THE EXISTENCE OF CONSISTENT ESTIMATE IN A
REGRESSION MODEL( II )*
MingZhong Jin
The Dept. of Scientific Researthes
Guizhou College for Nationalities
Huaxi, Guiyang City, 550025 CHINA
Let
Y, = x:,8 + ej ,l L i I n ,be a linear- regression model, Where
X, ,
x2
l
l
l
are known p-vectors, p is
the tmknown vector of regression coefficients and e, , e2 6+. are the random errors. Under what conditions there
exists a consistent(in this note, “consistency” means “weak consistency”)estimate
if e1,e2 qq. are iid. Ee, = 0,O < var(e, ) < 00,e, possesses
of fl ? K.C Lil’] proved that
a density f which is positive and absolute
continues on R’ , and that the Fisher information
is finite, then the necessary and
’
suficient
condition for the existence of a consistent estimate of fl is kr=, ‘ix: f + 0. This condition,
according to Drygaslzl, and T.L. Laif31,is
. exactly the same for the consistency of the Least Squares estimate. It is
easy to understand that the condition for {xi} (to guarantee the existence of a consistent estimate of fl) is
closely related to the assumptions imposed on {ej >. It is well-known in the theory of parameter estimation that
under the same sample ske, the accurancy which can be attained in the estimation of parameter of trancated
distribution family (such as the farnily of rectangular distributions) is higher than that in the regular distribution
family. So we naturally think that if the distribution of ei is of a trancated type, the condition (on {x~}) for the
existence of consistent estimate of fl should be weaker than specified in Li’s result. Chent41,in the special case
Y, = a + & + ei ,
(1)
lli<n
showed that this is indeed the case. He showed that if e, , e2 *. . are iid. Ee, = 0, e, possesses a density f
which vanishes outside the interval
[D,,a2],
[ol, 0~1, and on
f is positive
and satisfies the Llpschitz
condition, then the necessary and sufficient condition for the existence of a consistent estimate of p is
H,=
-
where
’
i=l
zl
xi
-x,
-
-+m,
us
+
00,
I
xPt = cX, + - . + 4, jl n , and that for a is X, /H,
l
Li, for, according to Li’s result, if e, ,e2 +-
l
+ 0 This condition is indeed weaker than that of
satisfies Lik assumption, than the condition for p should be
+ 00, as n -+ 00, which is stronger than H, + 00 .
C:=,(‘i -xnr
The model (1) includes
Yi = & + ei,
(2)
lliln
as a special case, but Chen’s result does not apply to this case. Of course, in model
(1)) under the same
fe, > as in Chen‘s result, H, --+0 is still the suficient condition for the existence of a
consistent estimate of p , but H,, -+ 0 is not necessary. Indeed, take X, = x2 = - +*= 1, then H, = 0, but
assumptions on
according to the law of large numbers,
k
is a consistent estimate of /?
In this note we give the necessary and sufficient condition for the existence of a consistent estimate of p
inmodel(2).
W e al so improve Chen’s result somewhat by weakening the assumptions imposed on {e, }
Theorem
density fwhich
Suppose that in mode (2) the random errors
vanishes outside the interval
e1 , e2 ;*-,
are iid.
Ee, = 0, e, possesses a
[ni, 0~1, and [oI, 02] the following conditions are satisfied: 1.
* Word Supported by the National and Guizhou province Natural Sciences Foundation of China.
There exist constants
c, , c2 ,O < c1 I c2 < 00 , such that c, < f(x)
exists function g defmed on
[a,,OJ
And that for small r > 0 and x E
Then the necessary
< c2 for x E
[a,,oz ] .
2.
There
such that
[oi,02]we have
and sufficient
condition
for the existence
of a consistent
estimate
of fl
is
REFERENCES
113Li
K.C.(1984),Regression
models with infmitely
Ikctional Ann. Statist 604-6 11.
strong
consistency
many parameters:
of
LSE
in
Consistency
regression
of bounded
model
linear
PI
Drygas H.(1976),Weak
and
Verw.Gebiete,34: 119- 127.
PI
Lai T.L..Robbins H and Wei C.Z. (1979)Strong
Ana1.9:343-362
PI
Chen Xiru.( 1992). On a problem of existence of consistent estimate in the development of statistics recent
contributions from Chma(237-248), Pitman Research Notes in Mathematital Series 258, logman Scientific
and Technical, London,( 1992).
Z.
Wahrsch.
Consistency of LSE in Multiple regression II.J.Multivariate