August 1980
/
TESTING EXPONENTIALITY
AGAINST HNBUE
by
Bengt Klefsjö
American Mathematical Society 1970 subject classification: 62G10, 60K10.
Key words and phrases: Life distribution, HNBUE, HNWUE, exponential distri­
bution, TTT-transform, hypothesis testing, efficiency, consistency, power.
This is a revised version of Sections 5 and 6 in Statistical Research
Report 1979-9, Department of Mathematical Statistics, University of Umeå.
ABSTRACT
Let
F
be a life distribution with survival function
finite mean
00
—
y = /q F(x)dx. Then
F
than used in expectation (HNBUE) if
If the reversed inequality is true
F = 1 - F
and
is said to be harmonic new bet ter
00
—
/ F(x)dx < y exp(-t/y)
F
for
t > 0.
is said to be HNWUE (W = worse).
We develop some tests for testing exponentiality against the HNBUE (HNWUE)
property. Among these is the test based on the cumulative total time
on test statistic which is ordinarily used for testing against the IFR (DFR)
alternative. The asymptotic distributions of the statistics are discussed.
Consistency and asymptotic relative efficiency are studied. A small sample
study is also presented.
1.
1.
Introduction
Several classes of life distributions (i.e. distribution functions
with
F
F(0 ) = 0) based on notions of aging have been proposed and studied
during the last decades. The most well-known of these are the
(i)
IFR (increasing failure rate)
(ii)
IFRA (increasing failure rate in average)
(iii) NBU (new better than used)
(iv)
NBUE (new better than used in expectation)
(v)
DMRL (decreasing mean residual life)
classes with duals. For definitions and relations see the Appendix and for
further details see e.g. Haines (1973) and Barlow and Proschan (1975).
Rolski (1975) introduced a new class of life distributions named HNBUE
(harmonic new better than used in expectation) with dual HNWUE (W = worse).
The purpose of this paper is to suggest some test statistics for testing
the hypothesis
HqI
F
is the exponential distribution
H^:
F
is HNBUE (HNWUE) but not exponential.
against
In Section 2 we summarize the definitions and some basic facts about the
HNBUE and HNWUE classes of life distributions .
In Section 3 we study some test statistics. Among the tests which are
consistent against the class of continuous HNBUE (HNWUE) life distributions
we shall find the test based on the cumulative total tiiae on test statistic
(see e.g. Barlow et al. (1972), Chapter 6).
In Section 4 a small sample study is presented.
2.
2.
The distribution class HNBUE and its dual cl ass HNWUE
We have the following definition.
DEFINITION 2.1
—
F = 1 - F
A life distribution
with finite mean
F
and its survival function
OO —
y = /q F(x)dx
are said to be harmonia new
better than used in expectation (HNBUE) if
(2.1)
OO
/ F(x)dx < y exp(-t/y)
for
t > 0.
•f-
If the reversed inequality is true
F
and
F
are said to be harmonic new
worse tha n used in expectation (HNWUE).
•
The reason for the name HNBUE is the following. Suppose for simplicity
that
F(t) > 0
for
t > 0 and let
OO
e„(t) = / F(x)dx/F(t)
t
denote the mean residual life of a unit of age
t. Then the inequality (2.1)
can be written
(2. 2)
—
t
-
/ (e„(x)>
F
0
< u for
t > 0.
1dx
This inequality says that the integral harmonic mean value of
less than or equal to the integral harmonic mean value of
is
e (0) (i.e. of
r
a new unit).
The class of HNBUE (HNWUE) life distributions was introduced by Rolski
(1975). Klefsjö (1977, 1980a, 1980b) studied the HNBUE (HNWUE) class in more
detail. Here we only summarize some basic facts which we need later on.
For further facts see e.g. Klefsjö (1980b).
3.
The HNBUE (HNWUE) class properly contains the NBUE (NWUE) class of
life distributions.
If
F
is HNBUE (HNWUE) with mean
y
then
00
(2.3)
/ FV(x)dx > (<)
0
for
v = 2, 3, 4, ...
and
00
v 1
/{I - FV(x)}dx < (>) y E -4
0
j-1 J
(2.4)
for
v = 2, 3, 4
This is a consequence of Theorems 7.3 and 7.4 in Barlow and Proschan (1975),
Chapter 4; cf. Klefsjö (1980b), Theorem 2.1. Furthermore, it can be proved
that if
F
is HNBUE (HNWJE) with F(0) f 0 but not exponential equality can
not hold in any of (2.3) and (2.4). In fact the inequalities (2.3) and (2.4)
are true also for non-integers
v. This can be proved in the same manner
as Theorems 4.8 and 4.8' in Barlow and Proschan (1965), Chapter 2.
Some of the aging properties
translated to properties of the
(i)-(v)
on p. 1 (and their duals) can be
scaled Total Time on Test (TTT-) transform
defined by
(2.5)
ttuCt) =
Cx(t)
• •.
Hp (1)
for
0 < t < 1 ,
" "
where
_i
F_1(t)_
H (t) =
/
F(x)dx
F
0
and
F_1(t) = inf{x: F(x) > t} .
For instance
and
F
F
is IFR (DFR) if and only if (P_(t)
is NBUE (NWUE) if and only if
r
cp (t) > (<) t
r
—
—
is concave (convex)
for
0 < t <
—
—
1; see
Barlow (1979), Bergman (1979) and Klefsjö (1979). Using (2.1) and the fact
OO
that H Cl) = / F(x)dx = y we get that a strictly increasing life distriF
0
bution F is HNBUE (HNWUE) if and only if
4.
(2.6)
<Pp(t) > (<) 1 - exp(-F ^tj/y)
for
0 < t < 1
(see Klefsjö (1980b)).
3. Some tests against th e HNBUE (HNWUE) alternative
In this section we shall study four different test statistics for testing
the hypothesis
Hq5
F
is the exponential distribution
H^:
F
is HNBUE (HNWUE) but not exponential.
against
Statistics for testing
against IFR (or IFRA, NBU, NBUE, DMRL) have
been suggested by several authors. See e.g. Bergman (1977), Hollander and
Proschan (1972, 1975), Koul (1977, 1978), Klefsjö (1979) and references
given in these papers.
Suppose that
0 = t(0) < t(l) < t(2) < ... < t(n)
from a continuous life distribution
F
is an ordered sample
with finite mean
y =
00
Further let
Dj = (n-j+1)(t(j)-t(j-l))
for
j = 1, 2,
n
denote the normalized spacings and let
k
S, = E D.
K
• i JJ
j=i
Note that
for
k = 1, 2, ..., n.
Sn s £?
-D.
,t(j).
We shall also use the notation
J
j=l
j = E?
j=l
Sk
u^ = —
n
for
k = 1, 2, ..., n.
—
F(x)dx.
5.
3.1
The test statistic
A^
Let
00
(3.1)
Then
B(x) = — / F(s)ds - exp(-x/y)
^ X
MB(x)
< 0
for
x > 0"
p. 2). Further, if F
least some
(3.2)
for
is equivalent to
x > 0.
"F is HNBUE" (cf. (2.1),
is HNBUE but not exponential then
B(x) < 0
for at
x > 0. Therefore
A (F) = / B(x)dF(x)
1
0
is a measure of deviation from
(1966) the sample analogue of
Hq
to
A^(F)
H^. Following an idea by Crouse
is the basis of our first test sta­
tistic. Substituting the empirical distribution function
F^
for
F
in
(3.2) we get
(3.3)
00
n
1
A. = VF) = / B (x)dF (x) = - I B (t(k)) ,
JL
X ri
* n
n
n .k=l
- n
0
where
B (t(k))
B
~ f
F (x)dx - exp(-t(k)/u ) ,
n t(k) n
1 n
yn - ± i t(j)
n
n j-i
and
oo
n
f
F (x)dx •ED. - —(S - S, )
n j=k+l J
n
n
k
t(k) "
(cf. Figure 3.1, p. 6). From this we get the test statistic
(3.4)
=
n
^
~\~
exP(~fc(k)•
Here a negative value of large magnitude of
A^
indicates that
is HNBUE. In a similar way we get that a large positive value of
indicates that
F
is HNWUE.
F
A^
/ F (x)dx » (t(k + 1) - t(k)Yl
V
/\
t(k) O
'WW-
W
+ ... + (t(n) - t(n - 1)Yl -—
\
A
"
ün-i)
Figure 3 .1
One disadvantage with
A^
is of course its rather complicated struc­
ture. For instance it seems difficult to get a simple expression for the
distribution of
of
A^
A^
under
H^. Furthermore, the asymptotic distribution
is still an open question. The consistency against the class of
continuous HNBUE (HNWUE) life distributions will now be proved.
If
F
is continuous then
1
—
n
n
Eu.
k-l^
1
/ cp_(t)dt
0 F
(see Koul (1978)), where (p
r
almost surely (a.s.)
when
n
00
denotes the scaled TTT-transform of
F
defined by (2.5).
00
Further,from the Law of Large Numbers and the fact that
y
-»•
n
a.s. when
n
1
1
—
00
we
get that
n
"" X /"Li
£ exp(-t(k)/y ) ->• / e
dF(x)
k-1
m
= / F(x)dx
0
00
a.s. when
n -* °°.
Therefore
A.,
1
with
B(x)
1
1 - / cp (t)dt - / e
0 F
0
as in (3.1).
/
^dF(x) = f B(x)d F(x)
0
a.s. when n
°°,
Since
/ B(x)dF(x) is strictly positive when
0
HNWUE (but not exponential), equal to zero if F
strictly negative if
F
F
is continuous and
is exponential and
is continuous and HNBUE (but not exponential)
the consistency follows.
Here we reaark that
A^
could be motivated in another way too. From
(2.6) it follows that
1 - tGp(t) - exp(-F-1(t)/y) < ( >) 0
if
F
for
is HNBUE (HNWUE), with inequality for some
t
0 < t < 1
if
F
is not expo­
nential. Therefore we expect the empirical scaled TTT-transform (p^
defined
at j/n by
Hn1(j/n)
<P_(j/n) = —:
H~ (n/n)
for
j =0, 1, ..., n,
for
0 < t < 1
-
where
F"1(t)
-1
H (t) =
/ F (x)dx
n
n
0
to satisfy the corresponding inequality, i.e. that
(3.5)
1 - <PQ(j/n) - exp(-Fn1(j/n)/yn) < (>) 0
Adding the left hand sides in (3.5) and using that
F 1(j/n) = t(j) (cf. Klefsjö (1979)) we get
n
3.2
The test statistic
Now let
J
j =0, 1, ..., n.
<PQ(j/n) = Uj
and
A.
i
A^,
be a function which is increasing and continuous on [0,1]
Further let
T(J,F) = / J(<f>_(F(t)))dF(t)
F
0
where
for
is the scaled TTT-transform of F.
8.
From (2.1) it follows that
> G(t) for
(3.6)
if
F
F
t > 0, where
F
is HNBUE if and only if
G(t) = 1 - exp(-t/y). Since
J
t
f F(x)dx/y >
0
is increasing
T(J,F) > I J(G(t))dF(t)
" 0
is HNBUE (and the reversed inequality is true if
F
is HNWUE). When
is a life distribution which is HNBUE (HNWUE) but not exponential we
00
therefore expect
T(J,F)
to be larger (smaller) than / J(G(t))dF(t). Accor­
ti
dingly, we get a family of test statistics for testing exponentiality
against HNBUE (HNWUE) by studying
oo
n
T(J,Fn) = / J((p&(Fn(t)))dFn(t) = £ I
Since
un = 1
J(Uj) .
we can equivalently study
_1
1 nI
T (J) = i
J(u.) .
j"l
Statistics of this form are named soove statistias
and are studied e.g.
by Barlow and Doksum (1972) and Koul (1978) for testing exponentiality
against IFR and NBUE, respectively.
Koul (1978) proved that
00
T_(J)
n
if
J
/ J(<p„(F(t)))dF(t)
F
o
a.s.
when
n -*• °°
is continuous on [0,1]. This means that a test based on
is consistent against every life distribution for which
00
(3.7)
T(J,F) > / J(G(t))dF(t).
0
We shall now prove that the inequality (3,7) holds if
F
is a life distri­
bution which is continuous and HNBUE (but not exponential) and
increasing.
J
is strictly
9.
Let
1 t _
K(t) = - / F(x)dx - G(t)
y 0
Since
F
is continuous
and in particular
K
K
for
has the derivative
is continuous. If
we can find an interval [a,b]
K(a) = K(b) = 0. Then
F
t > 0.
F
such that
K'(t) = (F(t) - G(t))/y
is HNBUE but not exponential
K(t) > 0
for
a < t< b
and
is non-constant in [a,b] since otherwise
K'(t)
is strictly increasing
in [a,b] which is impossible. Accordingly, we
can find
tg
for which
Since
is continuous there is a subinterval
F
F
and
t^
^Ct^) < F(t^)
I
and K(t) >0
of [tg,t^]
in
]tg,t^[.
in which
is strictly increasing. Accordingly,
i ^ _
cp(F(t)) = — / F(x)dx
y 0
for
t C I.
Therefore
00
/{ J(tp(F(t))) - J(G(t))}dF(t) > /{ J(tp(F(t))) - J(G(t))}dF(t) > 0.
0
I
It is well-known that, under H0, un, u0, ..., u ,
ct
1
2'
n-1
have the same distri­
bution as an ordered sample from a uniform distribution over [0,1] (cf. e.g.
David (1970), p. 80). Therefore,
E? } n., where
j= 1 j
for
n,,
1
..., n n
n-i
L
J(u.)
has the sane distribution as
are independent and
P(n-<x) = J
j-
1(x)
X € [J(0),J(1)]. Of course, it is desirable to have a simple expression
for the distribution of
i° order to get the rejection region of
the test. However, there is no family of distributions which is closed under
convolution when
J
continuous on [0,1]
this gives
is continuous on [0,1]. If we do not require
a good choice of
""1
u/2
J (u) = 1-e
(i.e.
J(Uj) = -2
An(l-Uj)
is
J
is
is
A
= -2
2
n-1
E jLn(l-u.)
J
j-i
to be
J(u) = -2£n(l-u) because
2
X (2)-distributed) and
x^(2(n-l))-distributed.
•
•
.
"S
As a consequence of this discussion we^propose
(3.8)
J
10.
as a test statistic for testing exponentiality against the HNBUE (HNWUE)
alternative. Here we reject
Hq
when
has a large positive value
(a negative value of large magnitude).
2
From properties of the x -di stribution we get that, under
,A - 2(n - 1)\
eXA
1 ->-N(0,l)
^ /4(n- 1) 1
The asymptotic normality when
Since
F
n -*•
00.
is not exponential is more difficult.
Uj, j = 1, 2, ...,n-l, behave asymptotically like order statistics
from the inverse TTT-transform
asymptotic properties of
the same as for
from
when
Hq,
Tn(J)
(see Barlow and Doksum (1972)) the
are, when J
J(zj)» where
is continuous on [0,1],
z^, j = 1 , 2,
n-1, is a samole
<p ^ (which is a distribution function on [0,1]). With
F
H(u) = J(ip (u))
"
we therefore obtain from Corollary 3 by Chernoff, Gastwirth and Johns
(1967) that
A2
(3.9)
v/?/>/n(——
~~0
y9)\
J
N(0,1)
when
n
°°,
where
1
p. » / H(u)du
0
f
a2 = / a2(u)du
0
and
a(u) =
! 1
i',u / H1(s)(l-s)ds.
u
Sufficient conditions for (3.9) to hold are (with the same notations as in
Chernoff, Gastwirth and Johns (1967)):
A*.
H
is continuous on ]@,1[
and satisfies a first order Li'pschitz
condition in every interval bounded away from
0
and
and is continuous except on a set of Jordan content 0.
1. H'
exists
11.
E.
There exists a
exists a finite
M
<5g, 0 < < 5^ < 1, such that for every
K > 0
there
such that:
(i)
if
0 < Up U£ < < 5
and
K ^
(ii)
if
1 - <5q < u^, Uj < 1
and
< K
then
M * < H'(Uj)/H'(Uj) ^ M.
K * < (1 - u.p /(1 - u2) < K
then
M_1 < H'(u1)/H'(u2) < M.
1
B**. / H'(u)/u(l - u)du
0
With
converges absolutely.
J(u)=-2£n(l - u), which is not continuous on
[0,1], things are
more complicated and we do not know if the asymptotic normality is true
in general.
The- consistency when
B**
F
is a life distribution for which
A*, E and
hold now follows from the fact that then we have strict inequality
in (2.6) for son»
t > 0
when
F
is HNBUE (HNWUE) but not exponential
and that therefore
3.3
> 2
if
F
is HNBUE but not exponential
y = -2 / £n(l-q> (u))du < = 2
F
0
< 2
if
F
is exponential
if
F
is HNWUE but not exponential
The test statistic
A-
Our third test statistic will be derived from quite another point of view.
Hollander and Proschan (1975) proved that the cumulative TTT-statistic
V =
uj
can
detect not only IFR (see e.g. Barlow et al.(1972),Chapter 6)
but also NBUE alternatives. They proved that
K* =
n
I J(j/n)t(j)/S
j-1
where
J(u) = I - 2u.
,
V
is a linear function of
12.
With
00
(3.10)
p(J,F) = / X J(F(x))dF(x)
0
(3.11)
a2(J,F)
(3.12)
J*(u) - J(u) -
OD 00
- f f J*(F(x))J*(F(y)){ F(min(x,y)) - F(x)F(y)}dxdy
0 0
and
00
y - / F(x)dx
0
it was proved by Klefsjö (see Hollander and Proschan (1980)) that
(3.13)
if
cf
(
^
2
/ X dF(x) <
0
00
and
N(0,D
when
n + ~
2
a (J,F) > 0.
In fact, it follows from (3.13) that a test based on
on
V, or equivalently
K*9 is consistent against the class of life distributions
F
for which
y(J,F) > 0. Now integration by parts shows that
(3.14)
y(J,F) = - ^ + / F^(x)dx.
z
0
From (2.3) it now follows that not only the class of continuous NBUE life
distributions (as was proved by Hollander and Proschan (1975)) but also
the class of
continuous HNBÜE life distributions is included in the con­
sistency class.
Since (2.3) holds for every
another function
V
> 2
we are wondering if we can find
which is continuous on [0,1]
and for which
13.
(i)
Ag =
Jß(j/n)t(j)/Sn
that of
(ii)
K*
has -°,n asymptotic distribution like
in (3.13);
00
y(J,,F) = - — + / FV(x)dx
V
0
(iii) A^
for some
has larger power values than
V > 2;
K*
against most of the common
life distributions.
Integration by parts shows that (ii)
is true for
2
v-1
J^(u) = -1 + v (1-u)
Furthermore, it follows from Slutsky's Theorem and Theorems 2 and 3 by
Stigler (1974), with
(3.15)
A3 =
J^(u) = -1 + v^(l-u)V *
and
n
E J3(j/n)t(j)/Sn
j=l
that
//n(A - {p(j„,F)/y})\
(3.16) ^
J-N(0,1) when
a(j3>F)/y
independently of
v > 2
(with
y(J3»F)
and
a(J^,F)
analogous to (3.10)
and (3.11), respectively). Particularly with
F(x) = Fq(x) = 1 - exp(-x)
(note that
A^
for
x > 0
is scale invariant) we get that
y(J3,F0) = 0
and
a2(J3,FQ) = (V-1)2/(2V-1).
From (ii), (3.16) and (2.3) it follows that a test based on
A3
is }
14.
independently of
v > 2, consistent against every continuous HNBUE (HNWUE)
00
life distribution for which
a^(J~,F) > 0 ( a^(J~,F) > 0
and
/ x^dF(x) < °°)
0
to get as good asymptotic efficiency as possible.
J
We now try to choose
v
When testing a simple hypothesis
sis
0 = 0q
0 > 0Q (say) by using a test statistic
mally distributed with mean
efficiency
y(0)
against an alternative hypothe­
T
and variance
which is asymptotically nor2
a (0)/n
the Pitman
is given by (see e.g. Rao (1965), pp. 390-396)
Ef(T) = (y,(60))2/a2(60).
In our case this means that
(3.17)
where
Ef(A3) = { y'(J3,F)9=e^}2/ G2(J3>F)0=e^,
0q
corresponds to the exponential distribution. We calculate
E^A^)
for linear failure rate, Makeham, Pareto, Weibull and gamma alternatives
given respectively by
F^(>'
for
0 > 0, x > 0,
F2(X) = 1 - exp(-(x + 0(x + e~X - 1)))
for
0 > 0, x > 0,
F3(x) = 1 - (l + Ox)"1^0
for
0 > 0, x > 0,
F^(x) = 1 - exp(-x^)
for
0 > 0, x > 0,
for
9 > 0, x > 0 .
F (X)
5
For
F^, F2
1 - exp(-(x+~ 0x2))
x
- —/ t0_1e-tdt
and
F^
we get
Hq
for
0 = 0q =» 0 and for
0 = 0Q = 1. Straightforward calculations give that
F^
and
F<.
when
Ew (A.) = (2V-D/ V 2
F1
3
E p ( A , ) = ( 2 v - 1 ) / { 4( v + l ) 2 }
2 3
E-, (A_) = (2V-D/ V 2
3 3
E - ( A , ) = ( I n v ) 2 ( 2 V- 1 ) / ( v - 1 ) 2
4 3
E t, (A,) =(v£nv-v+1 )2(2V - L )/(v-L) 4
5 3
It is easily seen that
E-, (= E-, ) and E_
are decreasing functions of
F1
T2
3
v (as a continuous variable), that Ehas a maximum between v = 3 and
4
v = 4 and that E-r.
has a maximum between v = 6 and v = 7 . Table 5.1
5
shows the Pitanan efficiency for some values on v.
TABLE 3.1
The Pitman efficiency of
V
E
= E
F
F
*1
3
2
3
4
5
6
7
8
/
E_
E_
0.083
0.078
0.070
0.063
0.056
0.051
0.046
5
1.441
1.509
1.495
1.457
1.413
1.367
1.324
S
0.448
0.524
0.556
0.576
0.582
0.583
0.580
\
Table 3.1 shows that the optimal value of
butions
v = 3
F
in
v.
E_
4
2
0.750
0.556
0.438
0.360
0.306
0.266
0.234
S
A^ for some values of
v
depends on the life distri­
H^. In view of the efficiency values in Table 3.1 we propose
as a compromise. Note that
suggest the test statistic
A^
V * 2
corresponds to
K*. Thus we
given by (3.15) with
J3(u) = -1 + 9(l-u)2.
3.4
The: test statistic
i
:
A,
4
After our discussion in Section 3.3 it is natural to examine if we can
find a function
J^(u) which is continuous on [0,1]
and for which
16.
(i')
A^ =
of
j)has an asymptotic distribution like that
K*
in (3.13);
00
(ii')
y(J/,F) = y eV , i - /{l-FV(s)}ds
^
J- 1 J
0
(iii9)
A^
for some
has larger power values than K*
v > 2 ;
against most of the common
life distributions.
The difference between
A^
and
A^
is that the consistency against the
HNBUE (HNWUE) class now is based on (2.4) instead of (2.3). Integration
by par ts shows that (ii*) is met with
T
/
N
J,
(u)
=
r
L
1
-T -
V U
V-l
j-i 1
Calculating the Pitman efficiency for the corresponding test statistic
A^ =
n
I J^(j/n)t(j)/Sn
j=l
we get Table 3.2.
TABLE 3.2 The Pitman efficiency of
V
E_ = E_
F1
3
E„
2
0.750
0.822
0.867
0.896
0.916
0.930
0.940
0.083
0.082
0.079
0.077
0.074
0.071
0.069
2
3
4
5
6
7
8
As in the
A^
case we propose
fourth test statistic is therefore
(3.18)
A4 =
n
E J4(j/n)t(j)/Sn ,
j=l
with
T
/ \
J4(u) =
11
T~
o
3u
2
•
A^
for some values of
E„
4
1.441
1.339
1.256
1.187
1.130
1.081
1.039
v = 3
V.
5
0.448
0.390
0.349
0.319
0.296
0.277
0.261
as a c ompromise. Our
17.
Also for
A
4
we have
)
^(A4 - {y(J4,F)/y})
a(J4,F)/y
with
y(J^,F) and
a(J^,F)
N(0,1)
when
n +
analogous to (3.10) and (3.11).
In particular it follows that, under H^,
/ /n Ay \
<C(— -)
—
->N(0,1)
when
n -> °°.
x/6l7T8Ö/
3.5
A remark
Comparing Tables 3.1 and 3.2 we note that a test statistic
J^(u) has larger asymptotic efficiency than
A^
A^
based on
based on
i-n the
linear failure rate case (and the Pareto case) but smaller efficiency
values when
F
is a Weibull or a gamma distribution. Therefore a test
statistic based on
J^(u) seems to be suitable when we expect a distribu­
tion like the Pareto distribution and a test statistic based on
J^(u)
is preferable when the alternative may be like a gamma or a Weibull
distribution. A consequence of this is that a test statistic
on a linear combination of
+ a J^(u)
for suitable
the statistics
A3
and
J^(u)
and
J^(u), i.e. on
A
based
J^(u) = J^(u) +
a > 0, could be a reasonable compromise between
A^. This family of test statistics will be studied
elsewhere.
4.
A small sample study
To get an idea of the quality of the test statistics
A^, A^, A^ and
A^
we made a small sample study. However, first we need the lower and upper
percentile points of
study for
n =10
and
A^, A^
n = 20
4.1, 4.2 and 4.3 on p. 18.
and
A^. We have carried out a simulation
with
20 000 replications each. See Tables
18.
TABLE 4.1
Critical values of the test statistic
and
A^n
1%
Upper tail
5%
10%
10%
5%
1%
n = 10
-1.559
-1.170
-0.977
0.182
0.295
0.470
n = 20
-1.999
-1.490
-1.223
0.415
0.605
0.915
Critical values of the test statistic
and
A ^/5n/4
when
n = 10
n = 20.
Upper tail
Lower tail
1%
TABLE 4.3
n = 10
n = 20.
Lower tail
TABLE 4.2
when
5%
10%
10%
5%
1%
n = 10
-2.079
-1.649
-1.405
0.930
1.311
2.033
n = 20
-2.198
-1.666
-1.389
1.056
1.434
2.166
Critical values of the test statistic
and
A ^/180n/61 when n = 10
n « 20.
Upper tail
Lower tail
1%
5%
10%
10%
5%
1%
n = 10
-3.041
-2.337
-1.946
0.571
0.877
1.409
n = 20
-2.836
-2.166
-1.787
0.754
1.083
1.682
We have simulated the power for tests at the significance level
a = 0.05
for some Weibull, Pareto and gamma alternatives (cf. p. 14) both for
and
n = 10
n = 20. We also included the life distribution
(4.1)
F(t) = (1 - e~3t)(l - e~7t)
for
t > 0
which is IFKA but not IFR (cf. Barlow and Proschan (1975), p. 83) and the
piecewise exponential distribution
F^
presented in Klefsjö (1980b) which
is HNBUE but not NB131. The power estimates are based on 2000 simulations each.
19.
In order to obtain comparisons with other test statistics we included
the following statistics in the power simulations:
A. = (D -D.)/S
1
n
1
n
A- = E?
a.D./S
2
j»l
J J
n
with
a. = i {(n +l)^j - 3(n + l)^j^ + 2(n + 1)j^}
B
= Z? , B.D./S
J"1 J J
ti
with
ß. = i {n + 3n' - 3nj - 3nj - 3j + j + 2n + 2j}
J
6
V
= L. - U.
J =1 J
TT
V* =
L
Here
A^,
v
n
=
w - w e
Yjt(j)/Sn
6
1
•
with
„ 2.
„ .
,.2 . . . „
. ..3,
4.3 . .2 - 2. 1 3 1 2 1,2 1 .
Yj = jJ - 4nJ +3n j -^-n +jn -j j+ ^ j
B
were introduced by Klefsjö (1979), V
tive TTT-statistic (see p. 11), V*
(1975) and
r 3 .2
t(j)/S
n
j=l
and
J
is the cumula­
was studied by Hollander and Proschan
L was presented by Gail and Gastwirth (1978) for testing goodness-
of-fit for the exponential distribution.
The power estimates can be read in Tables
4.4 and 4.5. The estimates are
calculated for all test statistics with the same simulated observations for
each distribution. With (at least) 95% confidence each power value is correct
within
± 0.02.
TABLE 4.4
Power estimates based on 2000 samples of size
ficance level
n = 10
with signi­
a = 0.05.
A1
A2
B
V
V»
A2
A3
A4
L
Weibull
0 = 1.5
0.30
0.23
0.37
0.35
0.19
0.36
0.24
0.37
0.34
0.35
Weibull
0.60 0.50
Q.74
0.75
0.38
0.75
0.53
0.75
0.72
0.70
0.31
0.37
0.33
0.17
0.33
0.22
0.35
0.31
0.32
F acc
to (4.1)
0.20 0.10 0.23
0.19
0.08 0.19
0.12 0.21
0.17
0.19
Weibull
6 = 0.8
0.15
0.21
0.17
0.22
0.19
0.24
0.20
0.22
Pareto
0 - 1/2
0.28 0.29
0.24
0.36
0.34
0.34
0.34
0.34
0.35
0.30
Fj_
0.66
0.24
0.14 0.13
0.20
0.28 0.21
0.14
0.15
6 = 2
Gamma
0.20
0 = 2
0.18 0.22
0.07
20.
TABLE 4.5
Power estimates based on 2000 samples of size
ficance level
n = 20
with signi­
0.05,
Ax
A2
B
V
V*
A1
A2
A3
Weibull
0 = 1.5
0.42
0.36
0.63
0.63
Weibull
0.78 0.75
A4
L
0.30
0.63
0.46
0.66
0.97
0.98 0.59
0.98
0.90
0.98 0.98 0.96
0.52 0.31
0.67
0.62 0.21
0.62 0.38 0.69
0.58
0.61
F acc
to (4.1)
0.36
0.13
0.43
0.32
0.09
0.31
0.18 0.40
0.28
0.34
Weibull
0 = 0.8
0.18 0.27
0.32
0.35
0.27
0.35
0.29
0.37
0.33
0.33
Pareto
S = 1/2
0.38 0.41
0.32
0.55
0.54
0.55
0.54 0.52
0.55
0.46
F1
0.69
0.48
0.48 0.14
0.63
0.58 0.65
0.38
0.21
0.62 0.60
6 = 2
Gamma
0 = 2
0.04
From Tables 4.4 and 4.5 we notice the following facts. A^
values of the four test statistics
Pareto case and for
of
A^
when
A^
A^, A2> A^
and
A^
except in the
n = 10. We also notice that the power values
are not less than those of
If we compare
that
F^
A^, Aj, A^
has the best power
and
A^
A^ (except in the Pareto case when
n= 10).
with the other test statistics we observe
in many cases has the best power values. One exception is the IFRA
distribution
F
according to (4.1). In this case the test statistic
B , which
is constructed to detect IFRA (cf. Klefsjö (1979)), has the largest values.
Particularly, A^ (and in most cases
the test statistic
A^, too) has better power values than
L.
In view of t he Pitman efficiency values and the power estimates it seems
to us that
A^
is a good test statistic, in
cumulative TTT-stati&tic
V
for example.
21.
For further comparisons between power estimates we mention that power
simulations were also made by e.g. Bickel and Doksuni (1969), Wang and
Chang (1977), Gail and Gastwirth (1978) and Klefsjö (1979).
Acknowledgements
The author is very grateful to Dr Bo Bergman, Professor Gunnar Kulldorff
and Dr Kerstin Vännman for valuable comments and stimulating discussions.
22.
APPENDIX
A life distribution
F (i.e a distribution function with
and its survival function
and
(i)
S = {t: F(t) > 0}
—
F « 1 - F
with finite mean
is a decreasing function of
t
for
x > 0;
increasing failure rate in average (IFE A) if
M
-Jin F(t)
:•
i
t
is increasing on
S;
(iii) new better than used (NBU) if
F(x)F(y) > F(x+y)
for
x > 0
and
y > 0;
new better than used in expectation (NBUE) if
F(x) / F(y)dy > f F(x+y )dy
0
0
for
(v)
OO —
J Q F(x)dx
increasing failure rate (IFR) if the conditional survival function
F(t)
(iv)
=
are said to be (or to have)
t~
(ii)
y
F(0 ) = 0)
x > 0;
decreasing mean residual life (IMRL) if
OO
t - ^— S F(x)dx
F(t) t
is decreasing on
S;
23.
(vi)
harmanio new better than use d, in expectation (HNBUE) if
00
/ F(x)dx < y exp(-t/u)
t
for
t > 0.
By reversing the inequalities and changing decreasing (increasing) to
increasing (decreasing) we get the dual classes DFR, DFRA, NWU, NWUE, IMRL
and HNUUE. Here
D « decreasing, I = increasing
and
W • worse.
The relations between the different classes are illustrated by the
following figure.
IFRA
(DFRA)
Y
—
NBU
(NWU)
X
NBUE
(NWUE)
V
=r
HNBUE
(HNWUE)
DMRL
(IMRL)
Different properties of t he classes IFR, IFRA, NBU, NBUE and DMRL
and their du als are discussed e.g. by Bryson and Siddiqui (1969), Marshall
and Proschan (1972), Haines (1973) and Barlo w and Proschan (1975).
24.
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