Isotonic regression refers to the problem of estimating a monotone sequence
based on a noisy observation vectorwhich is assumed to be an additive perturbation of ,
where the components of
are assumed to have zero mean and unit variance. It is commonly assumed thatare independent and identically distributed (i.i.d.) but we work with the more general assumption of exchangeability in this paper. A natural estimator for in this setting is the isotonic Least Squares Estimator (LSE), defined as
where denotes the usual Euclidean norm on and is the monotone cone of length non-decreasing sequences. As is a closed convex cone, as defined above exists uniquely; it can also be computed in time by the pool adjacent violators algorithm [4, 11].
The statistical properties of are typically studied in terms of the risk or the normalized mean squared error:
A key quantity in understanding is
where denotes the law of the noise vector . Indeed, it is clear that
where denotes the marginal distribution of and is the length of the block for all . We emphasize that (1.1) holds for arbitrarily dependent with zero mean and finite variance. It was also shown in  that also bounds the risk of the isotonic LSE in misspecified settings where does not lie in .
The quantity therefore crucially controls the risk of the isotonic LSE. The goal of this paper is to explicitly determine for every under the additional assumption that is exchangeable. Specifically, under the assumption of exchangeability, we show in Corollary 3.3 that, for all ,
where is the harmonic number and is the pairwise correlation. Combined with (1.1), our result provides a sharp non-asymptotic bound on the risk of isotonic regression for any exchangeable noise vector. In the special case when are i.i.d. with zero mean and unit variance, and thus (1.2) gives:
Here is the common distribution of the independent variables .
Previously, the formula (1.3) was known when is the standard Gaussian probability measure on . This was observed by Amelunxen et al.  who proved it by observing first that when and is the standard Gaussian measure, the formula
holds for every closed convex cone where is the intrinsic volume of . When is the monotone cone, the right hand side in equation (1.4) can be shown to be equal to by using the fact that the generating function can be computed in closed form. Amelunxen et al.  used the theory of finite reflection groups  to obtain the exact expression for this generating function. However, the exact expression for can already be found in the classical literature on isotonic regression (see Theorem 2.4.2 in Roberston et al.  and references therein).
The above proof does not work for non-Gaussian mainly because the expression (1.4) does not hold for general . In fact, the best available result on for non-Gaussian is in equation (2.11) of Zhang , who proved the asymptotic result:
This bound gives the right behavior as the right hand side of equation (1.3) but only as . We improve this result by proving for every that is always equal to the harmonic number for every probability measure having mean and variance .
We prove (1.2) by developing a precise characterization of the marginal distribution of each individual component of . Specifically, as long as is exchangeable, we show in Theorem 2.2 that has the same distribution as , the order statistic of the running averages . We prove Theorem 2.2 in Section 2, using a characterization of the components of the isotonic LSE as the left-hand slopes of the greatest convex minorant of the random walk with increments . This result and its continuous-time analogue may be of independent interest outside the study of isotonic regression, so in Section 2 we also address consequences for the greatest convex minorant of a stochastic process with exchangeable increments. The order statistics of the running averages can be fairly complicated even when is Gaussian; however, Theorem 2.2 easily implies results such as (1.2). In Section 3, we detail some risk calculations for isotonic regression and its variants which all follow from Theorem 2.2.
2 Main Result
Let denote the partial sums for , started at . Identify the random walk with its cumulative sum diagram (CSD) , where for integers
and linearly interpolated between integers. Letdenote the greatest convex minorant (GCM) of , i.e. the greatest convex function that lies below . See Figure 1 for a depiction of the GCM of the CSD. With this notation, we now recall the graphical representation of the isotonic LSE as given in Theorem 1.2.1 of Roberston et al. .
For any vector , the isotonic LSE is given by the left-hand slopes of the greatest convex minorant of the cumulative sum diagram. For all
For the remainder of this section let
denote the left-hand slope of the GCM at , so is equal to by the lemma. In particular, when we have . When , we have , and if then . Our next result generalizes this observation, showing that the slope is equal in distribution to the smallest running average if is exchangeable.
Suppose is exchangeable. Let denote the running average for and let denote their order statistics. Then
marginally for all .
As before, let denote the partial sum. Let be the last argmin of the sequence , and let be the amount of time the walk is non-positive . We will use Corollary 11.14 of Kallenberg , due to Sparre-Andersen, which says as long as is exchangeable.
Note that the slope of the GCM switches from non-positive to positive at time , since the horizontal line with intercept minorizes the GCM and touches it at time . Hence, no matter the sequence of increments , there is the identity of events
Also, for the time that the walk is non-positive, since if and only if , there is the identity of events
The equality in distribution then implies
If the sequence is modified to for some fixed , the modified sequence is exchangeable, and the values of and for the modified sequence are just and . Applying the above identity to the modified sequence gives
have the same cumulative distribution function, hence the same distribution. ∎
The proof of Theorem 2.2 has a straightforward generalization to the setting where is a continuous-time stochastic process. Knight  showed that the analogous distributional identity holds when has exchangeable increments and . Hence, by a similar proof, we find that the slope of the greatest convex minorant of at time has the same distribution as the percentile point of the occupation measure for the process . We record this result as the following corollary.
Let denote a real-valued càdlàg stochastic process on with exchangeable increments, such that . Define as the slope of the greatest convex minorant of at , and let denote the (random) cdf associated with the occupation measure of ,
where denotes Lebesgue measure. Then
marginally for all .
See Abramson et al.  for a general study of convex minorants of random walks and processes with exchangeable increments. In the special cases where is a standard Brownian motion or Brownian bridge on the unit interval, Carolan & Dykstra  derive the distribution of the slope , jointly with the process and its convex minorant at , for a fixed value . Given our corollary, their explicit formula for the slope provides the distribution of , giving new information about the occupation measure of for Brownian motion and Brownian bridge. The distribution of the percentile point of the occupation measure for has been obtained under the same generality as Corollary 2.3: see the introduction of Dassios  and references therein.
3 Consequences for Isotonic Regression
Since the identity of Theorem 2.2 holds marginally, it allows us to simplify expectations of functions that are additive in the components of . As long as is exchangeable,
Taking , we obtain our first corollary.
Suppose is exchangeable. For ,
Viewed through its graphical representation, is the left-derivative of the GCM at , so when the power , equation (3.2) yields the discrete arc-length formula
Closely related to this formula is the identity of Spitzer &
Widom , which takes to be a
sequence of i.i.d. random variables in
to be a sequence of i.i.d. random variables in(or the complex plane ) with finite variance. If is the partial sum and is the length of the perimeter of the convex hull , then
These formulas connect the geometry of the convex hull of a random walk to the magnitudes of the running means.
Consider the case when . Since is exchangeable, every pair of components has the same correlation . If we further assume has zero mean and unit variance, the right hand side of equation (3.2) can be computed explicitly
Summing over yields our next result.
Suppose is an exchangeable random vector with zero mean, unit variance, and pairwise correlation . Then
This result should be contrasted with other distribution-free identities, namely
provided has i.i.d. components with zero mean and unit variance. In particular, suppose we observe where has i.i.d. components with zero mean and unit variance, but it turns out that is constant. If we know is constant, we can estimate it by a constant sequence and pay a constant price in total risk. If we know nothing about the structure of and use , the risk is quite large by comparison. The monotone sequence estimate resides in the middle, with a much smaller risk of and knowledge only about the relative order. We explained in Section 1 how risk calculations when generalize to MSE bounds that are sharp in the low noise limit for arbitrary . For example, when has constant pieces, then (1.1), Corollary 3.3 and the fact that for every imply that
where is the number of constant pieces of the vector . These formulae (with the leading constant of 1 in front of the term on the right hand side) were previously only known when the distribution of was standard Gaussian.
Define the -risk of the isotonic LSE
so that . We can similarly employ Theorem 2.2 to explicitly calculate the -risk of the isotonic LSE when is constant and is Gaussian:
Suppose . Then for any ,
Note and apply the theorem. ∎
Corollary 3.4 should similarly be contrasted with the following identities when :
respectively. In particular, when , the bound holds for all , which is to say is bounded when whereas grows without bound as grows.
When is constant and , the risk of isotonic regression is
The continuous-time distributional identity in Corollary 2.3 applies to the asymptotic distribution of the isotonic least squares estimator. A standard model for studying the asymptotic behavior of isotonic regression is
where is non-decreasing. We observe , a noisy version of , and calculate by projecting onto the monotone cone. The function estimate is defined by and linearly interpolated between design points. Here, as before, the dependence on in is suppressed, but now we are interested in the behavior of isotonic least squares at a fixed point as .
Define the partial sum process by , linearly interpolated between design points. When the function is constant, the quantity
is given by the left-derivative of the greatest convex minorant of at . By the invariance principle, this converges in distribution to the left-derivative of the greatest convex minorant of standard Brownian motion at . This asymptotic result is well known and a similar result was noted for the Grenander estimator in Carolan & Dykstra , where Brownian motion is replaced with a Brownian bridge. Corollary 2.3 relates this asymptotic distribution to the percentile points of the occupation measure for .
Finally, Corollary 3.3 on the projection onto extends over to that of the set of non-negative monotone sequences . Theorem 1 of Németh & Németh  observes that the projection of onto is given by , the element-wise positive part of the projection onto . Hence the distributional identity Theorem 2.2 yields a similar set of identities for non-negative isotonic regression.
For any exchangeable noise vector ,
Furthermore, if is symmetric with unit variance, the generalized statistical dimension of the monotone cone is
where is the pairwise correlation.
Equation (3.8) is also shown in Amelunxen et al.  in the special case using the theory of finite reflection groups. The identity (3.7) allows us to show equation (3.8) for a much wider variety of noise vectors, and as before also allows us to obtain relations for the expected norms of the projection of the noise vector. All of our exact formulae follow from the distributional identity in Theorem 2.2, which exploits the geometric characterization of the isotonic LSE in Lemma 2.1. An interesting open question is whether similar characterizations— such as for convex regression —may yield exact non-asymptotic risk calculations in other shape-constrained estimation problems.
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