* This research was sponsored by:
Aerospace Research Laboratories
Air Force Systems Commond
United States Air Force
Wright Patterson AFB, Ohio
under the direction of Dr. fl. Leon Harter.
A GENERALIZED NOTION OF
RATE tHTI1 LIFE
r~ONOTONE
TESTIr~G
FAILURE
APPLICATIONS
Gordon Sil':ons*
Department of Statistics
University of North CaY'oUna at Chapel Hill
Institute of Statistics:'tlimeo.Series No. 968
December, 1974
A3ST~f\CT
New notions related to monotone failure rate are defined.
tions are found which guarantee that these notions hold.
Sufficient condi-
Applications result in
inequalities and sometimes equalities which relate ';power and significance level
for certain hypothesis tests.
These results generalize similar equalities found
by S. D. Dubey in connection with the exponential failure law and are closely
related to equalities found by E. Paulson, again in connection with the exponential failure law.
All of the tests considered involve
in the presence of a nuisance scale parameter.
a
minimum life parameter
Other applications are anticipated
but are not discussed.
AHS 1970 Subject Classifications.
Key Words and Phrases.
Primary 62NOS
!4onotone failure rate, life testing, minimum life, Mill's
ratio
A. GENERALIZED iWTIGN OF
[ilmmTm~E
FAILURE
RATE WITH LIFE TESTING APPLICATIONS
by
Gordon Simons*
1.
INTRODUCTIOli.
(l)
Y.~is
B(m)
paper is motivated by a remarkably simple formula,
=I
-n(IDO-m)/e
- e
which relates the power function
(1 - a) ,
o
B to the significance level ex for a number
of tests concerned with the location parameter
tial distribution.
m~ m
m of the two-parameter exponen-
a > 0 is a scale parameter, known or unknown, and rno arises
in the definitions of the hypotheses.
This formula appears in the work of Paulson
(1941) for two different tests and in the work of Dubey (1962, 1963) for several
classes of tests.
In each instance, the formula is a consequence of well-known
properties of the exponential distributions:
(i)
The exponential distributions "lack memoryH ("aging" does not occur)
when the location parameter is zero.
(ii)
The minimal order statistic in a random sample is exponentially distributed.
(iii)
The spacings between order statistics are independently distributed.
* This research was sponsored by: Aerospace Research Laboratories, Air Force
Systems Command, United States Air Force, Wright Patterson AFB, Ohio, under the
direction of Dr. H. Leon Harter.
-2The location para"leter
in
of the two-parameter exponential distributions
represents the minimum possible life lengtJl in a life testing context.
Analogs
of (1) can be obtained for certain other location-scale .?arameter families of
distributions when
ill
has tne same minimuIIl life interpretation.
We shall illus-
trate this for a simple class of tests considered by Dubey (1963).
Stated roughly, if a distribution has a linear ;,Jill is ratio l • an analog of
(1) will hold for the family of distributions ~roduced by introducin~ location and
scale parameters.
If the Hill's ratio is convex or concave, an'inequality analog
of (1) will hold.
The sense of the inequality will depend on whether convexity or
concavity holds and, also, upon tl:e nature of the null hypothesis, whether it
specifies a single value
ill
o
for
m or an interval of values
In demonstrating these results. we find it convenient to introduce and study
a generalized notion of monotone failure rate.
7his is done in Section 3.
Section 2 reviews the relevant literature associated with testing for the
location parameter of the two-paraneter exponential distribution.
two inequality analogs of (1) are derived.
In addition,
This material helps to motivate our
new notion of monotone failure rate, but it is not a lOJical .?rerequisite for
Section 3.
Analogs of (1) are obtained in Section 4 for more seneral location-scale
parameter families.
Here, the new notion of monotone failure rate is used.
In Section 5 J it is sl1o"m that
t~le
i!ill is ratio of each two-parameter Heibull
distribution and of eadl beta distributiol1 (types one and. t\'lo) is convex, linear
lEy ilill's ratio, we mean the ratio o~ the upper tail ( 1 - F(x) , say) of a distribution to the density ( fex) , say). This is t:le reciprocal of the hazard
function. The term Hill v s ratio originally VJaS applied to t;1e standard normal distribution, but has been used more generally in the statistical literature.
-3-
or concave.
This sets the stage for explicit applications.
We hope to discuss applications outside the life testing context in a later
paper.
2.
TESTING FOR nlE iVlIrHMlIM LIFE OF AN
EXPONENTL~L
DISTRIBUTION.
Let
,Xn be the order statistics based on a random sample whose common density is
feu) = !.e-(u.-m)/e
e
on its support
[m,ClO) •
the alternative
rn
~
mo
The problem of testing the hypothesis
m
= mo against
has been extensively studied with the following conclu-
sions obtained:
?fuen
e
a is known,
>
xl
is a minimal sufficient statistic and the likeli-
hood ratio tests reject the hypothesis
positive constant
c
= mo
Xl - rno ( [O,c] , where the
determines the significance level. These tests are strictly
Til
when
unbiassed (see Paulson (1941)), and, in fact, are uniformly most powerful (U.M.P.)
unbiased tests for their respective significance levels.
n
L x.) is a minimal sufficient statistic and the
1 1
likelihood ratio tests reject the hypothesis m = mo when the test statistic
Vfuen
s
= (Xl
e
is unknown,
(Xl'
n
- m0 )/L(X,1 - Xl) ~ [O,c] , where the positive constant
c
determines the
1
significance level.
significance levels.
These tests are U.i"]. P. unbiased tests for their respective
(See Lehmann (1959), page 202.)
The power function
8 for both cases is given by (1) for
is the significance level.
distinct for the two cases.
When
m < mo ,where
a
m > mo ' the forms of the power function are
They are given by Paulson, but are unimportant here.
-4-
In a life testing context, it is frequently convenient to work with test
statistics which depend on the smallest observations, say x ,x '
1 Z
2 5
r < n.
,x
For these, the likelihood ratio tests in the unknm"n
e
r
,
case are
given by Dubey (1962), and are based upon the statistic
(3)
Again the tests are U.i,1. P. unbiased tests (for the data used), and again (1) is
applicable.
Carlson (1958) considers the simple statistic
s
= (xl - mo)/(xn - xl)
which is a special case of the statistics
(4)
studied by Dubey (1963).
which reject
when
m = m
power function when
Equation (1) is applicable whenever
o
a
~
s~[O,c]).
a
=1
(for tests
While Dubey has derived formulas for
t:i~
1 , they are fairly complicated, and it would be instruc-
tive to derive the following analog to (1):
(5)
-n(m -m)/e
Pr(x > m ) > Pr(x > m ) = e
0
for m < mo ' inequality (5)
l
o
a
o
implies that a strict loss in power occurs to the left of mo when a ~ 1 •
Since
Derivation of (5).
Let
zl'zZ' ... ,zn
denote a generic sequence of standard
exponential order statistics, arising under a fixed probability measure
corresponding to a random sample of
Z 2:
0 .
m and
SiZ0
n
whose contmon density is
Then, without loss in generality, we may assume that
e
are the true parameter values.
In terms of the
x.1.
z'~,
e
= ez.1.
P,
-z
+
m when
-5-
1 - SCm)
= P((mo - m)/e
= p(Za > (wo -
<
za ~ (mo - n)/8 + c(zb - za))
m)/e) - p(za >
Pr(x a > mo ) = p(Za > (mo - m)/e) , and
is equivalent to
(5')
1 - a
(rn
o -
m)/e +
= p(Za
c(zb - Za))
~ c(zb - Za)) .
m < "0
m
P (za > u + C(Zb - Z )) < P(z > U)p(z > c(zb - Za)) ,
a
a
a
where u
= (mo
- m)/e .
Since
and
Z
b
-
Z
Thus, (5)
are independent, as property
a
(iii) in Section 1 makes clear, inequality (51) holds providing it can be shown
that
(6)
P(Za
>
u
+
v)
<
P(za
This is the case, in fact, when
u)P(za
>
a
~
1.
v) ,
>
za
0,
v
>
0 .
If the strict inequality between the
probabilities were replaced by a non-strict
that the distribution of
U >
inequality~
(6) would say, precisely,
satisfies the "new (is) better than used" property.
(cL, hollander and Proschan (D72).)
This property is easily deduced for any
distribution which has the more familiar increasing failure rate property.
is not difficult to show that the hazard function for
Z
a
It
is strictly increasing,
o
from which (6) easily follows.
It probably should be pointed out that the
~arginal
distributions of the
order statistics have increasing failure rates w11en the parent distribution does.
See Barlow and Proschan (1965), page 36.
We suspect that the new better than used
property is preserved by order statistics as well.
-6-
Consider now the one-sided hypothesis testing problem m ~ m
0
m< m
0
s
, with
8
unknown, and tests which reject
ill ~
m
when
0
versus
s < c , where
is anyone of t:1e above test statistics and the positive constant
the significance level.
~{hen
controls
These tests ar8 unbiased and quite good for the test
statistics defined in (3) and (4) when
tests.
c
a
= 1.
Formula (1) holds for these
the test statistic in (4) is used with
a
1 , (1)
7
does not hold
but an analog does:
B(m)
(7)
(Ccmpare this
wit~
>
1 - Pr(xa
(5).)
m0 )(1 - a) ,
>
m < mo
This can be derived in much the same way (5) is.
Specifically, (6) implies (7).
The right side of (1) provides an upper bound for
13 (m) •
This is because
n
the test statistic
s
= (x1- 0
m )IL
(x. - Xl)
1
1
class of tests which have constant power
produces a U.M.P. test within the
{emo ,8), 8 > O} ,
on the boundary set
and, for this, (1) holds.
3. A GENERALIZED NOTION OF MONOTOi1E FAILURE
with distribution function
function
h
= flF ,
where
F
F
u
Let
x be a random variable
and, if they exist, with density
denotes
1 -
equivalent definitions of the notion that
simple definition states that
~iTE.
h
r
~
f
and hazard
There are a number of essentially
has an increasing failure rate.
One
i.s an increasing function on the set of points
for which -F(u) > 0 . 2 Other definitions appear in Barlow and Proschan (1965),
2The convention tr.at Hincreasing li means ilnon-decreasing?' rather than ilstrict1y
increasing li is being followed. Likewise, lIdecreasing" means Hnon-increasing".
-7pages 23 - 25.
One suggestive definition requires
decreasing function of u for each v
conditional probabilities as
Pr(x > u) > 0
for each
positive values.
Ii
Pr(x > u
v > 0
Ii
o.
>
F(u
section, and be concerned \'I'ith wfiether
v
to be a
is decreasing in
Here, of course,
We shall want to view
v)/F(u)
T;lis can be reworded in terms of
vlx > u)
+
+
v
is a constant which assumes
as a fixed function of
Pr(x > u
+
u when
vex) Ix > u)
x, in the next
is a decreasing
function of u •
Let
w be any real (measurable) function of a single real variable.
random variable
x
and its distribution function
monotone failure rate
Pr(w(x)
>
ulx
>
u)
(i',1FR)
on
S c
{u: F(u) >
is a monotone function on
O}
F
The
will be said to have a
with respect to w
OJ:
1.L
The terminology increasing
S.
failure rate (IFR), decreasing failure rate (DFR) and constant failure rate (CFR)
on S with respect to w will be used according as
decreasing, increasing or constant on
Pr(w(x) > ulx > u)
S.
The value of these new concepts is tied to the next two theorems.
F is continuous everywhere and strictly increasing on an interval
image ander
THEOREIJI 1
P is
F
[0,1) .
or
For simplicity, let
has a 1-1PR on S with respect to
is monotone on
with respect to
on
(0,1)
I.
is
w
S
I
Suppose
whose
=I
if the ratio F(w(t)) /F(t)
The failure rate is increasing 3 decreasing or constant on S
w according as the ratio is increasing 3 decreasing or constant
I.
PROOF.
Let
P-l(s)
associated with
= inf{t:
pet)
F and let
z
<::
s},
° ~ s ~ 1 , denote the "quantile function"
denote a standard exponential variable.
Then
-8-
1 - e- z
is a uniform variable on
function
F.
[0,1]
and
P-ICI - e- z )
has distribution
We can, without loss of generality, take the random variable
P-ICI _ e- z)
used in the above definitions, to be
is equivalent to
z > - log F(u) .
Pr(w(x) > ulx > u)
Then the condition x > u
TI1US
= pr(w(p-1Cl
- e- z)) > ulz > log F(u)) .
given
Z > ~
z - log F(u) .
Hence
Now the conditional distribution of
unconditional distribution of
x
z
log F(u)
is the same as the
(8)
Suppose, for definiteness, that
=
l,Z
and
xl
is increasing on
I
and let
-1 (
-z)
z • be any two points in S . Set xi = F I - e F(u i )
-zIt easily follows that x.~ IS I and F(x.) = e F(u.) for i = l,Z
~
and
i
F(w(t))/F(t)
u1 <
U
~
::;
Xz
.
Thus
FCXI)F(U Z)
= F(X
2
)F(u 1 ) > 0
.
The following implications hold:
w(x z )
> U
z <~-> F(w(x z ))
<
F(u z )
(even if
w(x z ) 4 I )
F(X ) F (w(x Z)) < F(X l ) F(U Z)
1
<=--:> F(x )F(w(X )) < F(Xz)F(u )
2
l
l
<=>
=>
<=>
"f(w(xlJ)
w(x )
I
>
< F(u
l)
ul
The unidirectional implication uses the assumption that
on
I.
F(w(t))/F(t)
Consequently, from (3),
Pr(w(x)
>
uzlx > u z ) = Pr(w(x z ) > u z)
$:
Pr(vl(x l ) > u l )
= Pr(w(x) > ullx > ul) .
is increasinL
-9-
I.e.,
F has an IFR on
S with respect to
w.
The remaining DFR and CFR cases
o
follow similarly.
REI/lARKS
1.
If F(w(t))/F(t)
in
t.
tone.
is constant or decreasing in
If the ratio is increasing in
~fuenever
the ratio is monotone on
tioned into two intervals
on
2.
12 •
t
II
and
1
2
wet)
I
wet)
must be increasing
does not need to be mono-
the interval
with F(w(t)) > 0
I
on
can be partiII
and
=0
Either interval may be empty.
If the hazard function exists and
w is differentiable, then
is increasing, decreasing or constant on
h(t)
(9)
3.
t,
~
~
according as
= h (w (t) ) W (t)
or
While the assumptions that
II
F(w(t))/F(t)
I
F is continuous and strictly increasing are
explicitlY used in our proof of Theorem 1, we suspect they are unnecessary.
The crucial equivalence
li X
>
U <,"">
Z
>
log F(u) II
does not depend on either
one of them.
4.
For purposes of applications, it is desirable to weaken the assumption in
Theorem 1 that
F(w(t))/F(t)
necessary to assume that
is unknown.
is monotone on
w is continuous.
I.
In doing so, we find it
h~ether
it is really necessary
-10-
THEOREr1 2 F has a MFR on
S with respect to
the ratic. pG~(t))/F(t) is monotone on
w if w is continuous on
= {t
1*
€
I: wet)
I}
€
and
I
The faiZure rate.
is increasing decreasing or constant on S with respect to w according as the
3
ratio is increasing 3 decreasing or constant on
1*.
PROOF.
Refer to the proof of Theorem 1.
objective is still to show that
Assume the ratio is increasing on
w(x 2 ) >
U
z ~?
w(x l ) > u l .
with
U
z > ul
x*2 -- p-l (1
let
i
x(u)
= 1,2 .
and
w(x l )
- e-zp(uz)) , satisfies
= F- l (1 - e-zp(u)) , with
x(u)
is continuous on
S
But
xi
= x(Ui) ,
By working with the new
U
and
S
u*Z ,
z
x(u)
= w(x Z)
ul
= Uz €
w(xi)
Then there exists a new
= 1,Z
i
I,
z'
U
ui
say
To see this,
w(x 2) > u*2 and w(x 2) € I
,
z > 0 held fixed. Then x.1 = x(u.)
1
w(x(u 2 ))
Consequently, there exists a
€
x2 , say x*2 , defined by
and such that the new
is continuous on
Then, for
ul .
S
The
The argument used in
Theorem 1 to show this implication would apply here if w(x.)
1
Suppose w(x 2) 4 I
1*.
and
<
I
u*
2
and
=: I
for
u
€
S
.
Thus
w(x(u))
z > u l ;;:: w(x l ) = w(x(u l )) .
uz , such that w(x(u z)) = Uz
w(xi) = Uz > ui .
> U
<
.
x 2 with the superscripts deleted, the
required implication simplifies to:
(10)
Now, it has been noted that (10) hoJ.cl:3 if, in addition,
it suffices to show that the conditions
(11)
w(x,.,)
""
€
I ,
w(x l )
€
I.
Thus
-11-
lead to a contradiction.
is continuous on
w(x(u))
and
The argument again uses
x*
1
such that
S
w(xi)
the same, and are omitted.
which contradicts
w(xi)
and
Then
= ul
tie have shown that if
is increasing on
in order to find a new
ul
:::
u l < u*1 <
w(xi)
= UlE
S
u~
.to
u
.
and
1
xl , say
~~
l
w(x ) > u z
2
F has an IFR on
1, that
F has a DFR on
.
= I.
I
But
and the ratio
I
S
respect to
wi~h
than a set
1*
W
1*
w is increasing on
1*
can be partitioned into three subintervals
wet) < I
on
Ii
B if each point of
and
\'l(t) > I
on
w that the interval
and
Iii, where
(A set
pi
A is less than each point of
A is less
3.
is less than a set if it is less than each point in the set, etc.)
u
l
;:: S
=I
»
the condition
w(x l ) > u l
implies
In the first case,
the implication
It
W
For then it will follow, as in the proof of Theorem
(see Remark 1), and it follows from the continuity of
Ii < 1* < I",
F"(w(t))/F(t)
S with respect to
But if the ratio is decreasing or constant on
1*
I
The details are essentia1 1
remains to show that if the ratio is decreasing or constant on
W(x l ) > u
u*
.
w is continuous on
1*, then
and the fact that
x(u)
"w(x ) > u --=> w(x ) > u{
2
l
l
1, while in the second case,
W(X 0
"
)
~
I
Xl
E
1*
w(x i )
E
or
P'
I,
i
A point
Since
Since
= l,Z
>
and
follo\'ls as in the proof of Theorem
and, consequently,
o
-12S.
The set
is set equal to
S
I
the strength of the conclusions.
subset of
I
Their proofs only require
In the next section, two point sets
is allowed to range within
obtain.
in Theorems 1 and 2 in order to maximize
S
.
= {O,u}
S
to be a
, where
u
I , would suffice for the conclusions we
Expressed another way, our mathematics is most favorably described
in terms of a generalized notion of monotone failure rate, but our applications are in the spirit of the new better than used property.
I1rIew"
corresponds to the (boundary of the) null hypothesis and "used" corresponds
to alternatives.
4.
EQUATIONS AND HlEQUALITIES INVOLVmS
pm~ER.
Our objective in this section
is to illustrate in a fairly simple but substantive context a specific type of
of application.
The context will be that of testing for the value of a minimum
e.
life-parameter· m : in the presence of a"nuisance ·scale parameter
p(Ct - m)/e)
tions are assumed to have a distribution function of the form
where
F is known.
It is further assumed that
lutely continuous with density
interval
I
f, that
f
P(O)
m
, that
F is abso-
is strictly positive on some
whose left limit point is zero and that
We shall begin with the testing problem
=0
Observa-
~
mo
f
versus
is zero elsewhere.
m<
in
o
It is convenient to introduce three sets of order statistics which are
functionally related.
If
standard exponential random sample, then
is a set of order statistics for a
x.
1.
= P-
1
Cl - e
-z.
1) ,
i
is a set of order statistics for the parent distribution function
= 1,
... ,n ,
F and
-13-·
Yi
= 8x i
+ n
i
>
= 1,
... •n
is a set of order statistics
F((t - m)/o)
distribution function
rOT
the parent
Trte latter set represents actual
observations.
rej ects the hYFot;wsis
WDere
b
E
Tl130EEL.3
convex for
(12)
{2,3, ... ,n}
The test
t
E
I
3
C!
Wl1en
c > 0 are fixed.
and
is unbiased.
T
the
If the NiU
((ill
O
-
fa
ratio
get)
= F(t) /f(t) is
S satisfies
function
po~er
FTI
BCm) ;: : 1 -
where
m
o
D ;::::
m)/e) (1 - a.) ,
i8 the significance level of the test.
m ~ r:,o
Equality holds in (12) if
the Mill's ratio is linear.
13 (m)
= Fr(Y l - ma
s c(y,LJ
-
implies that t:le test
Since
where
u
T
-,
o·~
J.
is a ciecreasing function
SCm)
Y1)) - !?r (xl s (no
and equals
'"
is unbiased.
is the distributio,t function of
1 -
= (mo
m)/8
How,
(12) is equivalent to
an(l
~ecaU5e
(13)
But
quently,
is ir.depencient oE
zb - zl
Z,
.,)y
propeTty~...
(~;J.')
l.
is i:adepenclent of
xl'
of
_
~ectl.·on
1.
Conse-
In vieH 01 (13), inequality (12 1 )
-14-
holds if, for each
E >
0 ,
- F(xl)e-
E
~
Thus (12) holds if for each
function of
)
A1j)
-
Pr(x l > u)pr(x 1 > c ( F-1 (1 - -P(x 1)e -E ) - xl ))
E >
Pr(w(x 1 ) > ulx l
0,
>
u)
is a decreasing
u. where
In words, it suffices to show that
xl
has a IPR on
I
with respect to
This will be shown by validating the first inequality in (9).
Theorem 2 will be used.
Xl
hI
It will be shown that
has distribution function
= ni/F.
on
1*
= {t
It follows that
€
I: wet)
€
I}
=1
FI
-
To be precise,
w is differentiable.
pn
w.
Since
has the hazard function
FI(w(t))/F1(t)
is a decreasing function of
t
if
hI (t) ~ hI (w(t))w ' (t) ,
t
€
I,
€
I,
wet)
€
I .
Tnis is equivalent to showing
g(r)q' : :; g(q) ,
(14)
where
q
= w(r)
is a function of
r
rand
dated with the distribution function
Let
(15)
s
= P- I (l
F(s)
- F(r)e- E)
= F(r)e- E
,
,
f(s)s'
q
I ,
€
= F/f
g
is the nill's ratio asso-
F.
a function of
= f(r)e- E
,
r
g(r)sl
Then
= g(s)
.
-15-
Since
q
=
(1
+
c)r - cs
~
q;
= (1
w is differ-
(in rarticu1ar.
+ c) - s;
entiable) and (14) can be eXijrossed as
(16)
jmt
c
= (1'
(1 + c)g(r) s
&(~)
- '1)/(5 - r)
and, hence. (10) can be eX,iJr0ssed as
~(r)
(17)
It
+ CS(s)
•
1'-g,
s-r
s --- Z(q) + ~ . g(5) •
s-q
,,-q
r
€
l' E
I.
I •
q E I .
ocI.
re",ains to show that (17) folloHs from the assu:.T,ed convexity of
CTl
I
It is Cu.5i1y secn frcT:1 ti.c fi:'Gt equality in (1!;) tLat
l' €
I.
rn.crcfore (17) Lo1ci.s :J1'ovi(ih:; q
in (1::;) il;1plies
r < s
and, consequently.
r < s.
<
i.;ut
s
t:
t!.e first
I
b
if
e(~uality
q =.r r + c (1' - s) < r .
Thus (12)
Lo1ds.
Finally, if
c
co
is linear, equality i,olcls in (17).
111is translates into
an equality in (12).
If g
is concave, tli.e reverse inequality in (12) holds.
inequality is not very useful.
Such an
cave for a different testing problem.
nm~ely
ill
= mo
versus
m~
following Dubey's (1963) lead. we shall consider the simple test
accepts
m = tio
is con-
Eowever. it is useful to know that
Again.
ill
0
r:""*
L
if 0 s Y1 - mo s c (Yb - Y1) • The t,ower function
l3(m)
given by
Suppose
g
is concave.
T~e
reverse of inequality (12') holds.
I.e.,
is
-16-
where u
=
(m
o
- m)/e
~
O.
Combining (18) and (19) yields
m =:;
(20)
where
a
that
= S(O)
(20) if
g
From (18) it follows
m > mo , and from (20) it follows that
T*
This means that
o
o
is the significance level of the test.
SCm) > a when
m< m
I:l
is a strictly unbiased test.
B(m) >
a
when
Equality holds in
is a linear function.
5. NILL'S RATIOS.
In this section we are concerned with showing that some
common distributions have a i'lill's ratio which is convex, concave or linear.
The importance of such a property is explained in Section 4.
The simplest example involves t:w two-parameter Weibull distributions,
whose densitieS"-,are
f(t)
= Ayt Y-l e -At y
The i.:ill's ratio is
i. e.. ".,hen
when
f
get)
=
t
> 0
F(t) If(t)
=
A > 0 ,
Y > 0 •
t l - / (Ay)
is an exponential density.
y
g
g
is linear when
is convex when
y
~
1
y
=1
and conca-v"
0 < Y =:; 1 •
THEOREM 4
The MiZZ's ratio for the beta-one densities
o<
f(t)
are oonvex when
\l ~
1 and oonoave when
the beta-two densities
t < 1 ,
0 <
\l =:;
\l > J ,
1.
\I
> 0 ,
The MiZZ's ratio for
-17f(t)
=
r(~+v)
t~-l/(l +
r(~) ,rev)
are oonvex when
~ ~
t
)~+V
o<
~
and ooncave when
1
<
t
00
,
~
v
> 0
> 0 ,
0 < ~ ~ 1 .
PROOF.
Consider anyone of the above densities
bution function
F.
Form the Millis ratio
gil = (f2 fl + (2f i2 _ ffll)F)/f3
defined on the support of
k~ ~
0
= P/f.
= fgll/k = fl/k
£
except when
f
g
and its corresponding distriBy direct calculus,
Set
and
convex if
f
and concave if
kt
~
k(t)
+
= o.
F
Observe that
g
is
0 .
After some algebra, one obtains
£ i (t)
= -2f(t)(~-1)(v-l)
2(~-l)(v-l) +
k(t)
k 2 (t)t 3 (1_t)3
t(l-t)
v(v-l)
(l-t) 2
for the beta-one densities, and
£ 1 (t)
= -2f(t)(~-1)(p+v)
for the beta-two densities.
k(t) - 0 ,
roots in the support of
f
k
,
where
the function
.
(u+v) (]..I+v-l)
~"---'~-..,.-"-
(l+t) 2
always has a constant sign and
£1
Thus
monotone except across roots of
trivial case of
2(~-1)(~+v) +
t (l+t)
le(t)
k2 (t)t3(1+t)3
£
k
has asymptotes.
is
£
Except for the
clearly can have at most two
By conventional algebraic methods, it can be
shown that there is one such root when
£1
> 0
and no such roots when
9.' < 0
(For instance, for the beta-one densities, there are two real roots when
]..I + v < 1 , one on each side of the interval of support.)
The case
£1
=0
.
-18-
corresponds to
=1
1J
or
=1
v
for the beta-two densities.
Except when
0,
~'=
and
< 0
Type I:
t' >
o.
~(O + )
t(l - )
quent1y,
£ > 0
and
g
Type II:
~I
to
=1
and
£
=1
converges to zero at the other end
£
=0
t~pes
= sign(lJ
sign (k)
and
is convex or concave according as
£(0 + )
= 1,
£(1 - )
within the interval of support and
Consequently,
II
of cases to consider,
We shall illustrate them for the beta-one densities.
and
0,
densities, and to
1tlere remain' two basic
i' < 0
>
beta-01~e
This case is easily treated by direct methods.
of the interval of support.
~,
for the
= 0,
~
- 1)
II >
Conse-
0
1
or
lJ < 1
k has a root at some point
sign(k(t))
= sign((t o
- t)(lJ - 1)) .
has the general shape:
t
It
0
I
1
//
/
,/
.~
J
and
g
is convex or concave according as
Observe that
F
=£
- fl/k.
F
>
-fl/k
or
1J <
1 •
o
In the Type I situation (which is easily
characterized in terms of the parameters),
bound
1
1J >
£
>
0
and one obtains the lower
He have not investigated this bound to see whether it is
ever a good bound, nor have we searched the literature to see how it might
compare with other known bounds.
Both families of densities have a linear !'lill l s ratio when
every value of
v.
lJ = 1 , for
It will be recalled that when the fiill's ratio is linear
-19-
certain inequalities in the last section, relating power to significance
level, become equalities.
There is no simple relationship between the convexity or concavity. of a
fdll
I
ratio and the rnonotonicity of a hazard function.
S
The Weibull distribu-
tions suggest a close relationship but other examples do not.
when
jl
=1
For instance,
, both the beta-one and beta-two distributions have convex and con-
cave ['iill IS ratios.
But, when
jl
= 1 , the beta-one densities have strictly
increasing hazard functions while the beta-two densities have strictly
decreasing hazard functions.
support
tion.
[0,00)
It is not hard to see that a distribution with
and a concave tJill's ratio must have a decreasing hazard func-
Likewise, a distribution with support
[0,00)
and a convex Mill's ratio
must have an increasing hazard function unless some of its moments are infinite.
(If the hazard function ever strictly decreases, then
a
>
0
and all
t
2:
O.
get)
2:
at + b
for some
It follows that the distribution will have infinite
a
-1
moments of all orders
2:
6. ACKNOWLEDGEMENTS.
I am grateful to Ralph Evans on whose research contract
this paper had its genesis.
helpful contributions.
.)
Also, I am indebted to Norman Johnson for several
-20-
REFEREiiCES
Prosc~an,
F. (1965).
MathematicaZ Theory of ReZiabiZity
[1]
Barlow, R. E. and
Wiley, New York.
[2]
Carlson, P. G. (1958). Tests of hypoth~sis on the exponential lower
limit. Skandinavisk Aktuarietidskrift 47 - 54.
[3]
Dubey, S. D. (1962). A siillp1e test function for guarantee time associated
with the exponential failure Iffi~. Skandinavisk Aktuarietidskrift 25 - 38.
[4]
Dubey, S. D. (1963). A generalization of a sim?le test function for
guarantee time associated with the exponential failure law. Skandinavisk
Aktuarietidsb?ift 1 - 24.
[5]
Hollander, M. and Proschan, F. (1.972). Testing whet:ler new is better than
used. Ann. Math. Statist. 43 1136 - 1146.
[6]
Lehmann, E. L. (1959).
[7]
Paulson, E. (1941). On certain likelihood-ratio tests associated with
the exponential distribution. Ann. Math. Statist. 12 301 - 306.
Testing StatisticaZ Hypotheses.
Hiley,
i'~ew
York.