•
Some General Remarks on the De.finition
of the Concept of Biased Estimators
Prepared Under Contract No. DA-36-034-oRD..1517 (RD)
(Experimental Designs for Industrial Research)
by
.H. R. van dar Vaart
,
t
Institute of Statistics
r.1:iJneo Series No. 178
July, 1957
SOME GENERAL REMARKS ON THE DEFINITION OF THE CONCEPT OF BIASED ESTIMATORS
by
H. R. van der Vaart
Institute of Statis~ios, University of North Carolina
Raleigh, North Ce,rolina, U. S. A.
. . and
Leiden
1.
-
Universi~y,
Netherlands
Suppoee a parame'ter !.l. is estimated by a statistic (a ra.ndom variable) !!!'
statement of the problem
both of them real.
To avoid non-essential complications
- will be supposed throughtil
out to be distributed oontinuously • As is well known, m is called a
( positively biased
~
unbiased
negatively biased
1
estimator of !.l. i f
t
These various cases may be verbally described as follows (cf. for instance
V!ln der
Wa~rden,
~-~.-..-..-
·
1
1957, p. 29):
a positively biased
an unbiased
a negatively biased
1
estimator of !.l.yields an estimated value
which is
·';th;
r
on the avere.gelarger than
average equal to
on the average less than
1
the true value.
However, while this may frequently be an adequate description of what an
experimenter or a statistician has in mind when he resorts to the terms biased or
unbiased, it will not always meet his requirements. For instance, one may want to
know not so much whether a certain estimator will be right
Whether, say, the !re91;enCY E! ,£btainiJ1g
.!:.22
.2!!.~ ..
a_ve..,r..,a,o.:;;g-.e,
as
small estimate! will be .!!.nduly large
(an example of this situation is afforded by the estimation of the latent roots of
a determinant in connection with the problem of estimating the type of response
-2-
In this connection the remark made by Snedecor
(1948, p. 8) with
•
regard to the term unbiased: "In sampling from certain s;ymmetric populations, it
surfaces).
may be said that estimates made from unbiased samples are as likely to be in excess
of the population value as in defect", is interesting as he apparently felt the need
of adding something to the usual "true on the average".
This paper will explore a few alternative approaohes to the general concept
of bias together with the interrelations between various definitions given.
For
oonvenience, sake, only negative.:!?!!! will be considered in most cases, the treatment
of positive bias being then self-evident.
2 . Expectation-bias and median-bias
D e fin i t ion
-._--------
negatively .2 .! .E ! .£ ~ !
-
E(m)
1:
The random variable m will be said to be a
------------
! .! .2 .!!-~ ! ! ! ! ,g
of ~ i f
.estimator --
<~
{2,I}
This is the usual definition of bias.
D e fin i t ion 2: The random variable m will be said to be a
-.. _.... -----...-
--~----
ne~atively
m e d i a nab i a sed
-------------
estimator of
....
if it satisfies one of the
----.....-.--~
,equiValent conditions:
Med(!!!)
<~; P(~ S 1J.) >~; PC!!! ~ fl.) > P<.m ~ ~)
Replaoing the three inequality signs
<,
<, > ,
> in (2,2) by
(2,2)
=, =, = or
by>,
<,
one obtains the definition of median-unbiased, or positively median-biased
estimators, respeotively.*
* After this
paper had been written, the author read the reoent book: IINonparametric
methods in statistiosl! (Wiley, 1957) by D. A. S. Fraser, which on p. 49 defines
the oonoept of median-unbiased estimators without, however, giving any further
developments.
-.3-
Because of' the fact that for many probability distributions the expected
value and the median are unequal, these two definitions are not equivalent in general:
an expectation-unbiased estimator may be median-biased
(!), a median-unbiased esti-
mator may be expectation-biased (b), or even a negatively expectation-biased esti'"
n
2
mator may be positively median-biased (c). For example, 6 ... (n - 1)-1.2. (Xi - 'X)2,
i=l
the well-known estimator of cl from samples of size n from normal distributions
N(IJ., J), belongs to class (a), the median of a sample of size 2m + 1 from certain
-
-
skew continuous distributi. ons belongs to class (b) when used as an estimator of
the median of the corresponding distribution, and r, the well-known product-moment
of the underlying (supposedly bivariate normal) distribution
correlation coefficient, belongs to class (c) when used as an estimator of the correlation coefficient P
in case
P>
0 (see Appendix).
The last example is of particular interest as it
shows that the contention made by Tschuprow (19.39" p. 116) to the effect that the
estimator r systematically underrates P, is dubious in that this may be: taken to
-
.
mean that !
~
I
freemently
~ ~
.3. Bias co.ncepts based on comparing
underrates p -- which is not t:rne .
estimato~
Looking attentively at definitions 1 and 2 one sees they have one thing
in common:
all that is involved in using them is the distribution of the estimator
itself and the true value of the parameter.
This may seem natural enough, yet
remember the motivation of the present search for alternative definitions of biast
one may be interested in the question whether the frequency of obtaining tot) small
estimates will be unduly large and want to call an estimator with this property
negatively biased.
Now both the terms too small and unduly large imply some type of comparison
-
to be made. 'When would one call an estimate too smalU
A reasonable answer to this
-4-
tit
questior. seems to be:
if the estimate is smaller than a certain value, to be
denoted as comparing value.
The choice of this comparing value is, generally
speaking, arbitrary. Furthermore, when would one call a frequency (of obtaining
too small estimates) unduly large? A reasonable answer to this second question
seems to be:
if it is larger than it would have been if a different (presumably
lIbetterll) method of estimation would have been used; that is, if it is larger than
with a different estimator, to be denoted as comparing estimator.
Again the
choice of this comparing estimator is to a large extent arbitrary.
It is evident that a definition of bias ensuing from this approach
generally involves more than just the true value I..t. and the distribution of the estimator itself.
The next three sections will give three examples of bias concepts
based on this "comparing approach".
Take for the .comparing value the true value I..t. of the parameter.
-
Take for
the comparing estimator any estimator mc with Med(mc ) = I..t.. An estimator.!!! would
then be called negatively biased in the sense of section 3 i/f' P(m
~
jJ.)
> P(mc
~ I..t.)=
Hence this definition is identical with the definition of median-bias; cf. the
second inequality in (2,2), definition 2.
If the distribution of !!! is symmetric,
then of course median-bias and expectation-bias are equivalent and it is here that
Snedecor l s remark (see section 1 of this paper) seems to fit in the scheme.
The following argument is well in place here. One might want to look at
the idea of an unduly large frequency of obtaining too small estimates in another
way which might appear not to involve comparing estimators:
one might endeavour to
define the frequency of obtaining tOO6Illall estimates (Le., estimates smaller than
the comparing value) as being unduly large if the PEobability of obtaining estimates
-5-
e
smaller than the comparing value would be lar,Se as compared with the probability of
obtaining estimates larger than the comparing value. One might feel that with I.l.
for a comparing value this approach wouJd. lead automatically to the definition
( of. the third inequality in (2, 2) , definition 2), but it does not.
In fact, why
not use the definition
-
(3.1,lb)
say, with k r) l?
The simple fact that one is apt indeed to prefer the inequality
-
(3.l,la) to (J.l,lb) shows that in applying this approach one would be using
-
implioitly a symmetrically distributed estimator (or at least an estimator with
equal probabilities of values being smaller and being larger than the true value) for
e
a oomparing estimator.
An analogous remark holds true if' any other oomparing value
(than I.l.) would be used; always one would have to decide about questions like: how
-
large is k in (3.l,lb) to be chosen?
So this approach is essentially identical with
the approaoh based on oomparing estimators.
3.2
-~
Distribution-bias
----------------Take for comparing values all conceivable values
g
of the parameter ~
(the set of values ~ may be an interval, both finite and infinite).
As for the choice of the comparing estimator mo' there are often two or
more estimators with different, yet (almost) equally desirable properties. In such
-
oases there may be no point in selecting just one of these estimators for oomparison
so .that one may want to use all of them (in most practical cases the number of
estimators competing for the r~le of comparing estimator will not be too large).
The concept of bias will then have to be modified into bias
(a particular comparing estimator) m •
c
~
respect
!.2
-6-
e
To come back at the comparing values }, , note that if the set I of values
§
coincides with the union of the set of
5
with 0
and the set of ~ with 0
< P(m < ~) < 1
<P(mc < ~} < 1,
then the condition
P{m
s. $ ) > P{mc ~ ;; ) for
~e I
means exactly that!!! is stochastically smaller than mc {this term was introduced by
Mann and Whitney, 1947, p. 50}. However, in the definition to follow, I will be
--
allowed not to contain all these ~ -values.
The preceding considerations lead to
Definition
--.. -_ ...... _- 1.
Let I be the set of all values which the parameter
...,
l.l. might conceivably have (dependinfi; on the specific practical problem).
estimator m of
j..L
will be called negatively
res p e c t
..wit h ------~--
P(!!
~
5)
!!!! ~ ! ! .2 ~ ~ 1: !!. .!!-,E ! ! .! !
-
~
_---
to a
~ '(mo $ ~)
Then the
mat 0 -r
....est i ...........
tor
t e I,
the equality sign not holding true for at least one { e I.
Note that whereas the definition of positive distribution-bias is selfevident after definition 3, the definition of distribution-unbiasedness with respect
to mo presents difficulties.
If the
~
sign in (J .2,1) would be simply replaced by
an equality sign, then it would be easy to find estimators which are neitmr distribution-unbiased nor distribution-biased. Alternative definitions of distributionunbiasedness will easily appear to be unsatisfactory in other respects, and will not
be attempted here.
1:~~
_... -!?!!~
In certain problems one may be particularly interested in one special
comparing value, not the true value of the parameter
-
j..L
{for example,in response
surface theory the value zero plays an important rale in oonnection with the
-7-
estimation of the canonical regression coefficients).
value
t c'
so that
section 3.2.
$c.
is a fixed and known constant as
Call this special comparing
g
opposed to
occurring in
Handling the question of comparing estimators in the same way as in
section 3.2 one arrives at
est i
......
mat
0
r
_---~--
The definition of ~c-unbl~sednass presents no diffioulties.
Note that est.:mators which a!'e both expecta:'ion-unbiased and medianunbiased may be
5"c·biased
with I'espect to some estimator me'
-
In fact, due to the
asymmetry of $c with respect to lJ. ( Sc cannot be both larger and smaller than lJ.),
even an estimator m that is distributed s J1nmetrically around lJ. may be t. -biased
c
with respect to another estimator me that is also distributed symmetrically around
For example, let m be di3tribu~d N(lJ., 2(2 ) and m be distributed N(lJ., (i),
c
..and take ge = lJ. .... 20', then!!! is negati",ely gc-biased with respect to me' Hence,
I-L.
from the standpoint of 5c-bias an estimator which is both expectation-unbiased and
median-unbiased
~
E! !E£P~
than even an estimator which is negatively expectation-
biased and negatively medien-biased, but which has a smaller variance.
concepts of bias and of efficiency merge into each other.
Here the
Another possible reason
(also a mixture of bias and efficiency considerations) for preferring certain
biased estimators to certain unbiased estimators was brought forward by Olekiewicz,
. .-..-:--IJtl.at"4'. . . . . .
1950.
At this point the author remembers with pleasure a discussion with D. Hurst,
Department of Experimental Statistics, N.C. State College, Raleigh, who emphasized
that there is little practical value in proving that a certain estimator possesses
-
just one out of the whole long list of properties which estimators may be desired to
have, and that the main interest ought to be in estimators wi th several useful properties.
-8-
e
l!'
A lemma which is useful in provins
oert,ai~ tY!'.~~ 5~i.!!
An almost trivial lemma whioh, however, sometimes
provides an easy proof
of an estimator being (negatively) median-biased, or distributir.m-biased, or
f 0-
biased is
-
Lemma 1.
-~---
-
Whether the random variables m and z are independent or dependent, i f
pc!
~ ~
-
) = 1,
-thenPC!!! S g)
(c)
<: ),
~ P(mc ~ ~ +
-
where me = E + .!. A nec;essary and suffioient oondition for the equality sign
to hold in (4,1) is
P[ (m + .!
>g
+
S ) ()
(E ~
g )]
:I.
0
-Proof
---Drawing a picture will bring home the triviality of t1'e proof.
= (because
of oondition (0) in the lemma) =
:I
PC!!! +.! S
~
pemc ~ 5.
s+ C; ] + P[(.m + .! > g
+ ;
+ ( )" (!!! ~ ~) J ~
] ,
whioh oompletes the proof both of (4,1)
and (4,2).
It should be remarked that the equality (4,2) will represent a rather peculiar
-
property of the joint distribution of m and z, yet it is olear that (4,2)
-
cannot
be proved or disproved from the oonditions of the lemma alone. Three extra. oonditions
-
each of them sufficient for (4,2) not to hole, are
..<) !!! and .! are independently distributed and P(.!
(3)
eaoh (measurable) set in the half-plane.!
probability,
:I
(
.> ~ in
)
< 1,
(~.t~)-spaoe has positive
-9-
y) some set
[(o!
>Co)
('l
(!!!
>.f0)]
with ¢o +
to
>?: + ~
has positive
probability (implied by, hence weaker than ~).
Rem ark.
---_
......
Lemma 1 is useful in connection with certain cases
of negative bias.
The analogue for positive bias is:
t e m mal'
---_
.. - •
-
-
Wnether the random variables m and z are independent or dependent, if
:
P<,~ ~
,) = 1,
(ct)
-then
where me
lIS
1! + !. A necessary and sufficient condition for the
(4,1:"> !!
equality sign to hold in
P [(m +!
< t + <;
> (l (m ~
5 )] = 0
-
When applying Lemma 1 or l'
in order to prove that m is a biased estimator
.,
of IJ. in one of the senses discussed in this paper, take ~
lIS
0 and
for median-bias· take ~ ... IJ. and try to choose the random variable! in such a way
that m
c
=m +
z has
-
IJ.
for its
median~
for distribution-bias with respect to (m
-
+ !) take
t arbitrarily from the set I
of
conceivable values of IJ. and see if .! can be chosen in such a way that (!!! + !)
is' an interesting comparing estimator,
for ~c-bias with respect to (m + .!) take
~
=
f"c and see again if ! can be chosen
in such a way that (m + !) is an interesting comparing .estimator.
- -
In all three cases m and z may be mutually independent or dependent.
APPENDIX
41'
At the end of section 2 certain statements were made concerning expectationand/or median-bias of certain estimators.
In this appendix these statements will
be proved and illustrated.
A.l. The estimator s2 of the variance
a2
Be Xi,x2, .••. ,xn an n-fold sample from a normal distribution N(~,02). It is
well known that
1
n-l
is an expectation-unbiased estimator of 0 2 . It will be shown that ~ is, however,
negatively median-biased, i.e., that
(A .1,1)
n-l
j
1
e
-
n-l
y
~
y
"2-
1
dy
=
o
~(n-l)
=
=1
Y
1
- r{E:;})
_ (
,
0
Q n-l ; n-l
where the functions
n-l _ 1
t
e- t 2
dt
)
= y(n-l
"2 '
n-l,) /
"2'1
r (n-l)
"2
)
_~ r
= rr(f>J-
l
• 2
•
J2
X
cf.
Pearson and Hartley (1954), p. 122;
x
'Y(~x) = /
e-t t <><
o
=
I
(' I n-l' n-l
If 2 ' '""2' - 1 ) ,
and I are defined by
Q, y
CD
2
Q(x 1 '.j
=
-
1 dt,
1. ~ _ 1
e 2 y 2
dy,
_
A-2
e
cf.
Higher Transcendental Functions (Vol. 2," 1~?3}, p. 133;.
= I(
I(u,p)
cf.
-
= y(p +
, p)
1, m/r(p + 1) ,
(1946) or Jordan (1950), p. 56.
K. Pearson
Jordan
,_ m
Vp+l
(1950), p. 57 mentions that "it can be shown by the tables of the
• p)
function I(u,p) that ••• I( 1/
y P + 1,
analytically.
>
1
~
".
However, this can be proved
By means of the asymptotic expression for y( -<
given by Tricomi
<It
1, -< +
VU
y)
(1950) one finds that
y( ..<, 0<)/ r( 0<) = ~ +
3
vk-
+ o( 0<-1),
(A .1,2)
2TTo<
a formula which for real 0< can easily be proved by Laplace t s method.
This formula
implies that for -< large enough y(..<, o<)/r( 0<) exceeds ~ and that
y(..<, -4/r( 0<)
~ ~ if 0< ~
CD
Therefore, the inequality (A .1, 1) will evidently be proved (even for small values of'
n) if it can be shown that
y("<'O<)/ r( -<)
> y( -< +
1, 0< + l)/r(.,( + 1) for any .,(
> o.
Now this inequality is equivalent to
0< y( ..<, .,(»
or
y{ 0< + 1, .,( + 1)
0<
0< •
J e-t to< -
co<
Idt
o
+ 1
>/
e-t to< dt
0
.,(
or
or
[e-t t O<J ~
+
J
-< + 1
e-t to< dt
o
>
J
e-tt 0< dt
0
0< + 1
e--<o<O<
>/
e-t to< dt.
0<
As the maximum of the integrand e-t to< is reached for t
is clearly correct, whereby the proof is complete.
= ..<,
the last inequality
A-3
Rem ark.
------
Line 6 from the bottom of p. 141 of the above-cited
Volume 2 of Higher Transcendental !Unctions contains an error: it
x)lr(~)
,
is stated. that r(..<,
,
~ for ~
function
> C"
;It
of~.
> 0,
is a monotonic decreasing function of
wharaa" in reality y("", oz1/r(..<) is a decreasing
(Note that r(..<, x) + y( ..<,x) ... r(.-<) '; cf. Jordan
(1950), p. 57, equation (3) ).
The following table is computed from Pearson and Hartl,& (1954), p. 122
-
seqq., at some places checked by ....
K_
......P....ear_,.·!2!l!! table of the incomplete gamma function.
It is interesting to note that the asymptotic expression (A.l,2) yields results
which are accurate to 3 significant decimal places for n - 1
n ... 1
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
98
p(s2 ~ rl)
-
...
0,683
0,632
0,608
0,594 .
0,584
0,577
0,571
0,567
0,563
0,560
0,542
0,534
0,530
0,527
0,524
0,522
0,519
~
4.
A-4
e
~
The sample median as an estimator of the median of the distribution.
Be F(x) a continuous distribution function with dF(x)/dx different from zero
in
2E2. (finite or infinite) x-interval.. and !J." 3:.2' ••• ,
~2m+1 a sample of odd size
from the corresponding dis tribution • Rearrange the :values in the sample" and use
the notation x(l)
C#I
< x(2) =< ••• =_
< x(2m+1).
__
As F(x) is assumed to be a continuous
distribution function, the occurrence of equality signs has probability zero;
~(m+1)
..
is the sample median.
It is well knoTtJD (of. for instance Wilks .. 1948" equation (16)
-~
on p. 16) that
~
(A.2,1)
a < F < 1)
Denote by G the inverse (deftned. fo::.'
median of the distribution conside=:-ed.
(G(~'l)
(A .2,1) and use the equality F
of the function F so that G(~) is the
Substitute F(y) by F in the integrand of
III
~"
then
(A .2,,2)
1/2
Because of /
a
Fm(l_F)m dF
.1
1Il./
(as appears from substituting F
/.
1/2
o
Ifl(l_H)m dH
1/2
m(l-F)
m
dF
1
III
2"
J1
a
III
1-H), one obtains
m(l-F)
m
dF:&
The equations (A.2.. 2) a.l1d (A.2,3) prove that
1
'2
B(m+l,m+l) •
~(m+l)
is a
d..j;o.;;;a.n_-u...n.b::.;i;;;;;a..
se..d;;...;e_s...
t;;;im;;,;a;;.;t;;;o.;"r
,;;;,!flEl
..
--
of the median ..
G(1/2)
...... whether F(x) represents a skew distribution or a symmetric one •
'
f !
.,(m+l) equ al s
However, the expec t a t J.on
o
./+
r(2m+2)
.. 00
1
10I
B(m+l,m+l)
00
Y [F(y)]m [l-F(y)
r
dF(y) =
A-5
=
NOW;l
o
1
1
G(~) + r3(m+l~m+i'
[G(F) - G(~)
/1
0
1
[G(F) - G(~) )
(A,2,,4)
• pl'l(l-F)m <iF
J 2)m
• ('4..
h
J
dh
=
Hence the second term in the third member of (A.2,,4) equals zero for distributions
which are symmetric with respect to the median, since for these distributions
G(}h) -
G(~) is an odd function of h. Further this same second term in (A.2,,4)
J
is {Positt~Ve} for those distributions for which
nega J.ve
1 ) - G(~
1)
G(~h
e
>
< G(~1)
1
- G(~
..h)
(0
< h <~)
(A.2,,6)
-
Hence for such skew distributions as satisfy (A.2,6) the statistic x(m+l) is an
expectation,:biased estimator of the median G~L (though not only for this type
of skew distribution, of course.)
Rem ark.
------
~(!(m)
If' the sample size is even, 2m say" the statistic
+ !(m+l)} is customarily used as an estimator of tke sample
median. For certain skew distributions this is a median-biased
estimator of the median. If ¢ (x(l), x(2)" "" x(2m») is to be a
median-unbiased estimator of the median, then the function ¢ will
not be independent of the distribution function F. So for even
sample sizes there is no exact analogue to the sample median in samples
of odd size.
A-6
e
A.).
The estimator r of the correlation coefficient p
Be (~'Yl) JI (x2 'Y2) ""
(xn,Yn ) an n-fold sample fI'oma bivariate nonna1
distribution with correlation coefficient p. Define the sample correlation coefficien t in the usual way by
~l
{ n
r
/)
n
n
~ l~ (~-'X}(Yi-'iJJ / ~ ~ (xi -i)2 ~
~_~
(Yj_1>2 }
.
It is well known (cf. ~~n~~.J 1947, p. 344, eq. (14.55) JI and !lgm~g}g!~~, 1925,
p. 42, eq. (128) ) that
1 1 n+l
2
• F(!,~; 2; p ) •
(A .3,1)
The last equality follows from
~.~tJ~
integral representation for the hypergeometric
function (used in this context as early as 1925 by •••=====ii'e
Romanows kv , __
Le.) I
F ( a., b;c;z )
I:
r(c~ c-b)
r(b)
r
t
t b-l ( l-t)e-b-1 ( 1-tz)-8 dt,
a
see for instance Higher Transcendental Functions (vol. 1, 1953), p. 59, eq. (10), and
p. 114, eq. (1).
From
1
1
< (l-t)"-t for
2 -t
(l-tp )
2
any p ,,1, t r)
0
:lilt follows that
j
-1'
!..a..
n.,J.
(l-t) ~
t2
o
.1
2
(1_tg ) '[ dt
<j
~..a.
1
t
2
n-l. 1
(l-t)"'"
dt
0
1 n-l
r(~) r(n~l)
2
= B(~, -2) = - - - - - - for any p rj. 1,
~
r(~)
=
A-7
hence by the fourth and second members of (A.)>>l) that
g(n,
i)
<1
for any p2
lherefore r is a ne$apivelY
rJ
1.
expectation-bl~seq~stimator
of p if
e is
positive,
as was stated at the end of section 2.
Rem ark.
------
-
The well-known asymptotic series for E{r} may be derived
readily from (A.3,1).
-
In order to investigate whether r is a median-biased estimator of p, we use
fonnula (25) of Hotelling (19531, p. 200 (note that Hotellin~f~. n denotes the number
which is one less than the sample size), which entails
(1_r2)
n-2
=-
V2
n-4
"2
(A.),2)
t
11
(l:.t1T) n...
By means of the substitution
=
y,
suggested by section 5 of ~~~~1~~~~ (1953), the second member of equality (A.3,2)
reduces after some patient algebra to
u:,g,
'V2ff
r(n-l)
r(n{)
J 1 (1- 2) 2n,.4 (l+py) 2'1
o;YO
F J1
2
I, n. 1 • 1... I-I?..-' dy
\~,~, I'll'
As p increases from 0 to 1, the function (l+py)2 increases for any y
2
the function
hence 1 ...
* The
i;~y
-
(A.3,2a)
m"'-PYtl
decreases for any y
> 0,
> 0,
1 2
rit+
PiT increases for any y > 0,
author wants to thank Dr. R. J. Hader" Institute of Statistics, Raleigh, for
drawing his attention to this paper.
2
hence
F(~, ~; n-}; 1 .. 21i;~») increases for any y > 0:;
hence the integrand of (A.3,2!> increases with p for any
> O.
y
As
P<,~. ~ p)
PC::. ~ p)
= ~ if
P
=0
(the direct verification of this elementary fact from
(A.3,2a) is straightforward, though rather intricate) J this means that
> ~ if
p
> o.
Therefore!..!!!. positivel;r median-biased estimatg;:,
2!. .e !£
E 1! positive, which completes the proof of all statements made at the end of
section 2.
LITERATURE CITED
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~ + 302 pp., Vol. 2" 1953, xvii + 396 pp; McGraw Hill.
H0' New light on the correlation coefficient and its transforms;
Journal of the Royal Statistical Society, Sere B (Methodological)15, 1953,
193-232.
~g!~
-
Jordan, Ch. Calculus of finite differences; 2nd ed., New York, Chelsea, 1950,
xxi + 652 pp.
Kendall, M. G., The advanced theory of statistics, Vol. 1; London, Griffin, 1947,
xii + 1.67 pp.
Mann, H. B. and Nhitney, D. B." On a test of whether one of two random variables is
stochastically larger than the other; Ann. Math. Stat. 18, 1947, 50-60.
-
Olekiewicz, M., On the efficiency of biased estimates; Ann. Univ. Mariae CurieSktodowska" Sect. A., 3, 1950, 103-140.
-
e
Pearson, E. S. and Hartley, H 0., Biometrika Tables for Statisticians, Vol. 1,
Cambro Univ. Pr., 1954, xiv + 238 pp.
0
Pearson, K., Tables of the incomplete f-fwactions; Cambr. Univ. Pr. Re-issue 1946,
xxxi + 164 pp.
Romanowsky, V., On the moments of standard deviations and of correlation coefficient
in samples from normal population: Metron, Vol. 5, N. 4; December 31, 1925;
31-46.
Snedecor,G. W. Statistical methods, applied to experiments in agriculture and
biology; 4th edt (3rd print). Iowa State College Press, 1948, xvi + 485 pp.
Tricomi, F. G. Asymptotische Eigenschaften der unvollstandigen Gammafunktion;
Mathematische Zeitschrift 21, 1950, 136-148; cf. also Math. Rev. 11, 1952, 553.
Tschuprow, A. A., Principles of the mathematical theory of correlation (translated
byM. Kantorowitsch); London, W. Hodge and Co., 1939, x + 194 pp.
~en,
B. L. van der, Mathematische Statistik (Grundleilren der Mathematischen
lrJissenschaften Bd. ~) Berlin, Springer, 1957, ix + 360 pp.
Wilks, S. S., Order statistics; Bulletin of the American Mathematical Society Zll,
1948, 6-50.