJ’) of scale when
77 has
(l~J67’ include to for a \\ciglllc[l series of IT-ciistrihuli(ins. cxtcrldil]g the wwrk of Kot7. (’( (7/.
\vidc
usay:
subject to mt-dcl validation.
proprrtics
Ivith stoch:ls[ic
bounds
of
nonstandard
/2 distril>utions,
iu]d \vitll constructing
for thcw
dislrihutiolls ~ncicr
ill terms of stand:lnl dis[ribuliolls. is conccrnml with tlw wnstandard
hold in terms of scaled versions of the standard P distribution,
RESULT 4. If P has the cdf GP(t;v, Y1,... . yP,0), and if y* = (ylyz ... yP)l/~is the geometric mean, then G(I/yl; p, v, 1) S for each t e R\, with J = 0’0. RESULT 5. If c. is the critical value for a P test at the nominal lCVC1 and actual level a a*, then bounds on a* are given by GP(t; v, y], ... . yp, O) s G(~/Y*;P, v, ~) (2.2)