•
Some General Remarks on the Definition
of the Concept of Biased Estimators
Prepared Under Contraot No. DA-36-034-0RD-1517 (RD)
(Experimental Designs for Industrial Researoh)
/
by
,H. R. van der Vaart
~··:f
.
';!~~tr.
n','-'-
.
' .
.'I'
.,'f-.':
'Ipstitute of Statistios
'T-fimeo Series No. 178
July, 1957
SOME GENERAL REMARKS ON THE DEFINITION OF THE CONCEPI' OF BIASED ESTIMATORS
by
ce
H. R. van der Vaart
Institute of Statistics, University of North Carolina
Raleigh, North Carolina, U. S. A.
. and
Leiden University, Netherlands
1.
-
Statement of the problem
Suppose a parameter Il is estimated by a statistic (a random variable) .!!!,
both of them real.
-
To avoid non-essential complications m will be supposed through-
out to be distributed continuously. As is well known, m is called a
{ positively biased
~
unbiased
negatively biased
1
estimator of Il i f
t
These various cases may be verbally described as follows (cf. for instance
,
I
van der Waerden, 1957, p. 29):
--~=
--
a positively biaSed}
an unbiased
a negatively biased
estimator of Il yields an estimated value
which is
'
f
on the average larger than}
on the average equal to
the true value.
on the average less than
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 2.U
whether, say, the ,£regaency
E!.. .Q.~aining !22
2 ..a...ve,;;;r...a;.:; :g..e"
as
small estimate! will be unduly 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-
surfaces).
In this c,)nnection the remark made by Snedecor (19h8, p. 8) with
•
regard to the term unbiased: "In sampling from certain symmetric populations, it
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 ll •
This paper will explore a few alternative approaches to the general concept
of bias together with the interrelations between various definitions given.
conveniences sake,. only negative
~
For
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 1: The random
-----------nesatively .2 ~ .E ! .£ ! ! ! 1 .2 .!!-,'2 ! ! ! ! ~
-
E(m)
variable m will be said to be a
----------
estimator
2! .!: .!!
< IJ.
(2,1)
This is the usual definition of bias.
D e fin i t ion
---_
..... _--negatively !!:! l!
2:
The random variable m will be said to be a
--------
! ! .n-.2 ! ! ! !.!!
estimator
2! J:: l:!.ll
satisfies 2.n2 2!
~
eqpivalent conditions:
(2,2)
Replacing the three inequality signs
<,
<, >, > in
(2,2) by
=, =, = or
by>,
<,
one obtains the definition of median-unbiased, or positively median-biased
estimators, respectively.*
* After this paper had been written, the author read the recent book: /INonparametric
methods in statistics" (Wiley, 1957) by D. A. S. Fraser, which on p. 49 defines
the concept of median-unbiased estimators without, however, giving any further
developments.
-.3-
Because of the fact that for many probability distributions the expeoted
value and the median are unequal, these two definitions are not equ1valent in general:
-
an expectation-unbiased estimator may be median-biased (a)" a median-unbiased estimator may be expectation-biased (b)" or even a negatively expectation-biased estin
mator may be positively median-biased (0). For example, 82 • (n - 1)-1.
(xi _ 'i)2,
'2
~l
-
the well-known estimator or (} from samples of size n from normal distributions
N(I-L,02), belongs to class (a), the median of a sample of size 2m + 1 from certain
-
skew continuous distributi. ons belongs to class (,2) when used as an estimator of
the median of the corresponding distribution, and r" the well-known product-moment
-
correlation coefficient, belongs to class (c) when used as an estimator of the correlation coefficient p of the underlying (supposedly bivariate normal) distribution
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 ! systematically underrates P, is dubious in that this may be: taken to
mean that! e ! frero;ently
~
.D2.'!:. underrates
P .- which is not true.
3. Bias concepts based on comparing estimators
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 frequenoy of obtaining
tOG
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 small? A reasonable answer to this
-4question 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 i f a different (presumably
IIbetter ll ) method of estimation would have been used; that is, i f 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 \.l. and the distribution of the estimator itself.
The next three sections will give three examples of bias concepts
based on this Ilcomparing approach".
Take for the .comparing value the true value IJ. of the parameter.
Take for
the comparing estimator any estimator me with Med(mc ) = IJ.. An estimator !!! would
-
then be called negatively biased in the sense of section :3 t'f P(! S.
IJ.)
> P(mc
~
.
IJ.)= i
Hence this definition is identical with the definition of median-bias; cf. the
-
second inequality in (2,2), definition 2. If the distribution of m is
s~etric,
then of course median-bias and expectation-bias are equivalent and it is here that
Snedecor1s 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 tooemall estimates (i.e., estimates smaller than
the comparing value) as being unduly large if the 'Probability of obtaining estimates
-5smaller than the comparing value would be lat'ge 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 automatioally to the definition
(cf. the third inequality in (2,2), definition 2), but it does not.
In fact, why
not use the definition
say, with k ~ I?
-
The simple fact that one is apt indeed to prefer the inequality
-
(3.l,la) to (3 .1, Ib) shows that in applying this approaoh 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
a comparing estimator.
An analogous remark holds true if any other comparing value
(than I..l.) would be used; always one would have to decide about questions like: how
-
large is k in (3 .1, Ib) to be chosen?
So this approach is essentially identical with
the approach based on comparing estimators.
___
3.2
_
~a_~
_
Distribution-bias
Take for comparing values all conceivable values ~ of the parameter I..l.
(the set of values ~ may be an interval, both finite and infinite).
As for the choice of the comparing estimator me' there are often two or
-
more estimators with different, yet (almost) equally desirable properties.
In such
cases there may be no point in selecting just one of these estimators for comparison
se .that one may want to use all of them (in most practical cases the number of
estimators competing for the r~he of comparing estimator will not be too large).
The concept of bias will then have to be modified into bias
(a particular comparing estimator) me.
~
respeot
12
-6To come back at the comparing values
coincides with the union of the set of
and the set of
5
I. , note that
with 0
S with
0
< P(m < ~) < 1
< P(mc <!) < 1,
-
then the condition
P(l!!
S. $ )
> P(mc
~ ~ ) for
if the set I of values
~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
12 ! ! ! E. .! .:!? .! .2 !l 1·
Let I be the set of all values which the
tJ. might conceivably have (depending on the specific practical problem).
estjptator m of tJ. will be called negatively ~
with
----
1!
.:!?
~
5) ~
- ~ ~)
'(me
_
tor
Then the
! E 1! .:!? 1: .2 .!!-,E! ! ! !
e s-t .......
i mat
......
to a
P(]!
!
0
...
paramet~
~
-r
g e I,
the equality sign not holding true for at least one
g e I.
Note that whereas the definition of positive distribution-bias is selfevident after definition 3, the definition of distribution-unbiasedness with respect
to me presents difficulties.
If the
~
sign in (3.2,1) wo.uld be simply replaced by
an equality sign, then it would be easy to find estimators which are neitlBr distribution-unbiased nor distribution-biased.
Alternative definitions of distribution-
unbiasedness will easily appear to be unsatisfactory in other respects, and will not
be attempted here.
l:~~ -2!!~
--
In certain problems one may be particularly interested in one special
oomparing value, not the true value of the parameter tJ. (for example, in response
A
surface theory the value zero playa an important role in connection with the
estimation of the oanonioal regression ooeffioients).
value
ko~
so that!c -is a fixed and known oonstant as
section 3.2.
Call this speoial oomparing
opposed to
t
occurring in
Handling the question of comparing estimators in the same way as in
seotion 3.2 one arrives at
D
e fin i t ion
11 .
~e
estimator m of IJ. will be oalled negatively
(3.3,1)
The definition of kc~unbiasednass presents no diffioulties.
Note that estimators which
unbiased may be
aSj'11lJlletry of
§c
~e
5'o·biased with respect
both
expecta~ion~unbiasedand
to some estimator mo'
-
median-
In fact, due to the
with respeot to IJ. ( Sc cannot be both larger and smaller than IJ.),
even an estimator m that is distributed symmetrically around IJ. may be f.-biased
c
with respeot to another estimator me that is also distributed symmetrioally around
IJ..
For example, let m be
-
and take
go = IJ.
distribu~d N(IJ.~
." 20', then!!! is negatiYely
20'2) and m be distributed N(IJ., 0'2),
c
£c·biased-with respect
to me'
-
Hence,
from the standpoint of ~o-bias an estimator which is both expeotation-unbiased and
median-unbiased may
~
biased and negatively
worse than even an estimator whioh is negatively
median~biased,
but which has a smaller variance.
expeotation~
Here the
conoepts of bias and of efficiency merge into each other. Another possible reason
(also a mixture of bias and efficiency considerations) for preferring certain
-
biased estimators to certain unbiased estimators was brought forward by Olekiewioz,
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
e
that there is little practical value in proving that a certain estimator possesses
just
~
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 with several useful proparties.
-8-
e
l!.
A lemma which is u?2!~ in Rroving o~tai~~~.~f-2~~
An almost trivial lemma whioh, however, sometimes
provides an easy proof
f c-
of an estimator being (negatively) median-biased, or distribution-biased, or
biased is
Lemma
------
1-
-
Whether the random variables m and z are independent or dependent, if
P(~
~ ~ )
-
= 1,
(c)
then
PC!!! ~ ~) ~ P(mc ~
where me
f
<: ),
+
= !!! + o!. A necessary and suffioient condition for the equality sign
to hold in (4,1) is
P[ (,m + !
>S
+
~
)
n (!!! ~
g )] = 0
Proof
---Drawing a picture will bring home the triviality of tre proof.
pc!!! ~ ~ )
=
t
P [(o! ~ ~ + ( - !!!) () (!!! ~ ; )] v [(o!
= (because
= P[!!! +
~
of condition (c) in the lemma)
o! ~ S + t.; ] + p[ (m
pemc ~
;
>g + ,
+ ;
]
+
o!
>g
... !!!) II (m $. g)]
t
=
+ ( ) i't (!!! ~ ~)] ~
,
which completes 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 !!! and.!, yet it is olear that (4,2)
oannot
be proved or disproved from the conditions of the lemma alone. Three extra. conditions
each of them sufficient for (4, 2)
~
to hole, are
..<)
mand .! are
~)
eaoh (measurable) set in the half-plane.!
probability,
independently distributed and P(.!
= ( ) < 1,
>~
in (m,2)-space has positive
-9-
[(o! >Co) n (!!! >to>]
y) some set
with
;0
+ §"o
>(; + g
has positive
probability (implied by" hence weaker than ~).
Rem
ark.
_-=t
__
"'_
Lemma 1 is useful in connection with certain cases
of negative bias.
L e mma
-----
The analogue for positive bias is:
-1'.
-
-
Whether the random variables m and z are independent or dependent" i f
-then
pc!!! ~
where mc
$" ) ~ P(mc ~ g + ~
=!!! + !.
>
-A necessary and sufficient condition for the
equality. sign to hold in (4,,1:")
P
(ct)
r(!!! +! < t + c ) n (!!!
!!
~ $ )]
.
= 0
-
When applying Lemma 1 or 11 in order to prove that m is a biased estimator
"
of IJ. in one of the senses discussed in this paper, take' = 0 and
for median-bias· take
that m
c
-
=-m + -z
g=
IJ. and try to choose the random variable! in such a way
has IJ. for its median,
for distribution-bias with respect to (!!! + !) take
£ 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 + z) take
-
-
~
=
~
c
and see again if z can be chosen
-
in such a way that (!!! + !) is an interesting comparing .estimator.
- -
In all three cases m and z may be mutually independent or dependent.
A-~
APPENDIX
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 ~
-
Be JIl'x2, •••• ,~ an n-fo1d sample from a normal distribution N(lJ., ( 2 ).
well known that
It is
1
n-l
is an expectation-unbiased estimator of
0
2
It will be shown that
•
£ is,
negatively median-biased, i,e., that
n-l
1
n-l
n-I)
2
r( T
.2
Jo
~(n-l)
...
1
. 1"2
en....1 )
... I - Q( n-l
Jl.
e
-""2
y
n-l
T
-
I} )
...
~r
J
0
21
of>J. -1
-'2
• 2
•
I'
J2
X
cf.
Pearson and lIartley (1954), p. 122;
x *'
y(~x) = e-t t..( - I dt,
J
o
=
dt ...
(n-l
I, n-l ) ... y"2
' n-l,) / r (n-l)
"2 ... I
[r(
dy
E:! _ 1
e-t t 2
where the functions Q, y and I are defined by
'Ii
co
Q( X21
I
~-l
2 y 2
dy,
_:l.
e
<V"2'""2
n-l' n-l
-
)
1 ,
however,
A-2
cf.
Higher Transcendental Functions (Vol. 2, 1953), p. 133;
I(u,p)
= I(
,_ m
, p)
Vp+l
= y{p +
1, m/r(p + 1) ,
cf. K. Pearson (1946) or Jordan (1950), p. 56.
Jordan (1950), p. 57 mentions that "it can be shown by the tables of the
function I(u,p) that ••• I(
analytically.
Vp + i,
p)
>~
However, this can be proved
If.
By means of the asymptotic expression for y( 0< <to 1,
0<
+
VU
y)
given by Tricomi (1950) one finds that
(A.l,2)
a formula which for real
implies that for
0<
can easily be proved by Laplace t s method.
This formula
large enough y(..<" .t..)/r( 0<) exceeds ~ and that
0<
y(..<" .t..)/r(.t..)
i
~
if
0<
~
Q)
Therefore, the inequality (A.l,l) will evidently be proved (even for small values of
n) if it can be shown that
y( d.,.-4/ r(.t..)
> y(.t.. +
1, .t.. + l)/r(.t.. + 1) for any .t..
> o.
Now this inequality is equivalent to
0<
y( ..<, 0<) > y( 0< + 1, .t.. + 1)
or
.t..
.t.. •
.t.. + 1
J e-t t.t.. - ldt > /
o
e-t t.t.. dt
0
.t.. + 1
.t..
or
or
+
0<
e-.t...t...t..
>/
Jo
e-t t.t.. dt
J
>
e-tt.t.. dt
0
+1
e-t t.t.. dt •
.t..
As t h e maximum of the integrand e
4
t
.,(
is reached for t
is clearly correct, whereby the proof is complete.
= ..<,
the la.st ;inequality
A-3
Rem ark. Line 6 from the bottom of p. 141 of the above-cited
------Volume 2 of Higher Transcendental Functions oontains an error; it
.
is stated. that r(..<,
x)jr(.,() is a monotonio decreasing funotion of
,
.,( for .,(
> G,
X ) OJ where&!
function of.,(.
in reality y(.,I,
(Note that r(..<, x) + y( ..<,x)
XI/r(.I.}
= r( <><)
';
i6
a decreasing
cf. Jordan
(1950), p. 57, equation (3) ).
The following table is computed from Pearson and Hartley (1954),p. 122
seqq., at some places checked by K. Pearson's table of the incomplete gamma funotion.
It is interesting to note that the as;ymptotic expression (A .1, 2) yields results
whioh 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 ~ (l)
- =
0,683
0,632
0,608
0,594
0,584
0,577
0,571
0,567
0,,56:3
0,560
0,542
0,534
0,530
0,527
0,524
0,522
0,519
~
4.
A-4
e
~
The samRle median as an estimator of the median of the
distributio~.
Be F(x) a oontinuous distribution funotion with dF(x)/dx different from zero
in
2!!! (finite or infinite) x-interval" and :£:1." !2" •••" !2m+l a sample of odd size
from the oorresponding distribution.
the notation x(l)
~
Rearrange the values in the sample, and use
« ... ...
< x(2m+l).
•«-x(2) =
As F(x) is assumed to be a oontinuous
distribution funotion" the occurrence of equality signs has probability zero; !(m+l)
is the sample median.
It is well known (cf. for instance Wi12, 1948, equation (16)
on p. 16) that
p (x(m+l) ~ p)
-
{(iz,n1'~L~
l1li
(A.2,1)
r m+ ) r{'iii+!T
l1li
u
< F < 1)
Denote by G the inverse (defined for 0
of the funotion F so that G(~) is the
median of the distribution considered. Substitute F(y) by F in the integrand of
(G(~})
(A.2,1) and use the equality F
a
~, then
(A.2,2)
1/2
.1
FU(l-F)m d F . . /
If(l_H)m dH
1/2
Because of /
o
(as appears from substituting F
,
/
1/2....In
Jr
o
(l.F)
m
dF
1
III
2'
J
rl
= 1-H),
one obtains
yn(l..F)
m
dF
1
l1li
2' B(m+l,m+l) •
(A.2,3)
0
\\
--
The equations (A.2,,2) and (A.2 J 3) prove that x(m+l) is a median-unbiased estimator
-
~
of the median G(1/2) whether F(x) represents a skew distribution or a symmetric one.
~
J
/'+ co
-CD
....
1
B(m+l"m+i)
Y [F(y)
r [l-F(y) r
\
\
However" the expectation of !(m+1) equals
r(2m+2)
\
\
dF(y)
iii
A-5
=
NOW;l
o
1
1
/1
1
G(~)
+ :B(m+i,m+i)
[G(F) - G(~) J • F'l(l-F)m <iF
(A.2" 4)
o
[G(F) - G(t>
J
J •
• ('4..h
2)m
db •
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
1
1
G(Th) - G('2') is an odd function of h. FUrther, this same second tem in (A.2,,4)
is {~~:;~~:} for those distributions for which
(0
e
< h <~)
(A.2,6)
expectation-biased estimator of the median G(~2.. (though not only for this type
Hence for such skew distributions as satisfy (A.2,,6) the statistic x(m+l) is an
of skew distribution, of course.)
Rem ark.
------
~(!(m)
If the sample size is even, 2m say, the statistic
+ !(rn+1)} is customarily used as an estimator of tlD:e sample
median. For certain skew distributions this is a median-biased
estimator of the median. If ¢ (xC 1 ), x(2), "'J 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.3. The estimator r of the correlation
Be
coef~ic~ent p
(~'Yl)"
(x2 'Y2) ..., (Xn,Yn) an n-£old sample froms bivariate nonnal
distribution with correlation coefficient p. Define the sample correlation coe£ficien t in the usual way by
I'
II
n
~
t1=1
~ / / i~n
n
(x._x)2 ~
(Xi-X)(Yi-Y)
I
1=1
J~
1
(y .-y>2
J
j'J.
It is well known (c£. ~~~~~~" 1947" p. 344, eq. (14.,5), and !1g~~gl1~~;r:" 1925,
p. 42, eq. (128) ) that
-
=
E(r)
p .g(n" p2)
a
1 1 n+l
2
• F(~" 2'; 2; p )
p •
II
(A .3,1)
The last equality follows from
fMl~tJ ~
integral representation for the hypergeometric
function (used in this context as early as 192, by Romanowskv , 1.c.):
"11I.==I:I===~
F( a, b;c;z )
1:1
r(c)
r(b) r(c-b)
Jl
o
t
b-l (
loot
)c-b-l (
1-tz
--
)008
dt,
see for instance Higher Transcendental Functions (vol. 1, 1953), p. 59" eq. (10), and
p. 114, eq. (1).
c:;
1 n-l
B(-, - )
2
2
r(~) r(n~l)
= -~__
. r(~)
2
for any P
r 1,
A-7
e
hence by the fourth and second members of (A.)"l) that
g(n,
~erefore
i)
<1
for any p2 ., 1.
r is a ne$atively expeotation-biased estimator of
p 1£ p
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.),l).
-
In order to investigate whether r is a median-biased estimator of p, we use
--"
formula (25) of Hotelling (1953)*, p. 200 (note that Hotelling,tf!. n denotes the number
which is one less than the sample size), whioh entails
PC!.
By
n-2
2)
~ p) = - r(n-l)
· i (1
._p
V2 Tt r(n.~)
n-l
1
21
p
n..4
(1_r2)
•
2
(A .),2)
(~.Pl") not
means of the substitution
0=
y,
suggested by section 5 of ~~~~l~~g~ (1953), the second member of equality (A.3,2)
reduces after some patient algebra to
~ r(n-aF
V2n
r(n-ir)
n-4
1
J 1 (1_...,2)2
(1+,.,...,)2 FJ~, 1
"
t"J
1 1
\.:::~; n-~;
1
0
2
l-e \ dy
... 2(1+ AY)1
As p increases from 0 to 1, the function (1+AY)2 increases for any y
1 2
the function 1;~
> 0,
decreases for any y
> 0,
hence 1 - 2(1+AYT increases for any y
> 0,
l-i
* The
(A.3,2!>
author wants to thank Dr. R. J. Hader, Institute of Statistics, Raleigh, for
drawing his attention to this paper.
hence
F(~, ~;
n,,;!-; 1 ..
2(i~t»)
> 0,
increases for any y
hence the integrand of (A.3,2.:) increases with p for any
>O.
y
As P(~ ~ p)
r:a
~ if p = 0 (the direct verification of this elementary fact from
(A.3,2,!) is straightforward, though rather intricate), this means that
P(;:
~
p)
> ~ if
.e ~ ;;;o,p, ;, ;os_i..t..i;,,;.V',;.B,
section 2.
p
> O.
Therefore::..!!! positivell median..biased estimato:r
~ .e !!
which completes the proof of all statements made at the end of
.
LITERATURE CITED
Higher Transcendental Functions, (Bateman Manuscript Project), Vol. 1, 1953,
:xxvi + 302 pp., Vol. 2, 1953, xvii + 396 pp; McGraw Hill.
~~~ng,
H., New light on the correlation coefficient and its transforms;
Journal of the Roya.l Statistical Society, Sere B (Methodological)1;2, 1953,
193-232.
Calculus of finite differences; 2nd ed., New York, Chelsea, 1950,
xxi + 652 pp.
...Jordan,
Ch.
Kendall, M. G., The advanced theory of statistics, Vol. 1; London, Griffin, 1947,
xii + 457 pp.
Mann, H. B. and Nhitney, D. R. J 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 effioiency of biased estimates; Ann. Univ • Mariae CurieSki!odowska, Sect. A., J., 1950, 103-140.
Pea3:'son, E. S. and Hartle!, H. 0., Biometrika Tables for Statisticians, Vol. 1,
Cambro Univ. Pr., 1954, xiv + 238 pp.
Pearson, K., Tables of the incomplete r-fwllctions; Cambr. Univ. Pr. Re-issue 1946,
xxxi + 164 pp.
Romanowsky' V., On the moments of standard deviations and of correlation coeffioient
in samples from normal population: Metron, Vol. 5, N. 4; December 31, 1925,;
31-46.
Snedeoor, G. w. Statistioal methods, applied to experiments in agriculture and
biology; 4th ed. (.3rd print). Iowa State College Press, 1948, xvi + 485 pp.
Tricomi, F. G. Asymptotische Eigenschaften der unvollstandigen Gammafunktion;
Mathematische Zeitschrift 53, 1950, 136-148; cf. also Math. Rev. 13, 1952, 553.
-the mathematical theory of correlation-(translated
TsohuEOW, A. A., Prinoiples of
byM. Kantorowitsch)} London, W. Hodge and Co., 1939, x + 194 pp.
Waarden, B. L. van der, Mathematische Statistik (GrundleiJren der Mathematisohen
t.vissenschaften Bd. ~) Berlin" Springer" 1957, ix + 360 pp.
Wilks, S. S." Order statistics; Bulletin of the American Mathematical Society 2l!,
1948, 6-50.