ON FECHNER'S THESIS AND STATISTICS WITH NORM P TOKIO TAGUCHI (Received Oct. 9, 1973)

1.

Introduction and summary

Fechner's Potenzmittelwerthen and Abweichungssumme (see Fechner, G. T. [1]) express respectively generalized measures of location and dispersion. The author's original motive for taking up afresh these classical notions comes from the fact t h a t t h e y are in line with the notion of the least mean error regression with norm p (abbreviated as LMER with norm p or briefly p-LMER). However, as a result of reexamining these notions, the author arrived at a new idea, pan-probability-theoretic one, which results in an extension of the conventional probability theory. According to this idea, the probabilistic notions such as expectation, general moment, mean, variance, covariance, correlation and regression are all generalized ~vithout losing their fundamental characteristics. Also, by using this new concept Fechner's Abweichungssumme is extended to a multiple dimensional situation. An unbiased estimator of location parameter with norm p is also defined. The greatest merit of this theory seems to be t h a t it can supply a unified method of analysis for both the family of non-skew distribution represented by Pearson's system and the family of skew distributions represented by Paretoan's income distribution which has infinite variance. In this case, each of the families will have its own norm as was anticipated by Fechner. These norms can be chosen within the limits determined by their characteristic exponent ='s (see Gnedenko, B. V. and Kolmogorov, A. N. [2]). Generally, p is restrained to be less t h a n a and a is restrained to be more than i in this theory*. If the given density function is symmetric and has finite p t h absolute moment, the values of its location are all equal independent of p. But in the general case, the values of location, dispersion, correlation and other m o m e n t s constitute functions of p. To study the nature of these functions and t h e relationships among t h e m or to deny t h e existence of * By the way, this theory can also be applied, in the inverse form, to the case a<:l, such as Zipf's linguistic distribution, which does not have finite mean and variance, (see e.g. Taguchi, T., On Zipf's law, Proc. Inst. Statist. Math., Vol. 17, No. 2, 1969). 175

176

TOKIO TAGUCHI

location leads to a new subject. But this subject will not be pursued here. For a more precise account of the above-mentioned, the author examines the existence, the uniqueness and t h e satisfactory equation of Potenzmittelwerthen in the following section. In Section 2, the most fundamental notions of the pan-probability-theory are discussed. In Section 3, some new pan-probability-theoretic operators are introduced and some fundamental relations among them are proved. These relations are compared with those of the conventional probability-theoretic ones. By using these operators, the results of Section 2 are simply described. The two-dimensional mean deviation around the median and the correlation coefficient based on the above defined deviation are newly defined. Section 4 is devoted exclusively to the study of pLMER and its residuals. Many results in this section (e.g. the generalized normal equation) are obtained analogously as in Section 2 by the notations of the operator introduced in Section 3. The Paretoan distribution adopted as an example will be helpful in the understanding of the significance of this new theory. In the final section, the estimation theory with norm p is treated. A new definition of unbiasedness with norm p is proposed and an unbiased estimator of location with norm p is given. On the other hand, Cebysev type inequality is extended to statistics with norm p in this section.

2.

Generalized location and dispersion Let

(1)

G(')(a)= {E IX-alP} */'

exist for any given random variable X having density function f ( x ) and for any a and p>-l. DEFINITION 1 (Fechner, G. T. op. cit). If ao is a minimum of G(a), a~ is called location* with norm p of X (or mean value with norm is of X) and denoted by LcP~(X) or briefly lcp,. And in this case minimal value Gr of G(')(a) is called dispersion with norm p of X and denoted by /YP)(X). Then, it holds THEOREM 1. I f p > l , lC" exists uniquely. In particular, i f f(x) is defined in an i n t e v ~ I and nan-zero valued almost everywhere in L Ic*) is also d ~ r m i n e d uniquely. * Generally, Lc~)(X) is not equal to the power mean of order ~--1 by M~-I(X)= ~-t~/~, except in the cases of p----1 and 2.

ON FECHNER'S THESIS AND STATISTICS W I T H NORM p

PROOF.

177

Applying Minkowski's inequality (see e.g. Shisha, O. [3]), to

(2)

G(P'(la, +/~a~) = [E {1X-/a~-/~a21 p}l '/p

~_[E {21X--a, I+ p IX--a2 I} P]'/p , w h e r e 2, ~ > 0 and 2 + p = l ,

we have easily

G(~)(~a,+pa~)<2G(P~(a,)+pG(p~(a2) for p>l

(3 )

~_lGCP~(a,)+~GC~)(=,)

for p = l .

Namely, G (p) is a strictly convex function for p > l ; therefore, t h e existence and uniqueness of lcp' is obvious in this case. F u r t h e r m o r e , if f ( x ) satisfies t h e above given condition, G(~)(a) is also a strictly convex function for p = l and therefore l ~ is d e t e r m i n e d uniquely. (q.e.d.) Now, w e define ASSUMPTION 1. f ( z ) is an almost e v e r y w h e r e continuous and nonzero valued function defined in an interval /. And E [ X I p exists for any p>__l. T h e n we have THEOREM 2.

uniquely for p>=l.

I f f ( z ) satisfies Assumption 1, l cp~ and D~P)(X) exist Furthermore, i n this case l (~' satisfies

E (I X-l(~)[P-'/X~- l'p,) P (X<= lop') - E ([ X - Z ~' [~-yX > Z~,) P ( X > Z ~ ' ) = 0 .

(4)

PROOF. F r o m Assumption 1 G(~)(a) exists and f r o m T h e o r e m 1 1
(5)

~d- [E {(a-X)P/X<=a} P ( X <= a ) ]

+lira

I"

{(a+ Ja--x)"--(a--x)"}/,taf(x)dx

da~O .J--

= (a-- a)~'f(a) +

p(a-- x)"-lf(x)dx

= p E {(a-X)"-*/X<=a} e (X<=a) . Similarly

( 6 ) d [ E { ( X - W / X > a } P ( X > - ) ] = - p E{(X--a)~-yX>.} P ( X > a ) . Therefore, t h e equation

(7)

d EIX_aI~=0 d~

178

TOKIO T A G U C H I

has a solution which coincides with (4).

(q.e.d.)

COROLLARY i. I f f(x) is a symmetric function with respect ~o x = a and satisfies Assumption 1, it holds (8)

l'"=a

far any p~_l.

Example 1. If Assumption 1 holds, l c1~ gives t h e median (Fechner, G. T. op. cit.) and l r gives the arithmetic mean. On the other hand, Dc'(X) expresses t h e mean deviation ~=,~d of X around the median and Da'(X) expresses the standard deviation of X. F u r t h e r m o r e ,

(9)

{

where a expresses the standard deviation of X and r~ expresses Pearson's skewness p~o~, (see Kendall, M. G. and Stuart, A. [4]), because from (4) in Theorem 2 we can easily show t h a t l "~ is the real root of the following equation : (10) zs + 3 o ~ - / ~ = O .

3.

Pan-probability-theoretic notions and their characteristics

In this section, we will extend some conventional probability-theoretic notions. DEFINITION 2.* Let E~'~(X) be the expectation with norm p of X. Then E cp~(X) is defined as follows:

(ii)

E cp' (X) = E {(sgn X) fX ]p-l},

where

sgn X =

t

-1

as X < 0

0

as X =O

1

as X > 0 .

Generally t h e r t h m o m e n t p~P>(a) and the absolute m o m e n t ~P)(a) around a with norm p of X is defined respectively as follows:

(12)

p~P)(a)= E [ {sgn (X, a)} r[ X - a ]p+r-2] ,

where

sgn (X, a)=

i

--i

as X < a

0

as X = a

I

as X > a

and

r = l , 2, . . . .

* A n improved version of Definitions 2 and 3 can be found in another p a p e r to be published in Volume 2 of The Proceedings of the Institute of Statistical Mathematics in 1974.

ON FECHNER'S THESIS AND STATISTICS WITH NORM p

179

and

(13)

/~p)(a)=E (I X - a I"+'-~) . In this case, we directly h a v e T~.OREM 3.

It holds g?)(0) = E (') (X)

(14) and

~/tt?)(l (p') = D(P,(X ) .

(15)

Furthermore, it holds

(16)

E (~) ( a X + b ) = a E (p) (X+b/a)

and (17)

D(P'(aX+ b) = [a l D(~)(X+b/a) . COROLLARY 2.

It holds

(18)

={

g~+~_2(a),

i f p is even

fl,+~_2(a) ,

i f 7 is even

and

(19) where gp+~_~(a) and ~p+~_2(a) express respectively the conventional moment and absolute moment around a of order p+T--2. On t h e other hand, f r o m T h e o r e m 2, t h e following t h e o r e m m a y be deduced. THEOREM 4. t f Assumption 1 holds, ICP) satisfies the following equation with respect to x (location generating equation, abr. l.g.e.). (20)

E (~) ( X - x) = 0 .

COROLLARY 3. I f Assumption 1 holds and E ( I X I s) exists for any real number s, it holds

(21)

1 p+r-1 ~ ~p~2~

Furthermore, it holds

d dx f te~P-)'(x)dx

for r + p > = 4 .

180

TOKIO TAGUCHI r--1

(22)

.++'(+)=(-tv-'(+++-2)! (p'D! f - I

Definition 2 can be easily extended to any dimensional random vector X.

DEFINITION 3.* The conditional expectation E ~p~(XI Y ~ S) with norm p of X under Y e S, the expectation E Cp~{~(X, Y)} with norm p of ~(X, Y), the conditional moment p~)(alY ~ S) with norm p of X around a under Y e S, the s-t moment ,,cp) la b) with norm p of (27, Y) around (a, b) on X and so on are defined respectively as shown below: (23)

E Cp'(Xl Y E S) = E {(sgn X) IX [-"-'tIY ~ S } ,

(24)

EC~)ip(X, Y)} = E [{sgn ~(X, Y)} Ip(X, Y)I'-+I,

(25)

/J~)(alY ~ S ) = E [{sgn (X, a)}+lX-al++'-'lY ~ S]

and

(26)

/~) /a b) = E {(Y--b)~,u(1)(alY)} = E [ {sgn (X, a)}+(Y-- b)'l X - a ]++'-"l and

•

~a( b)=E{(X-a)'plP)(blX)} P ) = E [ {sgn (Y, b)}' (X--a)' [Y - b [P+'-2]

therefore, in particular (27)

~+,mz(a, cp) b) =/~IP)(a)

and

~.~,)l~(a,b) =/~Ip)(b),

etc. Then, considering Minkowski's inequality and HSlder's inequality (see Shisha, O. op. cit.), we easily have THEOREM 5.

(28)

If p>l,

~/IE'~'(X+Y)I~_ ~/ECP'(IX[+lYI) <

Furthermore, i f u r

(29)

it ho/ds

~/E(~)(]XI) + ~/E(~'(]YI) .

u > l and 1/u+l]v=l, it holds

IE(P)(X, Y)[_~E 'p'(IX, Y[)
and * A n improved version of Definitions 2 and 3 can be found in another paper to be published in Volume 2 of The Proceedings of the Institute of Statistical Mathematics in 1974.

O N FECHNER'S THESIS A N D STATISTICS W I T H N O R M

p

181

I ,,-,,., ,,,(a, b) l__
(30) Iff,.,,,(a, b) I< E (I x - a I' I Y- b 1,+'-') < ~/fl~(a) V/3,(p+,_s,(b). F u r t h e r m o r e , we can append in this case DEFINITION 4. The codispersion w i t h n o r m p b e t w e e n X and Y, codisp. (~) (X, Y) is defined by

I codisp.(P) (X, Y) Ip= ]p[P~),~{L(P)(X), L(P)( Y)} ]

(31)

X

I /~,.,uv{L (P) (P)(X), L(P)(Y)}I

and (32)

The sign of codisp, cp~(X, Y) = t h e sign of [~,,,~{L (P) (p)(X), LcP)(Y)}-.,(P)

F u r t h e r m o r e , t h e two-dimensional dispersion p of (X, Y) is defined as follows: (33)

D'P'(X,Y)=~ det( {Dcp'(X)}p

{LCP'(X)LCP'(Y)}]

D(P~(X,Y)

with norm

[c~

\lcodisp. (~) (X, Y)I p/'

{DCP)(Y)}p

I, "

On t h e other hand, for r a n d o m vector X=(X,,..., X,) h a v i n g density function f(xl,..., x,), t h e following assumption is v e r y often useful as a substitution for Assumption 1. ASSUMPTION 2. f(xl,..., x,) is an almost e v e r y w h e r e continuous and non-zero valued function defined in a convex domain D,. And E[X~I p exists for any p ~ l and i = 1 , 2 , . . . , n. In this case, we have THEOREM 6.

(34)

If Assumption 2 holds its follows: Icodisp, c') (X, Y)[
and therefore det (' {DCP'(X)} p \ I codisp .c~' (X, Y)[pn

(35)

Icodisp. C~'(X, Y)I~n~ > 0 ]

[DCp'(Y)} p

and

(36)

D'P'(X, Y) = ~/ {/VP)(X)} p

Icodisp, c~' (X, Y)I p''

_
If p = l

Icodisp.(~ (X, Y)I~'I L

{~'(Y)}~

182

TOKIO TAGUCHI

(37)

Icodisp, c" (X, Y) I= IE [ {sgn (X, L"'(X))} [ Y - LC'( Y)} ] I x IE [ {sgn (Y, Lm(Y))} { X - L";(X)} ] J ~EJ Y - L " ' ( Y ) I EIX-LC"(X) I

and (34) holds. (38)

But, if p > l , we have from Definition 4 and Theorem 5

JcodispH' (X, Y)IP< [V/~,~p_,,(LCP'(X))~/fl,(L(~'(Y))] x F~/fl~,(LC"(X))'~/flr

where u , u ' r u, u ' > l , 1/u+l/v--1 and 1/u'+l/v'--1. (p--l) and u'=p. Then we have v=p

and

v~_

]

Now, let u--p]

P p-1

Therefore (39)

IcodispH' (X, Y)I'<~p(LCP'(X))~,(LC~)(Y))= {D~p'(X)}P{DCP'(Y)lP

and (34) holds. Letting p = l , we have easily COROLLARY 4. I f Assumption 2 holds, we have the two dimensional mean deviation ~=,y~d of (X, Y) around the medians as follows (40)

I~.. ~d Icodev. (X, Y; med)11"2I /~,.y,reed= i i codev. (X, Y; med) ],/2 ~; reed __~ E

E I {sgn (XI, reed x)] {sgn (Y~, reed y)}

(x~rD (x~ r2) L

)(1-- med 9 X s - reed ~ • Yl-medy Ys-medy J where ~ , ~ d = E I X - m e d

xl, 8 r ~ d = E J Y - m e d yl

Icodev. (X, Y; med) l = IE [ {sgn (X, reed ~)} ( Y - reed y)] • [E [{sgn (Y, med y)} ( X - r e e d x)][ and the sign of coder. (X, Y; med) = the sign of [E [{sgn (X, reed x)}(Y- reed y)]

+E [{sgn (Y, reed y)}(X-med ~)]] (see Taguchi, T. [5] and [6] on the two dimensional mean deviation of (X, Y) around the means).

DEFINITION 5. The correlation coefficient with norm p between X

O N F E C H N E R ' S THESIS A N D

STATISTICS W I T H N O R M

p

183

and Y (abr./~)(X, Y) or briefly /P)) is defined as follows :

p(P)(X,Y ) = c~

(41)

(X, Y) D(~,,(X)D,,,)(y)

Then we have THEOREM 7. I f Assumption 2 holds, it follows

--1__/p'_1

(42)

for any p>=l. In particular, i f Xjj. Y, it holds

(43)

p(~)=O

.

But, i f X and Y are completdy positive (negative) correlated, it holds

p(.)=l(-- 1).

(44)

PROOF. (42) is obvious from Theorem 6. (45)

If XJ_[Y,

m,,~(L (~) (~)(X), L(~)(Y)) = E {Y - L(~'( Y)} E (~' [ X - L(~)(X)} = 0 .

Similarly ,I.%(L(~)(X), L(~)(Y))=O and therefore (43) holds. But, if X and Y are completely correlated, it can be expressed almost everywhere by

(46)

Y=aX+fl

a~ O .

Applying (46) to EcP){Y-L(P)(Y)} =0, we have from Theorem 3 (16) a

and therefore from Theorem 4 (20), we obtain

L(P)(X)-- L(p)(Y ) - 19

(48)

It also holds almost everywhere

Y - L(p'(Y) = a {X-- L(p'(X)}.

(49) Then (50)

l, l I ~ -_[ *_L

E[I

X-L(p)(X)I p} = a {/VP)(X)}p

Similarly (51)

Consequently, it holds

s

184

TOKIO TAGUCHI

(52)

codisp, cp' (X, Y ) = (sgn a)D(P~(X)D(~'(Y).

and therefore (44) holds.

(q.e.d.)

Considering p = l , we have, COROLLARY 5. I f Assumption 2 holds, we have the correlation coe~icisut p=.~8,.d between X and Y based on mean deviation around the median as follows:

(53)

p..y:.m~=

codev. (X, Y, reed)

(see Taguchi, T. op. cir. on the correlation coe~icient based on the mean deviation around the mean). Example 2.

L(P)(X) does not hold necessarily L'p'(X) = E r (X) .

If and only if p=2, the above equality holds. Furthermore, in this case ~P)(a)=~T(a). And {D'"(X, Y)}', codisp.'"(X, Y) and /2'(X, Y) coincide respectively with variance, covariance and the conventional correlation coefficient of (X, Y). 4.

Regressions with norm p

In this section, we treat mainly regressions of Y on X. Regressions in other dimensional cases and of the other kind can be easily~ obtained through the same kind of consideration. DEFINITION 6. The exact regression curve with norm p of Y on X (abr. p-ERC of Y on X) is defined as the curve y of x satisfying, (54)

E Cp'( X - x l Y = y ) = O ,

(see Kendall, M. G. and Stuart, A., op. cit). Then, from Theorem 1 we have clearly the following: THEOREM 8. I f f(x, y) satisfies Assumption 2 the p-ERC of Y on X exists uniquely for p~_l. DEFINITION 7. The least mean error regression line with norm p of Y on X (abr. p-LMERL of Y on X) is defined as the curve y = a z + ~ where a and ~ give the minimum values of ~ / E ( [ Y - a X - ~ I g .

Then we have THEOREM 9.

If

E(IX[ p) and E ( I Y [ 9 exist for any p>-l, the p-

ON FECHNER'S THESIS AND STATISTICS WITH NORM p

185

LMERL of Y on X exists also for p ~ l . In particular, i f Assumption 2 holds, the p - L M E R L of Y on X can be obtained uniquely. PROOF. By considering that ~/E (] Y - a ) ~ - ~ Ip) is a convex function of a and /3 for p ~ l , if E(]X] p) and E(I YI p) exist, the existence of the p-LMERL of Y on X is obvious. Furthermore, if Assumption 2 holds, ~/E ([ Y - a X - - ~ ]P) is strictly convex and therefore the above p-LMERL is obtained uniquely. (q.e.d.) Thus, we have the following: THEORE~I 10. I f Assumption 2 holds, the p - L M E R L of Y on X satisfies the following equations : E {XE (p' ( Y - a X - [ i [ X ) } = 0

(55) and

(56)

E (~' ( Y - a X - / ~ ) = 0

far p>__l.

These equations can be called the generalized normal equations.

PROOF. Considering Corollary 3 (22) and Definition 3, we have easily (55) and (56). Therefore, the above-mentioned theorem is proved. In particular, considering the case of p = l , we have COROLLARY 6. I f Assumption 2 holds, the regression line of Y on X minimizing E IY - a X - ~ ] satisfies the following equations : (57)

P ( Y > a x + ~) = P ( Y < a x + ~) = 1/2

and

(58)

E (X[ Y > a X + ~ ) = E (X[ Y < a X + ~ ) = E ( X ) . Furthermore,

COROLLARY 7. I f Assumption 2 holds and i f E([XI p+'-') and E ([Y[P+~-~) exist, the least mean error regression curve with norm p of n

Y on X (abr. the p-LMERC of Y on X ) y = ~ a~x~ satisfies the following : kzO

(59)

E{X k E ( P ' ( Y - a X - ~ [ X ) } = 0

for k=0, 1, 2 , . . . , n .

On the other hand, i f Assumption 2 holds, the p - L M E R L of Y on )(1, n

9.., X~, such as y = ~ a~x~+~, satisfies the following equations:

(60)

{

("

E X~E ~,' Y-Not~X,-~IX~

)1= 0

for h=l, 2 , . . . , n

186

TOKIO TAGUCHI

and

(61)

E(p)

_

_

=

,

.

.=

Now, let { be the residual of the p-LMEL of Y on X, namely (62)

~= Y-aX-~

9

In this case, we have THEOREM 11.

E(p, (~)=0,

(63) (64)

I f Assumption 2 holds, it follows: L(P' (~)=O

and

D'~'(~) = ~/ E I Y - a X - ~ ]P~ D(P'( Y) + [a ID(P'(X) PROOF.

for p~_l.

Considering Definition 3 and Theorem 10, we have directly

E r (~) = E c~' ( Y - a X -

~) = O.

Furthermore, because in this case

l~p)( ~) = ~ / ~

~_ ~/ E I~-- L C"( Y ) + aL 'P'(X ) + ~ I~

~_ ~/ E [ {I Y - L(~"( Y ) I+ Ial IX - L'p'( X ) I} ~] ~_ ~/ E{Y-L(i'(Y)I~'} + l a l ~/E{IX-L'P)(X)I p} for p>-l, we have easily (64) and the theorem is proved. Considering the case of p = l , we have COROLLARY 8. I f Assumption 2 holds, the regression line of Y on X m i n i m i z i n g E [Y - a X - fl [ satisfies the following : (65)

median of 5 = 0 ,

(66) (67)

a~--ay-- 2ap~ya~a~+v~a~

and (68)

where ~ and ~t~ express respectively, the mean difference of X and Y. PROOF. (65) and (66) can be obtained at once from Theorem 11. Now, considering,

(69)

~ _ 1 E (~,_~)2

ON FECHNER'S THESIS AND STATISTICS WITH NORM p

187

and (70)

,/r =

we have easily (67) and (68).

E

I ~- ~ I

,

Thus the above corollary is proved.

Finally we have directly from the Definitions 6 and 7: THEOREM 12. I f Assumption 2 holds and furthermore i f the p-ERC of Y on X has a linear form, the p-ERC of Y on X and the p-LMERL of Y on X are entirely equal. The orthogonal regression line with norm p between X and Y (abr. the p-ORL between X and Y) can be defined by the curve axW~y~-~" = 0 ; a~+~2=l, where a, ~ and T give the minimum value of ~ / E ] a X + ~Y+rl ~. COROLLARY 9. I f Assumption 2 holds, the p-ORL between X and Y satisfies the following equations: (71)

E (p) ( a X + ~ Y + r) = 0

and (72)

E [XiE Cp)(aX+ BY+fiX)} ] _ E [Y{E cp' (aX+ fir+ r IY)}I a f '

instead of the generalized normal equations (55) and (56). Example 3. For the symmetric density function, such as the normally distributed one and rectangularly distributed one, any p>=l gives the same location and regression. But the p-LMERC with logarithmic form of loguormally distributed density function as (73)

1

f(x, y)= 2 ~ a ~ j l - ~ , 2 exp _

[

1

2(1_p,2)

' 2p'(log z--/~)(log y--~y)' 4 ! ! a~ay

varies with p.

Namely

(74)

log y - ~ ' v = p ' ~ ( l o g x--~'~) a~

f(log x--/~') ~ a~~ (logy-r ayf g

}] )J

for p = l

and (75)

l o g y _ p ~ = p , 4 ( l o g x _ / j ~ ) + ( 1 - - p " ) ,2 a~ ' 2 a,

for p = 2

.

In these cases, the dispersions of the logarithmic residuals as given by

188

TOKIO TAGUCHI

(76)

v(~)=log Y - l o g (the p-LMERC of Y on X)

are different from each other.

Namely

/~.(v,,,)=E 1r

(77)

~ d 2),o'y

and a~.,,= E (v"' ~)= (1 -- p' ~)a~,.

(78) But

{/~"(v'2')}'= aL~= (1

(79)

-'"-'

--~

(1- ~")2

~"

;uu ~

4

,,

av

"

Example 4. Let us consider the bivariate Pareto distribution of type I (see Mardia, K. V. [7])

t [,~(o~+1) (o,o~)~+~} / (o,~ + o,v- o,o~)-+' (80)

f(x, y, 01, e2, a) =

for x~_O1>v and y~_0z>0, 90

for otherwise,

where a > l . The characteristic exponent of this density function is equal to a (see Taguchi, T. [8]). Then, the marginal density function of X is given by (81)

f(x)=

x";t

for x~_0t>0

0

for otherwise.

If p~_a, /~'(X) does not exist for f(x). mined uniquely as the solution of 1.g.e:

But if p
J,,I 'c') {lc"-x] "-Xx.+, dx= f;., {x_/,,,} ,-ix..+ x dx

(82) In particular, (83)

U"(x)=gx/~

and

U~(X)=

a

a--1

01.

On the other hand (84)

a--1 LFt'(X)=(2t/'-I) _-~al#~ and Dcz)(X)_/ -~/'a(-~--2)

Furthermore, considering

1 a -a1 Ox.

ON FECHNER'S THESISAND STATISTICS WITHNORMp y

Y

3.4 3.2

3"4 l 3.2 "

3.0

3.0

2.8

2.8 p = l , a--I

2,6

2.6

2.4

/

2.4

2.2

2.2

2.0

/

2.0

1.8

1.8 1.6

p = l , a=2

1.6 1.4

1.4

1.2

1.2

1.0

1.0 0.8

0.8 ~

x

0 1 2 3 4 5 Fig. 1. p=l, ~-~I, a=2, 0z=0~.=l

Fig. 2. p = l , 2 , a = 3 , Ot=02=l

3.4

3.4

3.2

3.2

3.0

3.0

2.8

2.8

2.6

2.6

2.4

2.4

2.0

2.0

1.s

1.~

1.6 [

1.6

1.4 1.2

1.4 p = l , a=5

1.2

1.0

1.0

0.8

0.8 x

0 1 2 3 4 5 Fig. 3. p = l , 2, a = 5 , Ot=0~=l

Fig. 4.

~ = I , 2, a=10, 01=0z=I

189

TOKIO TAGUCHI

190

I {ol(a+1)(o2~)-+q/ (o,x+ o,y- o,o=) "+' (85)

f(y/x)=

for x ~ 0 , > 0 and y~_0=>0 0

for otherwise,

we can easily obtain the following generalized normal equation for l
y>~+~ (02z + Oly--OL02)=++

ys.x+~ (Ozz+ O,y---oto~) -.'=+~

~>o=+~ (o~x+o,y-o,o2) ~'+2

~+p

and (O,:c+o,y-O,O.,Y §

In particular, for any a it holds (88)

y=(2'+<"+n-1)~x+02

as p = l .

But for only a > 2 , it holds (89)

y=O~+On a01

as p=2.

If a_-<2, the LMERL of norm 2 of Y on X does not exist (see Figs. 1-4). Moreover, we have in this case (90)

g'(X, y ) = l

for a > l

p<"(X, y ) = l

for a > 2 .

but (91)

By the way, the correlation coefficient p~.;~ based on the mean deviation around the mean is also 1/a.

Exam~Ze 5. The following density function:

seems to be most representative for norm p. 5.

Sample location with norm p Let XI,..-, X, be independent random variables having the same

191

ON FECHNER'S THESIS A N D S T A T I S T I C S W I T H NORM p

distribution. Then we can afford the following definition for any estimator II(~)=II(~)(X~,..., X,) of population parameter z(~). DEFINITION 8. The given estimator //~) is an unbiased estimator of ~cp) with norm p, if //c~) satisfies (93)

ECp)(//(~)_ ~(~))= 0 .

On the other hand, if it satisfies (94)

Ecp'{//(P'-Q (p)} = 0

for an unbiased estimator Q(~ of ~(p~,

H (p) is called semi-unbiased estimator of =(~) with norm p. Then, both types of unbiasedness are generally different from each other, except when p = 2 . Now, let us consider that the sample location 2(p' with norm :p can be given as a solution of the following equation: (95)

~ sgn (X,, ~(~')lX,-~(~)l~-'=0

for p>_-l.

i=l

Then we have LEMMA

(96)

E (')

1.

It holds far p > l

sgn ~ (sgnX~)lX~l "-I

,~='

--

9

PROOF. Applying successively Definition 3 (24) and Definition 2 (12), we have P-UI

=E{~=,

(sgn X~)

Ix, t =

"

~ E {(sgn X~)]X~ ]p-l] i=1

= 1 ~ E(p,(X,)=ECp, (X) .

(q,e.d.)

~=I

Hence, we have the following principal theorem: THEOREM 13. The sample location ~cp~ ezists uniquely and it is always a semi-unbiased estimator of population location f a r a n y norm p>l.

PROOF. Considering that d(P~= V~-. : / 1~=IIX~-2cP)]P__ is a strictly convex function of 2(p) for p > l and (95) gives a minimal condition of d (~),

192

TOKIO TAGUCHI

it can easily be seen that 1(p) exists uniquely. ing

Furthermore, consider-

~(I(.')=E'.' [{sgn~ (sgn(X..I(")),X.-2~"I"-~}

we have directly from (95) ~(,~(p))=O . On the other hand, from Lemma 1

~(2,)) = 1 ~ E'" ( X - 1'"). Therefore, E (p, (X,-~(P')= O. Namely 1(p) is a semi-unbiased estimator of l (p). (q.e.d.) COROLLARY 10. I f p = l , 1(p) is not always determined uniquely. In particular, i f 2(p, is determined uniquely, ~(*) is an unbiased and semi-unbiased estimator of l (') (see Mahamunu[u Desu, M. and Rodine, R. H. [9]). But, i f p=2, 2(p' is a sample mean and naturally an unbiased estimator of l (~). Finally, we can supplement the following Cebysev type inequality: THEOREM 14.

I f Assumption 2 holds, it holds f o r any positive real

numbers k

P (1X - l(2" [>=k) _~ [D(~'(X)} p kp

(97) PROOF.

We

have easily (97) from e.g. Cram~r, H. [10].

INSTITLvI~ OF STATISTICALMATIIEMATICS

REFERENCES [ I ] Fechner, G. T. (1878). Ueher den Ausgangswerth der kleinsten Abweichungssumme, dessen Bestimmung, Verwendung und Verallgemeinerung, Abhandlungen der K~nighch S~chsischen Geselischaft der Wissenschaften, mathemati~h.physische Klasse, X, introduced by Walker, H. M. (1929) ' Studie~ in The History of Statistical Method' The William & Wilkins Company, Baltimore. [ 2 ] Gnedenko, B. V. and Kolmogorov, A. N. (1949). Limit Distribution for Sum of Inde. pendent Random Variable, Moscow, English Translation Addison-Wesley, Cambridge, Mass., (1954). [ 3 ] Shisha, O. (editor) (1967). Inequalities, Academic Press, New York and London.

ON FECHNER'S THESIS AND STATISTICS WITH NORM

193

[ 4 ] Kendall, M. G. and Stuart, A. (1963, 1967). The Advanced Theory of Statistics, 1 and 2, second edition, Charles Griffin & Company Ltd., London. [ 5 ] Taguchi, T. (1972, 1973). On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional caseII and III, Ann. Inst. Statist. Math., 24, 599-619; 25, 215-237. [ 6 ] Taguchi, T. (1973). Concentration polyhedron, two-dimensional concentration coefficient for discrete type distribution and some new correlation coefficients, etc., Proc. Inst. Statist. Math., 20, 77-115 (in Japanese). [ 7 ] Mardia, K. V: (1962). Multivariate Pareto distributions, Ann. Math, Statist,, 33, 10081015. [ 8 ] Taguchi, T. (1960). Concentration-curve methods and structures of skew-populations, Ann. Inst. Statist. Math., 20, 107-141. [ 9 ] Mahamunulu Desu, M. and Rodine, R.H. (1969). Estimation of the population median, Skand. Aktuar Tidskr., 67-70. [10] Cram~r, H. (1937). Random Variables and Probability Distributions, Cambridge at the University Press.