•
Isotonic, Convex and Related Splines l
by
Ian W. Wright
and
Edward J. Wegman
Department of Statistics
University of North Carolina
lThis work was supported by the Air Force Office of Scientific Research
under Grant No. AFOSR-7S-2840. Dr. Wright's work was done while on
leave from Department of Mathematics, Papua New Guinea University of
Technology, Lae, Papua New Guina.
-e
Abstract.
In this paper, we consider the estimation of isotonic, convex or
related functions by means of splines.
It is shown that certain classes of
isotone or convex functions can be represented as a positive cone embedded in
a Hilbert space.
Using this representation, we give an existence and charac-
terization theorem for isotonic or convex splines.
Two special cases are
examined showing the existence of a globally monotone cubic smoothing spline
and a globally convex quintic smoothing spline.
Finally, we examine a regres-
sion problem and show that the isotonic-type of spline provides a strongly
consistent solution.
We also point out several other statistical applications.
Key Words and Phrases:
isotone, isotonic, convex, partial
isotonic spline, smoothing spline, interpolating
splin~
-e
62G05,
spline,
regression, strong
consistency.
AMS Classification Nos:
orde~
62Ml5, 65D05, 65010, 06AIO
1.
Introduction:
A good deal of
l~terature
concerning statistical inference
under order restrictions has appeared in the last 15 years.
Barlow et al (1972)
presents a rather convenient source book for this material.
The main body of
this work can be conveniently divided into two areas: inferences concerning
functions on a finite or countable set and inferences concerning functions on
the real line or intervals in the real line.
It is with this latter group of
inferences that we are concerned.
Examples of functions of interest would include regression functions,
probability
distribution and density functions, failure rate functions and
functions related to spectral analysis such as spectral densities, gain and
transfer functions and so on.
It is often known that these functions are
isotone (i.e. order-preserving) and, hence, should be estimated with a function
that preserves the order.
See Robertson (1967), Wegman (1970a,b), Barlow,
Marshall and Proschan (1963) and Marshall and Proschan (1965) for examples of
such estimation problems.
A characteristic feature of these isotonic estimators
is that they are step functions.
In most of these situations, smoothness is
frequently just as desirable as isotonicity so that while the step function may
be isotone, its lack of continuity prevents it from being widely accepted as a
satisfactory estimator.
More recently, there has appeared a good deal of literature concerning the
use of splines in statistical estimation problems.
review of some of these efforts.
See Wright (1977) for a
While a spline fit satisfies the requisite
smoothness properties, it may not be isotone as desired.
we present a combined approach.
In this present paper,
For technical convenience, the class of partial
orders is restricted to those compatible with the additive group of real func-
·e
tions.
Our framework is general enough to encompass estimation problems for
monotone, or convex or positive functions and other related families of functions.
2
2.
Partial Orders and Isotonic Splines:
In this paper we shall be dealing only
with abelian groups of real functions which can be added pointwise.
Definition 2.1.
The relation, »
, on the group, G, of functions is required
to satisfy the following conditions:
~
i.
f » f
ii.
f » g, g » h implies f » h
iii.
f » g implies f
iv.
f » 0, g » 0 implies f
v.
f » g and g » f implies f
+
h » g
+
+
h for all h c G.
g »
:::
0
g.
A group satisfying i to v is called a partially ordered group.
is dropped. G is called a pre-ordered group.
called~he
positive cone of the order. ».
The set. P ::: {g
If condition v
E
G: g »
O} is
The set, P, defines and is defined
by the partial order. ».
The partial order, »
• defined here is essentially distinct from the
partial order which usually occurs in papers on isotonic methods - (cf. Barlow
et al (1972. chapter 7)).
In the standard treatment, a partial order.
is induced on the real line.
the order.
say.~
•
A function f is said to be isotone if it preserves
That is. f is isotone iff f(x )
l
~
f(x ) whenever xl
2
functions may both be isotone and yet not comparable.
~
x .
2
Two
What is true. however.
is that the set of isotone functions forms a positive cone as just defined.
have in mind an essentially different use of the partial order, »
We
.
We let L represent the set of functions on [0.1] which are Lebesgue meas2
urable and square integrable with the usual Banach space norm.
We let W , m > 1.
m
-
be the set of functions on [0.1] for which f(j), j ::: 0.1 •...• m-l are absolutely
continuous and f(m) is in L .
2
<f.g> :::
I5=o
This is a Hilbert space with inner product
f~ f(j) (t)g(j) (t)dt.
k
Let C • k ::: 1.2 •...•
00 •
be the set of all
~.
3
functions on [0,1] which are k-times continuously differentiable, and finally
we let D be the differentiation operator.
•
In order to define a suitable positive cone, we let F be a continuous
linear map of Wm_l into L2 which commutes with the differentiation operator,
i. e. D(Ff) = F(Df) for all fEW
m
c
W 1.
m-
We define a partial order, »
, on
W by f » 0 i f and only i f (Ff) (t) ~ 0 for every t t [0,1].
m
There are several operators, F, of particular interest.
Example 2.1.
If F is the identity map, the set, P
= {g
c
W: g »
m
O} is just
the set of positive functions in W .
m
Example 2.2.
in W.
m
Example 2.3.
If F = D, P is just the set of monotone non-decreasing functions
Similarly, if F = -D, P is the set of non-increasing functions.
2
If F = D
, P
is the set of convex functions in W while F =
m
_D
2
yields the set of concave functions.
Example 2.4.
If F is defined by
Ff(t) = (Df(t)
(-Df(t)
o< t < m
m < t < 1,
then P is the set of unimodal functions with mode m.
Example 2.5.
If F is defined by
2
Ff(t) = (-D fCt)
2
D f(t)
1
t
l ..:: t ..:: t 2
o ..::
t ..:: t
l
or
then P is the set of functions which are concave on [t ,t ] and convex elsel 2
where. The applications of these sample characterizations should be
abundantly clear.
In principle we desire to find an estimating function which belongs to P.
Clearly, in general, there will be many possibilities.
In order to ensure that
our estimate is also as smooth as reasonably possible, however, we select a
-~
function satisfying the following criterion:
4
;1 (f(m)(t))2 dt
Minimize
o
subject to
2.1
(a)
f
(b)
a
W , Ff(t) > 0 for every t
m
-
f
_< f(t.) <
.
l
I
-
B.,
I
Constraints 2.la are, of course, simply that f
[0,1]
E
i = 1,2, ... ,n
E
P.
The second constraint 2.lb
deserves a bit more comment.
Suppose our data points are {(t.,y.); i = l, ... ,n}.
1
A function f in W
m
1
which coincides exactly with a polynomial of degree 2m-l on each interval
;1 (f(m)(t))2 dt is called a polynomial spline
0
[t.,t.
1] and which minimizes
1
1+
of degree 2m-l and the t., i = 1,2, ... ,n are the knots of this spline.
1
A well-studied problem in approximation theory is to find a solution of
the following optimization problem:
;1 (f(m)(t))2 dt subject to
Minimize
2.2
o
(a)
fEW
(b)
f(t.)=y.,
m
1
1
i
1,2, ... ,n.
The solution f, called an interpolating spline is a spline as just defined.
In contrast the problem:
n
Minimize
2.3
I
i=l
(yi - f(t )) 2
i
+
A!~ (f(m) (t)) 2 dt
with A > 0 fixed and subject to
(a)
fEW
also has a spline function solution called a nmoothing spline.
and Wahba (1970) or Cogburn and Davis (1974) for details.
rn
See Kimeldorf
e-
5
A smoothing spline intermediate between 2.2 and 2.3 solves the following
problem:
(f(m) (t)) 2 dt
Minimize
subject to
2.4
a. < f(t.) < S.,
l.-
l.
-
i
l.
1,2, ... ,n.
==
See Atteia (1968) for more details.
It will be seen that Problem 2.1 is the isotonic version of Problem 2.4.
Our results show that a solution to 2.1 exists and is a polynomial spline of
degree 2m-I.
The Hilbert space methods used to solve 2.1 cannot be used to
solve the isotonic version of 2.3 because the objective function is not a
Hilbert space norm squared.
Using convex programming one of the authors has
solved the isotonic version of 2.3 for F
=
D, but the general problem remains
unsolved.
3.
A Restricted Isotonic Spline.
We consider first a restricted problem.
Let
F be a continuous linear map of W _ l into L which commutes with differentiation.
m
2
Denote the partial order defined by F on W by», and assume
m
function g E W satisfying
m
(Fg)(t) ..:
E:
a. < g(t.) < S. for i
l.
1.
1.
==
there exists a
1,2, ... ,n with
> 0 for every t E [0,1]. The continuity of DF implies there exists
ad> 0 such that
for every fEW
2
m
12
Take N ..: d 11 D gl L
2
2
IE: •
m
In what follows, we shall typically assume n > m.
6
Theore~
3.1.
Under the conditions just given, the problem:
Minimize
3.1
subject to
a.
feW
b.
f» 0
c.
a. < f(t.) < S.,
d.
Ff(j/N) >
1
m
-
1
i = 1,2, ... ,n
1
£,
j
=
0,1,2, ... ,N,
has a solution which is a polynomial spline of degree 2m-l with possible knots
at t., i = 1,2, ... ,n and at j/N, j = 0,1, ... ,N.
1
In order to establish Theorem 3.1, we need to estahlish some preliminary
results.
The straightforward proof of Lemma 3.2 is omitted.
Lemma 3.2.
< IIDmg IlL
IIDmfllL
that f »
and FfU/N) >
L,
j = 0,1, ... ,N together imply
2
2
O.
We next define
Al
= {u.1
E W , i = 1,2, ... ,n: <u.,f>
=
-f(t.) for every fEW}
A
2
= {v.1
E W, i
= 1,2, ... ,n:
=
f(t.) for every fEW}
m
m
1 1 m
<v.,f>
1 1 m
and finally
= -F
f(j/N) for every feW}.
m
7
Take L
=
Al U A u A and define p; L
2
3
~
R by
p(u.)
=
-a.
u.
E
Al
p(v.)
=
B.1
v.
(
A2
p(w.)
= -E
W.
~
1
1
1
1
1
1
A
3
1
We define a convex subset C of W by
m
{f
C
E
W;
m
for every
~ E
L,
<~,f> ~ p(~)}
That is to say, C is just the set of functions in W which satisfy the constraints.
m
--
Let T be a continuous linear map of W onto L2 and suppose Ker T is of finite
m
m
dimension. (In the case of the application we have in mind, T = D so that both
conditions are trivially satisfied.)
fying
We thus have in mind finding s
€
C satis-
I ITsl I = min I ITfl I·
Such an element s will be called the generalized
fEC
spline function relative to T in the set C.
Next let
C = {f
E
W:
m
<~,f> ~ 0 for every ~ c L}.
We posit four structure
axioms and, in turn state two theorems.
HI.
C n (Ker T)
H2.
The subset LeW is weakly compact and the map p is a continuous
m
= {a}
map of L into R.
H3.
C n(Ker T) is empty.
H4.
I
= {f
E
W : for every
m
~, <~,f>
Under HI, Atteia (1968) proves Theorem 3.3.
<
p(~)}
is not empty.
8
Theorem 3.3.
Under HI, there is at least one
spline function relative to T
in the convex set C.
We refer the reader to Atteia for proof.
If, in addition, we consider the
remaining axioms, we have the following result due to Laurent (1969).
Theorem 3.4.
Under HI-4, the element
is a spline function (relative to T
SEC
in C) if and only if
where
F
s
= {~
E
L:<~,s>
p(~)}
and CC(F s ) is the smallest closed convex cone with vertex 0 containing F and
s
T* is the adjoint of T.
We are now in a position to apply Lemma 3.2 and Theorems 3.3 and 3.4 to the
present case.
Proof of Theorem 3.1:
At the optimum s,
II Oms II IJ2 .2
Ilomgll L
2
and so by Lemma
3.2, the spline function must satisfy all the constraints of L, and hence s »
O.
We need only to verify axioms HI-4 to guarantee the existence of s via Atteia's
existence theorem and its characterization via Laurent's characterization theorem.
In our case, T = Om , so that Ker T is the set of polynomials of degree m-I.
-
Every function in C is zero at all points t. , i = 1,2, ... ,n.
-
n > m-l, C n Ker T
1
= {OJ.
Hence HI holds for n > m.
finite hence compact in any topology.
is satisfied.
Hence as soon as
For the L described, L is
This implies p is continuous so that H2
e-
9
Next we consider H3.
C is the set of functions which satisfy the
constraints, while Ker T is the set of polynomials of degree
lIence if
m~l.
C n (Ker T) is not empty, then the optimal solution, s, independent of the
Theorems of Atteia and Laurent, will be a polynomial of degree m-l.
of degree m-l is trivially a polynomial spline of degree 2m-I.
A polynomial
Hence, we may
assume C n (Ker T) is empty.
Finally, I = {f
E
W:
m
<~,f>
<
p(~)
for every
~}
contains the function g
of the conditions to Theorem 3.1 so that I is not empty.
is empty, the characterizing Theorem 3.4 applies.
closed convex cone generated by all
~ E
Thus provided C n Ker T
Thus, -T*Ts belongs to the
L corresponding to active constraints.
In other words,
-•
T*Ts
with
d~ ~
0 and
d~ =
0 when
~
is not an active constraint.
are point evaluation functionals.
~ E
Thus T*T s is zero at all points except
those corresponding to active constraints (which become
2m
points, T*Ts = 0, i.e. D s = O.
Recall that all
knots).
Between knot
Hence, s is a polynomial of degree 2m-l
0
between knot points.
4.
The General Isotonic Spline.
We have already mentioned the general isotonic
spline with possibly an infinite number of knots able to occur anywhere.
The
following result makes the details precise,
Theorem 4.1.
exists g
-e
E
Let»
be the partial order defined by F: W _ l
m
W satisfying
m
~
L .
Z
If there
L
10
Fg(t) > 0
t
rO,l]
(
and
a. < g(t.) < B.
1
1
1,2, .. _,n,
j
I
then the problem:
Minimize
subject to
4.1
a)
f»
b)
cx. < f(t.) < B.
1
0
-
1
1.
i = 1,2, ... ,n
has a solution which is a polynomial spline of degree 2m-I.
Knots are located
(potentially) at data points, t., and, in exceptional cases, at a countahle
1
•
number of points elsewhere.
Proof:
The proof follows the lines of argument for Theorem 3.1.
m
We take T = D
and define
= Ff(t) for every feWm}
Let pew) = 0 for all w
E
A .
3
The key requirement that A be weakly compact holds
3
in this case and p is continuous.
The remainder follows as before.
o
e·
11
5.
Two examples.
Passow (1974) and Passow and Roulier (1977) presented some
initial results concerning monotone and convex splines.
•
i
=
If (t.,y.),
1
1
1,2, ... ,n represents the data points, they were interested in obtaining
piecewise monotone or convex interpolating splines.
used to extend these results.
Theorem 5.1.
Theorems 3.1 and 4.1 may be
We assume t l < t < ... < tn'
2
If Yl < Y2 < ... < Yn '
With arbitrarily small error bounds, there
is a globally monotone cubic smoothing spline of the form of Theorems 3.1 and
4.1 minimizing
Proof:
f~ (f" (t)) 2 dt.
We need establish that there exists a suitable g satisfying the assump-
tions of Theorem 3.1.
Let
ep(t)
={ ~XP
(1/(t
2
Itl
It I
- 1))
-e
•
< 1
> 1
ep(t) is known to be infinitely differentiable on the real line and hence
epEW,m>l.
m
Let epl (t) =
ft_00 ep(u)du.
00
epl is in C and it is clear that
t
fl
-1
and, of course, Depl (t)
hold,
f
00
€
0 for all t.
ep(u)du
t > 1.
When the conditions of this proposition
adding together suitably scaled translates of epl will give a function
E C with Df (t)
l
l
If
~
< -1
~
= y.,
1
0 for every t which satisfies f l (t 1.)
is the error bound on Yi' the function gl (t)
fies the hypotheses of Theorem 3.1 and 4.1 with F
f (t)
1
=0
+
and m
Et, t
= 2.
= 1,2, ... , n.
i
E
[0,1] satis-
o
12
Theorem 5.2.
<
If
for i
1,2, .. "n-2, and the y. have
=
1
arbitrarily small error bounds, then there exists a glohally convex quintic
smoothing spline of the form of Theorems 3.1 and 4.1 minimizing
Proof:
f~
(f " '(t))2 dt.
'
From the function ¢(t) of Theorem 5.1, we define
f~oo ¢(u)du]dx .
When the conditions of the theorem hold, adding together suitably scaled trans-
satisfies f (t )
2 i
=
i
yi'
Yi' the function g2 (t)
=
E
C with 0 f 2 (t) ~
° for all
1,2, ... ,n. If E is the maximum error bound on the
£t2
t c [0,1] meets the assumptions of
f (t) + -22
= 0
Statistical Interpretation.
2
and m
3.
D
In the foregoing sections, we describe the
isotonic spline purely in terms of its mathematical character.
its statistical nature.
We now turn to
We consider a statistical model of the form
Yet)
6.1
t which
=
Theorems 3.1 and 4.1 with F
6.
2
00
lates of ¢2(t) will give a function f 2
= f(t)
+
eft),
t
E
[0,1],
where yet) is an observation at location t, f(t) is a function to be estimated
and eft) is an error.
Of course, [O,lJ can be replaced with any finite interval.
We have in mind to discuss several possihle error structures.
In our first type of error structure, we consider a data set (t. ,Y(t.)),
1
i = 1,2, ...
such that the t. are dense on the interval [0,1] and such that
1
e(t.) form an i.i.d. sequence of independent errors.
I
1
We assume, first of all,
that the support of the common density of eft) is a finite interval, say
e·
13
[-el,e ] containing 0.
Z
There is no a priori need to consider the density to be
symmetric or with mean 0.
We note that yeti)
forms a 100% confidence interval for f(t.).
1
Lemma 6.1.
+
el = 8
i
and yeti) - eZ = a
We have the following lemma.
Suppose the model 6.1 holds with e(t.), i = 1,Z, ...
1
Ll.d. random variables with support as described above.
{t.: i
1
= 1,Z, ... }
is dense in [0,1].
i
Then for
n
a sequence
Suppose further that
> 0 and any interval (t,t'),
there is a t. and t. in (t,t') such that
J
1
B.1 -
f(t.) < n almost surely
1
and
f(t.) - a. < n
J
Proof:
J
almost surely.
We consider the subsequence of t!s falling in the interval (t,t').
1
We
relabel this sequence t. and we note that the subsequence, e(t.), of random
1
1
errors corresponding to the t.
1
E
(t,t') again forms an i.i.d. sequence.
Consider
the interval (-el,-e +n). Since the support of eft) is (-el,e ), the set (-el,-el+n)
Z
l
has positive probability.
=
Hence with probability one, for sufficiently large i,
f(t.) + e(t.) + e l < f(t.) - e l + n + e = f(t.) + n.
l
1
1
1
1
Similarly for a ..
J
We may now state a consistency theorem.
o
14
Theorem 6.2.
Let F
= D be the operator, so that we have a monotone
Suppose further that ex., S. arc as defined above and
non-decreasing spline.
1
1
Finally let f ~ W be non-decreasing, then s ,
m
n
that {t.} are dense in [0,1].
1
the isotonic spline based on (tl,Y(tl)), ... ,(tn,Y(t )), exists and
n
almost uniformly
s
> f with probability one.
n
Proof:
Since ex. < f(t.) <
1
tence of s
n
1
in Theorems 3.1 and 4.1.
Let E > 0 be given.
uous.
B.1 w.p. 1, f is the g needed to guarantee the exis-
Since f is continuous on [0,1], f is uniformly contin-
Corresponding to E/3 there is a 8 > 0 such that If(x) - fey)
whenever Ix-yj ~ 8.
I
<
E/3
We divide [0,1] into consecutive intervals, I., of length
J
6 except for possibly the last which may be shorter than 6.
Consider the interval I. and let its endpoints be x. < x ..1'
J
f and s
n
J
Since both
J+
are non-decreasing on [x.,x. 1]'
J
J+
Consider first f(x. 1) - s (x.).
n
J+
J
f(x. 1) - s (x.) < f(x. 1) - f(x.)
n
J+
+
since
Ix.J + 1
-
J
-
J+
J
+
f(x.) - s (x.) < £
. J
J
n
f(t.) - a. < E/3.
1
1
s (x.) > s (t.) > a.
n J n 1 1
-
3
J
every t. < x ..
1 -
interval, (x. l'x.], we may choose an i such that t.
J
J
f(x.) - s (x.),
x.1
= 6.
J
J-
n
.1
c
J
For the
(x. l'x.] and
J-
J
This happens with probability one by Lemma 6.1.
Thus since
15
f(x.) - 5 (X.) < f(x.) - 5 (t.) < f(x.) - a. < f(x.) - f(t.)
J
J
n
+
-
f(t.)
1
n
J
1
.1-
J
1
0. '
1
o
)
J
But f(x.) - f(t.) < £/3 since Ix. - t.1 < Ix.
x. 11 = 0 and f(t.) - a. < £/3
J
1
J
1
J
J1
1
with probability one by above. Hence f(x. 1) - 5 (x.) < 3£/3 = £ with proban
J+
bilityone.
Similarly
5
n
(x. 1) - f(x.) <
J+
J
-
£
J
-
with probability one so that for
n sufficiently large
If(t) -
5
n (t)1 ~
t
£,
E
1.,
J
w.p. l.
Clearly this convergence holds for all I. except possibly the first and last,
J
hence except on a set of measure less than 28.
The almost uniform convergence
o
is clear.
This result may be extended by the following Corollary.
Corollary 6.3.
Suppose the operator F in Theorem 6.2 is replaced by any of
those in Examples 2.2, 2.3, 2.4 or 2.5, and suppose Ff(t) > O.
If the remaining
conditions of Theorem 6.2 hold, then s n exists and
almost uniformly
s
Proof:
If F
If F
= -0,
with probability one .
the result follows in an obvious parallel to Theorem 6.2.
the result follows since f and s
then non-decreasing.
individually.
> f
n
n
..
will be first non-increasing and
The results of Theorem 6.2 can he applied to each part
Similarly for the remaining cases.
o
16
In the foregoing results, the convergence is uniform except possibly
near the end points a and 1.
behavior of s
n
We remark here that by suitably regularizing the
outside the interval (min t., max t.) we can extend the uniform1
ity of convergence to the entire interval, [0,1].
1
This can be done in several
obvious ways which we shall not detail here.
The regressicn model with e(t) having bounded support may he viewed with
suspicion by those used to conventional models with normal errors.
However,
much data these days are collected with digital instrumentation which almost
implies bounded range on the errors.
We also point out that at any given data
point, (t., Y(t.)), the error bounds a., 8. do not change as a function of n,
1
1
1
1
i.e., we are not supposing increasingly accurate bounds.
Of course, the fact that the support is bounded allows us to give 100% error
bounds which implies that f(t.)
always falls in (a.,
8.),
which, in turn,
1
1
1
guarantees the existence of s.
If the support is unbounded, as with normal
n
errors, finite 100% error bounds are impossible.
A natural suggestion is to
replace the 100% error bounds with a% bounds for some a < 100.
The problem is
that, with probability one, f(t.) will fall outside the error bounds for some
1
i, and, hence, we are not even guaranteed the existence of a sequence of splines
much less their consistency.
Hence, the consistency in the style of Theorem
6.2 for unbounded support is a moot point.
Not all is hopeless, of course, in this case.
For large sets of data, about
a% of the intervals are not going to contain f(t) and hence may
sistent with isotonicity.
~ven
be incon-
A practical procedure may be to discard a percentage
of the bounds (say no more than a%) which prevent us from fitting an isotone
spline, and use the remainder to fit the spline.
Alternatively, we could borrow
a leaf from Barlow et al (1972) and simply "pool adjacent violators".
Hence
extend the length of interval for a violating interval by "averaging" with an
adjacent non-violating interval.
17
We may also note that the situation where there are many data points is
not really the optimal situation for use of the isotonic spline anyway.
is much noisy data, a conventional smoothing spline is appropriate.
If there
And of
course in a noise-free data situation, an ordinary interpolating spline is
appropriate.
It is in the context of a relatively sparse data set that the
added knowledge of isotonicity will allow for the relatively largest improvement
in efficiency.
The regression model, 6.1 of course, is not the only possible statistical
application.
If (a. , 8. ) represent strongly consistent estimates of f(t.),
In
In
I
n~
i.e. a. , 8.
In
In
-+
w.p.I.
f(t.) with probability 1 as n
I
+ "",
then clearly s (t.)
n I
-+
f(t.)
I
Clearly if the t. 's become dense in [0,1] as n increases and if a suit-
able sequence of s
I
n
exists, then for Examples 2.2 through 2.5, s
strongly consistent pointwise estimator of f.
n
will be a
Using the methods of the
Glivenko-Cantelli theorem, the pointwise consistency can be extended to almost
uniform consistency.
Finally, we note in some circumstances it is possible to construct confidence bands (rather than just confidence intervals).
The cumulative periodogram,
for example, is an isotone function which admits confidence bands (c.f. Box and
Jenkins (1970)).
A confidence band amounts to an uncountable set of constraints,
but for which only a finite number need he active.
Clearly, if the confidence
bands converges to f as n increases, we again may construct a strongly consistent estimator of f using the appropriate isotonic spline.
18
REFERENCES
Atteia, M (1968), "Fonctions (spline) definies sur un ensemble convexe",
Math.~
Num.
12~
192-210.
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972),
Statistical Inference under Order
Restrictions~
John Wiley and Sons, New
York.
Barlow, R. E., Marshall, A. W., and Proschan, F. (1963), "Properties of
probability distributions with monotone hazard rate", Ann. Math.
34~
Statist.~
375-389.
Box, G. E. P. and Jenkins,
Control~
G. M. (1970), 7'1:me Series AnalyBis Forecasting and
Holden-Day, San Francisco.
CdgbU¥n, R.and Davis, H. T. (1974), "Periodic splines and spectral estimation",
Ann. Statist., 2, 1108-1126.
Kimeldorf, G. S. and Wahba, G. (1970), "A correspondence between Bayesian
estimation on stochastic processes and smoothing by splines," Ann. Math.
Statist., 41, 495-502.
Laurent, P. J. (1969), "Construction of spline functions on a convex set",
Approximation with Special Emphasis on Spline FUnctions, (ed. I. J.
Schoenberg) 415-446, Academic Press, New York.
Marshall, A. W. and Proschan, F. (1965), "Maximum likelihood estimation for
distributions with monotone failure rate", Ann. Math. Statist., 36,
69-77 .
Passow, E. (1974), "Piecewise monotone spline interpolation", J. Approx. Theory,
12, 240-241.
Passow, E. and Roulier, J. A. (1977), "Monotone and convex spline interpolation",
SIAM J. Numer. Anal., 14, 904-909.
e'
19
Robertson, T. (1967), "On estimating a density which is measurable with respect
to a a-lattice," Ann. Math. Statist. 38, 482-493.
Wegman, E. J. (1970a), "Maximum likelihood estimation of a unimodal density
function", Ann. Math. Statist, 41, 457-471.
Wegman, E. J. (1970b), "Maximum likelihood estimation of a unimodal density, II,"
Ann. Math. Statist., 41, 2169-2174.
Wright, 1. W. (1977), "Splines in statistics," Institute of Statisticis Mimeo
Series No.
114~
Chapel Hill, North Carolina.
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Ian W. Wright and Edward J. Wegman
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SUPPLEMENTARY NOTES
19, KEY WORDS (Contlnu. on r.v.r•• • Id. /I n.e. . ..." and Id.nllly by block numb.r)
isotone, isotonic, convex, partial order, spline, isotonic spline,
smoothing spline, interpolating spline, regression, strong consistency
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20,
ABSTRACT (Contlnu. on r.v. . . . .Id. /I n.e ••••ry . . d Id.ntlly by block numb.r)
In this paper, we consider the estimation of isotonic, convex or related
functions by means of splines. It is shown that certain classes of isotone or
convex functions can be represented as a positive cone embedded in a Hilbert
space. Using this representation, we give an existence and characterization
theorem for isotonic or convex splines. Two special cases are examined showing
the existence of a globally monotone cubic smoothing spline and a globally
convex quintic smoothing spline. Finally, we examine a regression problem and
show that the isotonic-type of spline provides a strongly consistent solution.
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We also point out several other statistical applications.
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