On the Concept of Tail-Heaviness*

By Javier Rojo

Mathematical Sciences DepartmentUniversity of TexasEl Paso, Texas 79968

Technical Report No. 175October 1988

Research partially supported by NSF Grants DMS 86-06420 and DMS 84-01388.

Department of StatisticsUniversity of CaliforniaBerkeley, California

On the Concept of TaillHeaviness*

By Javier Rojo

Mathematical Sciences DepartmentUniversity of TexasEl Paso, Texas 79968

American Math. Soc. 1980 Classifications: Primary: 62-02, Secondary: 62B15, 62E99

Key words and Phrases: Tail-orderings, quantile functions, exponential tials,polynomial tails, swiftly varying tails.

* Research partially supported by NSF Grants DMS 86-06420 and DMS 84-01388.

- 2 -

On the Concept of Tail-Heaviness

By Javier Rojo

Mathematical Sciences DepartmentUniversity of Texas

El Paso, Texas 79968

In the intuitive approach, a density f (x) is said to be lighter tailedthan g (x), when f (x) /g (x) -e 0, as x - oo. An alternative is toconsider the behavior of the ratio F-1 (u)/ G (u), in a neighbor-hood of one. The present paper examines the relationship betweenthese two criteria and concludes that, for a large class of distribu-tions, the intuitive approach is a refinement of the ratio of the quan-dle functions approach.

1. Introduction. The concept of tail-heaviness of a distribution function F per-meates both the theory and practice of statistics. Among others, the following exam-ples illustrate the importance of the concept. In the problem of estimating the locationparameter of a symmetnrc distrbution, tail behavior of the underlying probability distri-bution has a direct effect on the efficiency of the estimators of location. In extremevalue theory, the tail behavior of F determines the limiting distributions of the extremevalue statistics. In nonparametric density estimation, certain methods of selecting thesmoothing. parameter work well for short-tailed distributions, but the presence of amoderate outlier causes the methods to choose too large a smoothing parameter.

Until recently, the literature on tail ordering (e.g. van Zwet (1964), Barlow andProschan (1975), Lawrence (1975), Loh (1984), Caperaa (1988)) has been concernedwith orderings of the whole distribution, with the resulting ordering strongly affectedby the behavior of the distrbutions in the center.

The most common approach to a pure tail ordenrng defines tail weight in terms ofthe rate at which the density tends to 0 at infinity. More precisely, a density f is saidto have a lighter tail than g if

f (x)/g (x) -e 0, as x-*oo. (1.1)

With this "intuitive" definition, a normal density has a lighter tail than a logisticor double exponential density, which in turn are lighter tailed than a t distribution. An

altemative approach is suggested in Pazen (1979) and Lehmann (1988) -in terms of thedensity quantile functions. The latter author, defines the distribution F to be lightertailed than the distribution G if

(F-1 (u) - F-1 (v))/(G-1 (u) - G-1 (v)) .AM (1.2)

for 0 < v < u < 1, and some M > 0. When both F-1 and G-1 are differentiable, (1.2)reduces to gG-1(u)/fF-1(u) being bounded on (0,1). Then, as an altemative to(1.1), the following criterion which compares the densities at the same quantile, ratherthan at the same x, could replace (1.1),

gG-1(u)/fF-1(u) -* 0 as u -*1 (1.3)

When f or g do not exist, the ratio given in (1.1) and (1.3) may be replaced bytheir "integrated" versions,

(1 - F (x))/(I - G(x)) (1.4)and

F-1 (u)/G-1 (u). (1.5)

The definitions to be adopted in section 2 will be in terms of (1.4) and (1.5), and itwill turn out that, in a way to be made precise in sections 3 and 4, the definition interms of (1.4) is a refinement of that in terms of (1.5) for a large class of distributions.The purpose of the present paper is to give an account of this relationship. Section 2provides the definitions and considers some of their possible drawbacks. Section 3gives a result on the limiting behavior of the quantile function for distributions withpolynomially or exponentially decaying tails, and concludes that the definition in termsof (1.4) agrees with the definition in terms of (1.5) in the case of polynomial tails but,in the case of exponential tails, the former definition is a refinement of the latter. Thisraises the question of whether, in general, the two definitions agree except in the casethe definition given in terms of (1.4) is a refinement of that in terms of (1.5). Section4, in a way made precise there, shows that as long as the tail of the distribution doesnot vary too slowly, the answer to the question raised in section 4 is in the affirmative.Section 5 and 6 end the paper by comparing the two definitions in terms of invarianceunder monotone transformation and in the case the distributions have bounded support.

2. The Definitions. As pointed out earlier, the "intuitive" approach defines F aslighter tailed than G if (1.1) holds. More generally, if f or g do not exist,f (x) / g (x) may be replaced by its "integrated" version (1.4).

_ J _

- 4 -

DEFNITION 1. Let F and G be probability distribution functions. Then,

F < G if iii(1 -F(x))/(l -G(x))=c, c <.D x

F < G if F ' G but G i FD D

F - G if F . G and F t G.D D D

When the limit in (1.1) exists, the limit of (1.4), as x -+ oo, exists and they areequal. However, the limit of (1.4) may exist, e.g. when F and G are the Poisson andgeometric distributions, without the densities f or g even being defined.

Although definition 1 is satisfactory for some purposes, we mention two possibledrawbacks:

i) If F and G have finite support, both ratios in (1.1) and (1.4) are undefined forlarge x, and the definitions therefore do not apply.

ii) The definition is not location nor scale invariant. This is illustrated by the fol-lowing examples:

EXAMPLE 1. Let f (x) = ex, x >O and g (x) ex b, x > a, b >O. Then,f (x)/g (x)-e , as x - oo when b < 1, but f (x)/g (x) -+ 0 if b > 1. Whenb = 1, f (x)Ig (x) -+ e'.

-- (X-U)2-EXAMeLE 2. Let f (x)=(I/ acy)e 2 and g(x)

--(X -9)2/X2(1 /flit)e 2 . Then, the behavior of f (x )/g (x) is as follows:

0 if t2 > o2 or, if 2 = o2 and (0a2 > Lt;2limf (x)/g (x) = 00 if X2 < oY2 or, if r2 =C2 and 0a2 < 2

l 1 otherwise

Although for some applications, e.g. efficiency of point estimators, a normal distri-bution N (O,0 2) has lighter tails than N (0, 2a2), for other applications, e.g. selection ofthe smoothing parameter in nonparametric density estimation, a definition under whichall normal distributions are considered to have equally heavy tails, might be preferred.Also, the facts that f (x) / g (x) and (1.4) are undefined, when F and G have finitesupport, for large x, together with the lack of invariance under location and scaletransformations of definition 1, serve as the motivation for an alternative definition.One possibility is suggested by the work of Parzen (1979) and Lehmann (1988). Thelatter author defined F to have a lighter tail than G if

- 5 -

(F-(u) _F-1(v))1(G- (u) - G-(v)) M (2.1)

for 0<v <u < 1, and some M >0, where F1 (u)=inf(t:F(t) . u} and G-1(u)is similarly defined. When both F-1 and G-1 are differentiable, (2.1) reduces to

gG1-(u)/fF-1(u) < K, 0 < u < 1. (2.2)

Parzen (1979) calls the function fF-1 (u) the density-quantile function, and considersits limiting behavior as p -- 1 or 0 as a measure of tail weight. When M = 1, (2.1)has been named dispersive ordering in the literature and was originally proposed,under a different name, by Doksum (1969).

As our definition, we consider a slight variant of (2.1).

DEFINMON 2. Let F and G be distribution functions with inverses F-1 and G1respectively. Then,

F ' G if limFtI(,)/G-1(g) < 00q g1

F < G if F ' G but G I Fq q q

F -G if F < G andF k G.q q q

When G-'(p) CPO, as p - 1, and lim gG1 (p) /fF1 (p) exists, then

limF1 (p)/1G-1(g)= limgG-1(g)/fF1 (p), but again, F-1(p)/IG-1(g) may be

bounded without the density quantile functions even being defined. (e.g.F(x)= I -q' andG(x)=1 -qx with0< q I q2< l,x=X ,<3.A class of standard examples, illustrating definition 2, will be considered in the

next section. For now, the merits of definition 2 will be evaluated in light of thedrawbacks of definition 1.

i) If F and G have finite support, let pF = sup(x F (x) < 1) and1G = sup(x : G (x) < 1 ). Then, F-I (g) -+ pF and G1- (p) -> 1G, as 1 - 1,and hence F - G ifgF Q9G *.0. When, pF =0and 9G > 0, F < G, and

q qff PF = 0, JG < 0, F < G. In contrast, (1 -F (x))/(1 - G(x)) and

qf (x) /g (x ) are not defined for large x.

ii) When X - F, it can easily be shown that if Y = aK + b, a > 0, then, denotingthe distribution of Y by G, G-1 (p) = aF- (p) + b . Thus, if F-I () / G'1 (p.) isbounded in a neighborhood of 1, so will (aF 1 (,u) + b ) / (aG 1 (p) + b ), pro-vided that when b * 0, aG-1 (p) + b k+ 0. In contrast, recall examples 1 and 2.

- 6 -

A more detailed comparison of the two definitions, in terms of i) and ii) above,will be taken up in sections 5 and 6. For example, it will be shown that definition 2remains invariant under more general transformation than just scale and location.

This section ends with two examples that suggest a property of the q orderingwhich will be studied in more detail in sections 3 and 4. Namely, the q ordering maybe thought of as a smoothed out version of the D ordering in the sense that it ignoresthe low order terms in the tail of the distribution function.

EXAMPLE 3. (Parzen (1979)). Let 1 - F(x) = eX'SI and1 - G(x) = e-x2swith cl * c2, -1 < c1,c2 <1. Then,

(1 - F (x))/(l - G(x)) = eSiX(C2-Cl) (2.3)

Also, it is not difficult to see that F- (u) and G-1 (p) satisfy the equations-ln (1 - p) = F-I (p.) + c 1 sinF-1 (p.) and -ln (1 - .) = G-1 (p.) + c2sinG1 (p.).Therefore,

F-I(p) 1 + c2sinG-1(p) c1sinF-1(p.) (2.4)G-1()G-1 (g) G-1-(9)

Then, while (2.3) oscillates between eC2-C1 and ec-C2, (2.4) converges to 1.

EXAMPLE 4. Let f (x) = ¢(x), where ¢ (x) denotes the standard normal density

function, and letg (x)= 1Ix lex2/2 -00 <zx <00. Then,2

1-F (x) = 1-D(x) (l/Ii«x)exe2 -+0 a x -+ c.1-G (x) (1/2) e x212 (1/2)exx/2

Thus, F <G. On the other hand, gG.(g)=(1-p.)(-21n(2(1-p.))) 12 andD

fF1 (p.) (1 - p) (-2 In (1 - p))1/2. Then, lim F- ()/G- (p) = c > 0. It followsJ1-+

that F - G, and the q ordering ignored the low order term x present in the density ofq

G.

3. Distributions with polynomial or exponential tails. In this section the obser-vation made in examples 3 and 4 of the last section is studied in more detail for theclass of distributions with polynomially or exponentially decaying tails. Lemma 5gives the limiting behavior of F-1 (1p) and, as a consequence, theorem 1 concludes thatthe q and the D orderings agree for the polynomial tails case, but the D ordering is arefinement of the q ordering for the exponential tail case. Some examples are con-sidered to illustrate the results.

- 7 -

This section begins by recalling some definitions and results from the theory offunctions of regular variation. Recommended references are Bingham, Goldie andTeugels (1987), (BGT), and de Haan (1970).

DEFINMTION 3. A positive function h is said to be regularly varying at oo, withindex 8, -00 < 8 <oo, if for all k> 0,

lim h(x) -X (3.1)x 0h (x)

A regularly varying function h with 8 = 0 will be called slowly varying. If h is regu-larly varying with index 8, the notation h z R8 will be used. A rapidly varying func-tion h satisfies (3.1) with 8 = 00 or --.

LEMMA 1. A function h e R6 if and only ifx

h (x) = x6c (x)exp(f(e(t)/t)dt), (3.2)a

for some a > 0, some c (x) -* c e (0,00), and some e(x) - 0.as x -+ c.

When 8 = 0, so that h is slowly varying, and c (x) - c, h is said to be normalizedslowly varying.

LEMMA 2. (de Haan (1970)) If h is non-decreasing, h e R8, 0 86 C*, and ifh (x) -boo as x -+*, then the function h * defined by

h*(x) = inf(y: h (y) > x

is regularly varying with index 1/8. If h e Rsj, 8 < 0, and h (x ) -* 0 as x -+ co, withh nonincreasing, the function h ** defined by

h** (x) = inf(y : h (y) < 1/x)

is regularly varying with index -1/8.

LEMMA 3. (de Haan (1970)). Let hI, h2 e R& 0 < 8 < oo. Suppose h1, h2 arenondecreasing. Then, for 0 < c < oo,

lim =c <=> lncx^2(X) ~~~xIh2 (X)

For --c < 8 <0, an analogous result applies for nonincreasing functions.

- 8 -

Let h1 and h2 be nonnegative functions such that

hi (x)IxF- O and hI(x)x - o°°, as x- oo,for all E > 0, (3.3)

and h2(x)/ex o O and h2(x)ex e- 0, as x - oo, for all E >0. (3.4)

Let H1 be the collection of all hI satisfying (3.3), and H2 all h2 satisfying (3.4). Notethat in particular, H1 contains all slowly varying function, while H2 contains all regu-larly varying functions.

A large class of standard distributions may be described as having densities belong-ing to one of the following families:

f (x) h 1 (x)x--a a > 1 (3.5)or

f (x) h2(x)e-x, p > 0, (3.6)for some h 1 e H1 and h2 E H2, where g 1 (x) g2(x), as x -4 oo, will mean thatg 1 (x) / g2 (x) tends to a finite positive constant, as x -+ co. For example, the Cauchy,the t, the F and the Pareto distributions have densities of the form (3.5). The classdefined by (3.6) includes the normal, gamma, logistic, double exponential, and Weibulldistributions. One may speak of densities satisfying (3.5) or (3.6) as having polyno-mial or exponential tails respectively. These concepts are now defined more generallyin terms of the distribution function. The rest of this section assumes that F (x) < 1for all x.

DEFINMTION 4. A distribution function F is said to have polynomial tails if

1 -F(x) - hi Wx)x+l a> 1 (3.7)

for some h1 e H1. Also, F is said to have exponential tails if

l-F(x) - h2(x)e-xP, p >0 (3.8)

for some h2 H2-

It will be assumed that both hl(x)xa+l and h2(x)exP are nonincreasing. It isnot immediately obvious what the relationships between (3.5) and (3.7), (3.6) and(3.8), and their respective quantile functions are. To study this issue, it is first arguedthat, modulo a regularity condition on h, nothing is lost, within asymptoticequivalence, by assuming H1 to consist of all normalized slowly varying functions,and H2 to consist of all h such that h ((lnx)1/e) is slowly varying for all £ > 0.

- 9 -

LEMMA 4. Let h be nonnegative.

i) If h E HI and lim xh'(x) exists, the limit is zero and h is a normalized slowlyx-.40 h(x)varying function.

ii) If h e H2 and lim h'&) exists, for all e > O, theny e- h (y )yg"

the limit is zero and h ((Inx )liE) is asymptotically equivalent to a slowly varying func-tion.

PROOF.

i) Let c = lim xh'(x)/h (x). Suppose c < 0. Then, by L'Hopital's rule, theX -+00

- Inh(x)(upper) order of h (x), defined by imn ,is also c. By theorem 2.3.11 inXc Inx

BGT, there is g (x) e RC, with g (x) 2 h (x) such that lim g (x)Ih (x) = 1.x

Therefore,

xEh (x) = XSg (x)O(1)

= x + I (x)O(1), where 1 (x) is slowly varying at co.

It follows that limx, xh (x) = 0 for all e < -c contradicting (3.3). Hencec 2 0. A similar argument, considenrng limx-5h(x) instead, gives c <0, andhence c = 0.

To show that h (x) must be slowly varying, we follow BGT. Define 8(x) by

6(x) = h'(x) Integrating from xo to x, one gets that h (x) = k exp( 8(t) dt)l

where k = h (x0). It follows from lemma 1, that h (x) is generalized slowly varying.

ii). Let c = lim h')( . Again one argues by contradiction. Suppose c < 0.Y-+*0 h (y)ey5_1By L'Hopital's rule lim In (h ((mny)l1E)) = c and hence, by theorem 2.3.11 in

y-o. lnyBGT, there is g (x) e RC with g (x) . h ((lnx)l1) andlim g (x)/h ((ln x)111) = 1. Therefore, choosing <ce,x -*00

lim ex h (x) = lim e(lnY)vh ((lny)1E)X-400 y-*00

= lii e(In") eyc I (@) h ((lny)11-)y-** yC I (y)

= lime(InYy)'+clny I (y)O (1) (3.9)Y

- 10-

Now, for 0 <y < -c, e(lYy)"6+clny s y7 for all y sufficiently large, and hencesince 1 (y) is slowly varying, (3.9) equals 0, contradicting (3.4). Thus c > 0.Now suppose c > 0, and consider ex h (x), and carry a similar argument toconclude that c = 0.

Another application of theorem 2.3.11 in BGT gives that h ((lnx)l/) is asymptoti-cally equivalent to a slowly varying function. 0

As observed before, H1 contains all slowly varying functions, and, since everyslowly varying function is asymptotically equivalent to a normalized slowly varyingfunction, nothing is lost, within asymptotic equivalence under the assumption in i) oflemma 4, by assuming H1 to consist of all normalized slowly varying functions. Alsonote that any function h such that h ((In x)1/£) is slowly varying for all E > 0, satisfies(3.4) and hence, under the assumption of ii) of lemma 4, H2 may be assumed to con-sist of all h for which h ((lnx)l/) is slowly varying for all e > 0.

The following lemma connects the limiting behavior of, (3.5) and (3.7) and (3.6) and-(3.8), and their respective quantile functions. It will be instrumental in proving themain result of this section, theorem 1. In what follows, I will be the genenrc term fora normalized slowly varying function at oo.

LEMMA 5. Let F be a distribution function. Then,

i) (3.7) <=> F-1 (p) 1 ( )(1 - )-p/(.fl). If in addition the density f exists,

then (3.7) <=> (3.5).

ii) 1 - F (x) - h (x)e-, p > 0 for some h E H2 if and only if(F 1 (p.))P - (In! (1/(1-pl)) - ln(1 - p)) -e k, < k <0, as p - 1. It fol-lows that (3.8) => F1 (.)- (-In (1 - p.))l'P as p e 1

iii) 1 - F (x) - h (x)e'p <=>f (x) - h (x)e-xPxP-l.

PROOF.

i) To show that (3.7) holds if and only if F-1 (.) 1 (1/(1 - p.)) (1 p.)11l(a-1), it isfirst shown that lemma 3 remains valid when at least one of hI or h2 e R 8. The proofis a slight variant of the proof of lemma 3 given in de Haan (1970).

Suppose (3.7) holds and let U1 (x) = I (x)xa+l and U2(x) = 1 - F (x). Then,

lim U - c, 0 < c <00, by assumption. It is claimed that this impliesU2 (X)

lim - c/(),where U~ and Uj* are as defined in lemma 2. For supposeU~~~~*(x)~ ~ ~

not. Then, there is xn -+ co such that limUi (x -=b * c1/(a1) and, sincen oUj2 (Xn )

- 11 -

U1 e R,+l, it follows that

________ _ = c (3.10)n-,mU I(U2 (Xn))

On the other hand,

U1 (U;* (Xn)) 1 U1 \U1 (Xn))lim ** = lim ,n -U (U*2 (Xn)) C n -*oU2(U2* (Xn))

lim U I (U (xn))xn1 n______________ _ **(3.11)C lim U2(U2* (xn))xn

n -

Now, for U non-increasing and non-negative,U(U** (y)+0) 1 U(U** (y)-O) (3.12)U (U** (y)) yU (U** (y)) U (U** (y))

Also,ifU -g e RB,)8<0,thenforallx < 1,

1 . U(t - 0) < U (tx) U (tx) g(x) g (t)U.(t) U(t) g (tx) g (t) U(t) t-,

Hence, lim U (t - 0) = 1. In a similar fashion, it may be shown thatt-*0o U(t)

lim U(:t-O) = 1, and it follows from (3.12), since U* (x) -4 oo as x -+ oo, that

lim xU (U** (x)) = 1. This, coupled with (3.11) then implies thatx )00

lim U*1(U* (x) =c1- contradicting (3.10). Therefore, U* (x)- U* (x).n)-Ul(U2 (Xn))Lemma 2 then gives that (1 - F (x))** - I (x) xlI(al). But

(1 - F (x))** = inf(t: 1 - F (t) < l/x)= inf(t:F(t) >1-/x) - F-(1 -1/x).

Then, F-1 (j) - (1/(1 -p)) (1 - L)fl1(a1), as j -+ 1.

To show the converse, suppose F-1 (p) -1(1(1 - g)) (1 - as p 1.Then, (1-F (x))**- 1 (x)x 11-1). Define U I(x) = I (X)hI(a1) andU2 (x) = (1 - F (x))**. Using an argument analogous to the one above, it is con-

cluded that U"* (x) - U'* (x) as x -* oo. By lemma 2, U1 (x) = 1 (x)xal. It

remains to show that U (x) 1 1 .Now,U2 -F (x)'

U *(x) = inf(y:(1-F(y))* 2 x)

= inf(y :F1 (1 -1/y) > x).

- 12 -

But, F-1F (x) < x, so that U' (x) . 1/(1 - F (x)). Also, for any e> 0,

F-1 (F (x) + e) 2 x and hence U' (x)15 for 0<e< 1 -F(x).1l -F(x) -

Therefore 1-F (x) s U2 (X) . 1 F (X) - e. And setting £ = (1-F(x))2 andthen letting x oo, it is concluded that

u2 (x)lim U(x _

x C-1/(1-F(x))

That (3.5) implies (3.7) when the density f exists, follows immediately from an appli-cation of Karamata's theorem. (see, e.g., proposition 1.5.10 in BGT).

To prove that (3.7) implies (3.5), a variant of the monotone density theorem is used.The proof is short and follows that of BGT. It is included for completeness. In whatfollows, F (x ) denotes 1 - F (x).

Considerbx

F (ax)-F (bx) = Jf(t)dt for 0 < a < b <co.ax

Since f is eventually nonincreasing, for large x,

(b-a)xf (bx) < F(ax)-F(bx) < (b-a)xf(ax) (3.13)xpl (x) xfil (x) xp I (x)

where I is slowly varying at oo, and 3 = -a + 1. The middle term in (3.13) is

F(ax) aPI (ax) _ (bx) bP I (bx) c a-)(ax)P1 (ax) 1 (x) (bx)P l (bx) 1 (x)

Thus, the first inequality in (3.13) gives

li- f (bx) c (aP-bP) (3.14)x-*Ox l (x) b-a

Letting b = 1 in (3.14), and taking the limit as a 1 1, gives lim f (x) . -cr3. Ax-.30 xV11I (x)

similar argument, using the right inequality in (3.13) gives lim f (x) > -cx i+ x~1 (x)

Thus,f (x) I(x)x.ii) If 1 - F (x) - h (x)e-x , p > 0, some h E H2, then 1 - F ((lnx)11P) -

h ((lnx)11P)x-1 = (x)x-1. Let U1 (x) = 1 - F ((In x)11P) and U2(x) = I (x)x1.Since Ul(x) U2(x), using the arguments given in the proof of i),Ur (x) - Uj* (x), where Uj* (x) E R1. Since

U1j" (x) = inf{t : 1 - F ((In t)11P) < 1/x }

- 13 -

= inf(eyp : F (y) 2 1 - l/x

(F-1(I - lx))r=

then e(F1(111x))F -I(x)x and therefore, e(F1(1-1X) -ln(xI(x)) _C, 0<C <o; Itfollows that (F (1 - I/x)jP - {ln(x 1 (x))}- k, < k <00, as x -* oo, or

equivalently (F1 (gi))P - {ln (i (11(1 -p)) - ln (1 - p))} k, as p - 1. Conversely,if (F (p))P - (In(l (1/(1-p.))) - ln(1 - p)) -+ k, then e(F (11/Y)l/yi (y) . c

Letting U =e(F1(111Y)), U2=yi(y), it follows that U" (y) U' (y)=yi(y)But

Uj (y) = inf{t: e(F'(l/t)" 2 y)

= inf(- -: F-1 (s) 2 (In y)l/P} 11-s 1-F ((lny)11P)

where the asymptotic equivalence follows from an argument similar to the one used ini). Thus, 1 - F (In y)lIP) - 1 (y)y-1 and therefore,1 - F (t) - I (etp)e-tP = h (t)e-tp.To finish the proof note that (F-1 (p))P - {ln (1(1/(l -p))) - ln (1 - p)) -+ k implies(F-1 (,))P _ (ln (1(11(1 -p.))) - ln(1 -p1)) < 0

(-In1(1 - g)) {(-In(1 - g)) 0(Recalling that for slowly varying functions, I (y) 0O as y -+ oo, it follows that

Iny(F-1 (.))P / (-ln (1 -p)) -* 1.

iii) Let 1 -F(x)-h(x)e-xp, p >0, hE H2. Define I -F*(y)=1 - F ((In y)l/P) - I (y)f1 where I (y) = h ((In y)lIP). Then, lettingf * (y) = dF * (y) / dy, since f * is eventually non-increasing, another application of the"monotone density" argument, (see part i), gives f * (y) 1 (y)y2. Since

_1 1_- -1 1-f*(y) =f ((In y)lP)(lny)P y1l/p, it follows that f ((In y)11P)(Iny)P y _

h ((In y)l/p)y-2 and hence, f (t) - h (t)tP-l elP.Now suppose f (x) - h (x)xPl e'xP. Then for each £ > 0, there is A (e) so that for

all x > A (e), c - e ff(X)p - < c + e, some c, 0 < c <0. Thus, for allh (x)XP- X

large x,00 00 00

(c -e)fh (y)yP-l e-Ypdy < f (y)dy < (c + e)Jh(y)yP-l eY dy.x x x

But,co00

fh(y)yP'e-Ydy = ± | h ((lnt)1P)t-2dtX P e P

- 14 -

00

- 1i|jI (t)C-2dtP exp

where I (t) = h ((In g)lIP) is slowly varying. An application of Karamata's theoremthen implies that the last integral is asymptotically equivalent to 1 (exp)exp and hence1 - F (x) - h (x)exP. Q

As a consequence of the lemma, the main result of this section is obtained.

THEOREM 1.

i) For distributions given by (3.5) or (3.7), the q and D orderings agree.

ii) For distributions given by (3.8) or, distribution with densitiesf (x) - h (x)xP-' e-xp, p > 0, h E H2, the D ordering is a refinement of the qordenrng.

PROOF.

i) It is enough to consider distributions given by (3.7). Let 11 -G (x)- U2(x), where U1ERa and U2 ERp,lim sup (1 - F (x))/(1 - G (x)) = climsupUI(x)/U2(x),

X- X -4,

- F (x) - U1 (x) anda,4 0 < 0. Then,0 < c <o. Using

the proof of i) in lemma 5, limsupF-t(u)/G1'(u)=c * lim sup U** (x)/U** (x), where Ur and U2 are defined as in lemma 2,

and 0 < c*<<0. Thus, when a = , the result follows immediately from lemma3. If a <(, then, since 11(x)/12(x) is slowly varying, 1 (x)xPl2(x)12 00Xand hence G < F. By part i) of lemma 5, F- (u)/G-1 (u) -e o, as u - 1,

Dand hence G < F. The case a > ( is analogous.

q

ii) This follows immediately from ii) and iii) of lemma 5.

Theorem 1 gives an interesting relationship between the q and D orderings for alarge class of standard distributions, but it also raises the question of whether the Dordering is always a refinement of the q ordering except when they agree. To gainsome insight into the answer of this question, let us consider three examples. Eachexample will compare the q and D orderings but for different types of distributions.The first example looks at a pair of distributions with tails that decay more rapidlythan an exponential tail. The second one considers an intermediate case between poly-nomial and exponential tails, and the last one compares two distributions whose tailsdecay slower than a polynomial tail.

- 15 -

EXAMPLE 5. Let 1 - F (x) = eec and 1 - G (x) = e- , x > 1. Then(1- F (x)) / (1 - G (x)) -* 0 as x -* oo. Also, since F-1 (g) = ln (-ln (1 - i)) andG-1 (g) e('n (-In (1 - P))) , F-1 (.) / G-1 (p.) - 0 as p. - 1 and the q and D orderingsagree.

ExAMPLE 6. Let 1 - F (x) = e-(Inx ,x > 1 and 1 - G (x)= 1 - D(In x). Then,since f (x) / g (x) -* , for k > ' 2, F < G. Also, F-1 (p) = e(In()) and

DG-1 (p) = ebL Then,

F (1) = e(-In 1 - ))/k )-J.L)

G-1R

= exp{(-In (1 - p))1/2 ((-n (1 _.))l/k-lk2 -1 ))112

- 0 as p. - 1,

for k > 2, since I-1 (p) - (-In (1 -p))1/2. If 0 < k <2, g (x) andf (x)

G 0. Then again, the q and the D orderings agree.

F1 1

EXAMPLE 7. Let 1 - F (x)= x > e and 1 - G (x)= , x > e12.Inx 2lnxThen, (1 - F (x)) / (1 - G (x)) = 1/2 and, since F-1 (p.) = exp(1/(1 - p.)) andG-' (p) = exp(1/2(1 - p.)), G-'(p) /F- (p.) -* 0 as p. -* 1. Thus, while F G,

DG <F.

q

Examples 5, 6, and 7, suggest that perhaps the conclusions of theorem 1 hold truemore generally. Namely, the D ordering is a refinement of the q odering as long asthe tails of the distribution do not decay "too slowly". The next section will considerthis question in more detail. We end this section by illustrating the use of lemma 5with a table of probability distributions and their respective quantile functions.

- 16 -

Probability Distribution Density Quantile Function

ExponentialLogisticWeibull

Extreme ValueNormal

Double Exponential

CauchyPareto

c(1GX)C-, 0 <x <

e-X,x >0

eX/(l+eX)2, oo<x<Xocxc-1 e-X' IX > Ocx -e

,x>

exe -oo<X <

(1nGe-x212 -oo < x < °o(1/2)e7x/1+-,-o<x<oo

(1/7 1 (1 + X2), -°° < X <1o(X 1+1/0)-1l, X > I

1- (IRg)l/c-ln (1 -p)

ln (g / (1 -p))C(-l(l-g))I/c

in (-In (1 -J.)).In1(-I (I.-,)In2gp, p < 1/2

-ln(2(1 -p)),p.> 1/2tan (4- 1/2)

(1-jip

Example 1-9 are in Parzeneralized.

Probability Distribution

(1979). Using lemma 5, these examples may be gen-

Density Behavior of F-1 (L), as g -- 1

Gamma (a, 1)

t-distibution (n)

F-distribution (m,n)

Generalized Logistic

Generalized WeibullGeneralized Normal

Generalized Double Exp.Cauchy

+r/(al+) e x >0

n

-IAX2

M+m

(+nX) 2m

kixlIlex(1 +ex)2 ' 0@<X <0

kbc-le"% x > 1, c >0k.lae-X2200<X<C

kIxille, <<x <o

(1I/7)/(I+x2), -oo<x <00

Thus, for example, it follows from the table that, extreme value < normal < exponen-q q

tial < lognormal < Cauchy. Also, Cauchy < Pareto if > 1, but Cauchy > Pareto ifq q q q

, < 1. Otherwise, Cauchy - Pareto.q

1)2)3)4)5)6)7)

8)9)

10)

11)

12)

13)

14)15)16)17)

- -I (1-L)

- (1- )

_ (1 _g-21

- (-An (I _-))- (-ln (1 g))112

_ (i F-gl

- 17 -

4. The case of swiftly varying tails. After theorem 1, the issue was raised ofwhether the D ordering always agrees with the q ordering except in the case theformer is more refined. Examples 5, 6 and 7 suggested that as long as the tails of Fand G do not vary "too slowly", the answer to the above question is in theaffirmative. This section makes more precise this observation and concludes that, infact, for distributions whose tails do not vary too slowly, the D ordering is arefinement of the q ordering except where they agree.

We start by defining swift tail variation. The rest of this section assumesF (x) < 1 for all x.

DEFINITION 5. A distribution function F is said to have a swiftly varying righttail, (F e SYT), if for each e > 0, there is t > 1, such that

1-F(tx) < e foralllargex. (4.1)1 -F (x)

It is easy to see that if F has polynomial or exponential tails, then F e SYT. Twodistributions of practical interest that do not satisfy (3.7) or (3.8), but satisfy definition5, are the lognormal and extreme value distributions, which are considered in examples5 and 6. Examples of distributions with tails which are not swiftly varyinrg are givenby 1 - F (x) = I (x), where I is a positive non-increasing slowly varying function. Itwill be shown below that in fact, modulo asymptotic equivalence, these slowly varyingtailed distributions are the only examples of distributions which do not have swiftlyvarying tails. First, some terminology and results from the theory of functions of regu-lar variation are introduced.

DEFINITION 6. Let f be positive.i) (After Matuszewska (1964)). The upper index a (f ) of f is the infimum of thoseoc for which there is a constant c = c (a) such that for each A > 1,f(Qx)/f (x) < c (1 + o(1)}X, x -o o, uniformly in k E [ 1,A]. The lower index,f (f ), of f is the supremum of those [ for which, for some D = D ([) > 0 and allAt > 1, f (kx )/f (x) > D (1 + o (1)) XP, x -- c*, unifomnly in x e [ 1,A]ii) The upper order p (f ), and the lower order p (f ) are defined by

p(f) jim- Inf (x) p(f = lim Inf (x)mC nx x-+oo lnx

iii) f is said to be almost increasing if there is X and a constant m > 0, such thatf (y) 2 m f (x), y 2 x 2 X. Almost decreasing is similarly defined.

The following lemma relates the indices a (f ) and [3(f ), to the lower and upperorders p (f ) and p (f ). It also gives a characterization of a (f ) and [ (f ).

- 18 -

LEMMA 6. (See theorem 2.2.2, proposition 2.2.5, and theorem 2.3.11 in BGT)

i) 3(f) .C(f) < p(f) < a(f).ii) ca(f ) = inf(a: xaf (x) is almost decreasing)

13(f ) = sup([3: x-f (x) is almost increasing)iii) Suppose p (f) is finite. Then, there is g e R ,P with g (x) > f (x) for all x, andlim g (x)If (x) = 1.x )00

It follows from iii) of lemma 6, that if p (F) = 0, then F I (x), whereF = 1 - F (x) and I (x) is slowly varying, and hence F 4 SIT. On the other hand, ifp (F) < 0, then using i) and ii) of lemma 6, there is 1B < 0 such that x-PF (x) is notalmost increasing. Hence, for all m > 0, all A, and y > x > A,(y/xf)- F (y) IF (x) < m. Settang y = \x, X > 1, it follows that F e SVT.Recalling that slowly varying functions I (x) have p (1) = 0, and noticing thatp (F) < 0, the following theorem has been proven.

TBEOREM 2. Let F be a distribution function. Then, F e SVT <=> p (1- F) < 0<=> 1 - F not asymptotically equivalent to a slowly varying function.The following lemma will be instrumental in proving the main result of this section.

LEMMA 7. Let F and G be distribution functionsi) If F E SVT, then, for 0 < E < 1, F-I (1 - e(1 - g))/F (p.) is bounded in a neigh-borhood of 1.

ii) 1 - F (x) < 1 - G (x) in a neighborhood of oo <=> F-1 (p) < G-1 (p) in a neigh-borhood of 1.

iii) lim F1()G 0 (1 - F (x))/(1 - G (x)) < 1, go = G (xo).9--+0 x 4xo

PROOF.

i) Given 0< c < 1, there is t > 1, such that (1 - F (x))/(l - F (x)) < e for all largex. Since F-1 (ll) - oo, as - 1, set x = F-1 (±). Then,F (tF-1 (p)) > 1 - e (1 - FF-1 (p.)). Thus, F-1 F (tF1 (p.)) .F-1 (1 - e (1 - FF-1 (p))), which implies, since F-1 F (s) < s and FF-1 (s) > s, thatt > F-1 (1 -e (1 -p)) IF-1 (p).ii) -F (x) < 1 - G (x) for x > A <=> F (x) > G (x) for x > A

<() F-I (g) < G-1 () for 0 > Fsp (A ).

iii) Suppose lim F _))= 0 and suppose lim - F (x) > 1. Then, there is xai lloGt1(X X-sXO 1- G (x)

arbitrarily close to xo such that (1 - F (x))/I(1 - G (x)) > 1. It follows that

- 19 -

G1 (g)/F-' (p) s 1 contradicting the fact that F1 (9)/G1 (p.) -* 0, as p -- p. 0

The following is the main result of this section. It confirms the observation madeearlier. Namely, the D ordering is a refinement of the q ordering, except where theyagree, as long as the tails of the distributions do not vary slowly.

THEOREM 3. Let F and G be distribution functions with F or G e SVT. Then,i) F < G =>F < G andF - G => F -G

D q D q

ii) If lim (1 -F (x))/(1 -G (x)) exists,F < G =F < G.x-o q D

PROOF.

i) Suppose F e SVT. When lim 1 _ F (x) < °l F (x) < k for all large xX-.0 l -G (x) Il-G (x) .kfrallrex

some k > 1. (The case k = 1 was covered by lemma 7 ii)).It follows that 1 - F (x) < k (1 - G (x)), and setting x = G-1(), that1 - FG-1 (p) . k (1 - GG-1 (p.)) all p close to 1. Therefore,FG-1 (p.) 2 1 - k (1 - GG-1 (.)) F-1FG-1 (p.) 2 F-1 (1 - k (1 - GG-1 (p.))).

But F-1 F (x) < x and GG-1 (p) 2 g. Thus,

G-1 (p) > F-1 (1 - k (1 - 9)) for all p close to 1.

It follows that F- k)

- G 1 . The result now follows from lemma 7F-1(1- k (1 -p)) G~1(p.)

upon changing variable to s = 1 - k (1 - p.) in the last inequality.1 -F(x0)

Now suppose G E SVTT. If for xo l G < 1, then G-1 (po) 2 F-1 (p.o), where

po = F (xo). Thus, attention may be restricted to the values of x for which1< 1-F(x) 1 1Gx1< - G(x) < k, some k > 1. It follows that -< <1, and by the

argument above, G-' (p.) F-1 (p) is bounded, and so will F1 (p) / G-1 (p) provided

G-l(p.)1F-'(p) stays bounded away from zero. But if lim G (p1) -0,W-Fo F~1 (p.)

1-F (x) 1R-F(x)lim G 1 by iii) of lemma 7, contradicting the fact that 1 F(x)> 1.x--*xo 1- G (x) 1 -G (x)Therefore, if F or G e SVT, F < F < G; Reversing the roles of F and G, it

D qfollows also that F- G =F - G.

D q

ii) We first show that if lim F-(p) =0 then lim 1-F(x) =0 Suppose

G e SVT and F-1 (p) / G-1(p) - 0. Choose e > 0, t > 1, so that

- 20 -

(1-- G (at))I(1 - G (x)) < e for all large x. Also choose g0 so thatF-1 (p) /G 1 (p) < l/t for all p > go. Then,

F 1(p) < G-1(A)/t <-> F-1(g) . G *- ()<=> F (x) > G*(x)

where G * (x) = P (Ylt < x), and Y - G. Thus, F-I (p) < G-' (p) / t if and only if1 - F (x) . 1 - G (tx), and then

(1 - F (x))/(l - G (x)) . (1 - G (x))/(l - G (x)) < e

for all large x.

If F E SVT, a similar argument gives 1 - F (xlt) . 1 - G (x) for al large x, and thenl - F(x) 1 - F(x) 1 - F(xlt) 1 -F (x)1 -G (x) 1 - F (xlt) I -G (x) 1 - F (xlt)

for all large x.

We now conclude that if lim 1 F (X) exists, then F < G => F < G.x 1-OG (x) q D

F1 p. G-' (p)To see this, note that F <G implies lim F < oobut lim G = . Tus,q ±-+1 G-1(') W41 F-1 (p)

select a sequence Pk -+ 1 so that G1 (t . Then F-1 (p)/G1 (A) -+ and,F1I (p.k)'co%F

using the first half of the proof, it follows thatI F ) < e for all large k. Since1 - G (xk)

lim (1 - F (x))/(l - G (x)) exists, it must be 0. O

We end this section by illustrating, by means of examples, that in general the con-verse to ii) of lemma 7, and i) and ii) of theorem 2 do not hold.

EXAMPLE 8. Consider example 4 where 1 - F (x) = 1 - (b(x) and1 -G (x)= e-x2/2 Then F-1(p)/G-1(p) - 1 while2(1-F (x))/(l - G (x)) -+ 0.Since 4 and G e SVT, the converse to ii) of the theorem is not true. Reversing theroles of F and G, also shows that the converse to i) is not true. It also followsimmediately that the converse to ii) of lemma 7 does not hold.That the sufficient condition that F or G e SVT in theorem 2 part ii) is not necessary,follows from the following example.

EXAMPLE 9. Let 1 -F (x)=- and l- G (x) = , x >e. Thenlnx (lnx)2

- 21 -

F d SVT, G d SVT and both (1 - G (x))/(1 - F(x)) and G-1 (I)/F-1 (f) go to zero.HenceF <G andF <G.

q D

5. The case of bounded support. The motivation for a new tail ordering arosefrom the need to have a location and scale invariant ordering, and the nonapplicabilityof the D ordering when both F and G have finite support. The issue of invariance ofthe orderings under monotone transformation will be taken up in the next section. Inthis section, the q ordering, although applicable, is found to provide an unsatisfactoryanswer when F or G have finite support. An alternative ordering, q*, is defined tohandle this case.

For a distribution function F let hF and FA* be defined as follows:RF = sup(x : F (x ) < 1) and FA* = F (x) /F (A ), for x < A . Three different situationswill be of interest here: i) Comparison of two distrbutions with finite support, ii)Comparison of a distribution with finite support with one having infinite support, andiii) truncation of two distributions at a common point A.

i) Comparison of two distributions with finite support.Let F and G have finite support with RF * 0 and hG . 0. Then F - G. It is

qintuitively clear that this may not be satisfactory when RF <IG, since there is positiveprobability under G, and zero probability under F, of observing values in the interval(9F, IG ] and therefore one would like to conclude that F < G.

Even when RF = hG the q ordering may not give a satisfactory answer as illus-trated by the following example.

EXAMPLE 10. Consider the family of "triangular" distributions defined byfa(x) = ca(1 - Ix la), Ix I < 1, a > 0, and the family of "inverted" triangular distri-bution given by ga(x) = baIx la, Ix I < 1. Then, since IF = G' Fa - Gac, all a> 0.

qBut 1 - Ga (x) > 1 - Fa (x) for all a > 0 and all 0 < x < 1. Thus, eventhough thereis higher probability of observing values close to 1 under Ga than under F.a the qordering can not differentiate between Fa and Ga.

Since 1 - Ga(x) > 1 -Fa(x), 0 x <1 is equivalent to Fa1 (p.) < Gal (p.) for

all - < A < 1, one may consider an alternative to the q ordering defined as follows:2

DEFINITION 6. For distributions F and G, F < G if there is go < 1 such thatq

- 22 -

It is easy to see that when F or G are continuous and strictly increasing, the q * order-ing is equivalent to the D * ordering which is defined next.

DEFNTION 7. If RF < G, F < G; If RF = G,then F < G if there is b < RFsuch that 1 - F (x) < 1 - G (x) for all x E (b,4F).

Returning to example 10, it follows that Fa < Ga and Fa <* Ga. Also, accord-

ing to definitions 6 and 7, F < G and F < G whenever RF <PG. Thus, in the caseq* D

of both distributions having bounded support, definitions 6 and 7 provide a satisfactoryanswer.

ii) Comparison of a distribution with finite support with one having infinite support.If<00° with F= oo, then G < F, G < F and G < F. That this may not be

satisfactory is illustrated by the following example.

EXAMPLE 1 1. Let F = ', the standard normal distribution; then C* < F, whereq

C is the Cauchy distribution. But there is higher probability of observing values closeto A under CA than under F, for all large A, and thus one would like to say that, atleast for sufficiently large A, F <CA*.

It seems difficult to find a satisfactory solution in this case. Our suggested solutionconsists in truncating the normal distribution at the same point A and then applydefinition 6, in which case, for all large A, A > x0, where x0 is the unique positivesolution of x (1 + x2) = '2cexp(x2/ 2)), we will obtain FA* S CA*.

qThus, in the general situation with PF = , PG <, truncate F at gG. and concludeG < F if there is Pg < 1 such that G-' (;) < FU(* ) for all E (po, 1).

q

One interesting issue that arises from the discussion above is to determine ifF < G => FA* < GA* for distribution with infinite support. This problem is discussed

q qnext.

iii) Truncation of two distributions with infinite support at a common point A.Consider distributions F and G with 1F = RG = . It is of interest to determine

if

F < G= F < GGA (5.1)q q

To see that (5.1) is not true for all A, consider again the case F = (D and G the Cau-chy distribution. It can easily be shown that choosing A so that 0 < A < 1 andG(A)lD(A)>I4,then4(x)/D(A)>g(x)/G(A)forall0<x <A andtherefore1 - oI (x) / D (A ) > 1 - G (x) /G (A) which implies GA* < 4;. However, the

q

- 23 -

following is true.

THEOREM 3. Suppose F < G with lim f (x) /g (x) < 1, where f and g are theq x

density functions of F and G respectively. Then FA* < GA* for all A large enough.q

PROOF. Since lim f (x) /g (x ) < 1, there is a neighborhood B of infinity such thatx-40

f (x)/g (x) < 1 - c, some e >O, all x e B. This implies that F (x) > G (x) forx E B and therefore, truncating and rescaling at any interior point b of B gives,fb*(x) < g* (X) in some interval (b - 8, b) and hence Fb*(x) > Gb*(x) which in turnimplies Fb*' () < G 7' (,u) for g in a neighborhood of 1. It follows that Fb* < Gb*.

qThe theorem is now illustrated by the following example.

ExAMPLE 12. Let F = (, G is the logistic distribution, and H is the Weibull dis-tribution with parameter c > 2. Then, since 4(x)/g (x) -4 0 and h (x)/ (x) -4 0,then, for sufficiently large A, HA* < IDA < GA*.

q q

6. Invariance under monotone transformations. It was argued in section 2 thatthe q ordering remains invariant under location and scale transformations. This sec-tion concludes that both the q and q* ordenrngs also remain invariant under continu-ous nondecreasing transformation. Some examples of the use of the following theoremconclude the section.

THEOREM 4. Let X and Y be random variables with distribution function F andG respectively. Let h denote a continuous nondecreasing function.i) Suppose h (s (x)) = 0 (h(t(x))) as x -* 1, whenever s (x) = 0 (t(x)) as x - 1,and let H1 be the collection of all such h. Then,

X < Y <=> h (X) < h (Y) for all h E H1.q q

ii) Let H2 consist of all h e H1 such that lim h (s(g))/h (t()) =oo whenever1L-1

lim s (g)/t(g) = oo. Then, X < Y <=> h (X) < h (Y) for all h e H2.g--*l q q

iii) Let H3 consist of all h continuous and strictly increasing. Then

X < Y <=> h(X) < h(Y) forallheH3.q q

PROOF. The if part of i, ii) and iii) follows immediately by taking h (x) = x. Theonly if parts also follow easily from the conditions on h and the fact that if X F,Y = h (x), then G-1 (g) = h F-1 (p), where G is the distribution of Y.

- 24 -

Examples of functions satisfying the conditions in i) are functions with boundedderivative, and more generally, functions satisfying a Lipschitz condition withh (x) 2 x for all x. Part iii) of the theorem accepts the following generalization: Leth1, h2 be continuous increasing function, with h1 (x) > h2 (x) for all x, h 1, h2 nonne-gative in a neighborhood of oo. Then,

X < Y <-, h I(X) < h2(y)q q

The use of the theorem is now illustrated by the following examples.

EXAMPLE 13. Let F = D and G be the extreme value distribution. SinceG1 (j)/F1 (p.) e 0 as . -e 1, Then G < F. It follows that eX < eY, where

q qX - G and Y - F. Thus,

exponential < lognormalq

EXAMPLE 14. Let F = 4' and G be the exponential distribution with scale param-eter X. Since f (x)/g (x) -+ 0, X < Y, where X - F, Y - G. Again it follows that

X ~~~~~~~~~qeX < e . But eY has distribution 1 - 1/xX. In particular, when x = 1, this gives the

qreciprocal of a uniform (Parzen (1979) terminology). For general X > 0, it follows that

lognormal < Pareto.q

Acknowledgements. The author wishes to express his indebtedness to Erich L. Leh-mann for his many insightful remarks that led to a much more cohesive exposition.

REFERENCES

BARLOW, R.E. and PROSCHAN, F. (1975). Statistical Theory of Reliability and LifeTesting. Holt, New York.

BINGHAM, N.H., GOLDIE, C.M. and TEUGELS, J.L. (1987). Regular Variation.Encyclopedia of Mathematics and its Applications. Cambridge University Press.

CAPERAA, P. (1988). Tail ordering and asymptotic efficiency of rank tests. Ann.Statist. 16 470-478.

DE HAAN, L. (1970). On regular variation and its application to the weak conver-gence of sample extremes. Math. Centre tracts #32, Amsterdam.

DOKSUM, K. (1969). Starshaped transformations and the power of rank tests. Ann.Math. Statist. 40 1167-1176.

- 25 -

GELUK, J.L. and DE HAAN, L. (1987). Regular variation, extensions and Tauberiantheorems. Centre for Math. and Computer Science, #40, Amsterdam.

LAWRENCE, M.J. (1975). Inequalities of s-ordered distributions. Ann. Statist. 3413-428.

LEHMANN, E.L. (1988). Comparing location experiments. Ann. Statst. 16 521-533.LOH, W.Y. (1984). Bounds on ARE's for restricted classes of distributions defined

via tail orderings. Ann. Statist. 12 685-701.MATUSZEWSKA, W. (1964). On a generalization of regularly increasing functions.

Studia Mathematica 24 271-279.PARZEN, E. (1979). Nonparametric statistical data modeling. J. Amer. Statist. Assoc.

74 105-131.VAN ZWET, W.R. (1964). Convex transformations of random variables. Math. Cen-

tre Tracts #7, Amsterdam.