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R n ( V , p ) = inf T n sup m V E T n - m p p ,

where the infimum is taken over all measurable estimators T n of m . Similarly, we define the linear L p - minimax risk as

R n lin ( V , p ) = inf T n lin sup m V E T n lin - m p p ,

where the infimum is now taken over all linear estimators T n lin . Obviously, R n lin ( V , p ) R n ( V , p ) . We first state some definitions.

The sequences { a n } and { b n } are said to be asymptotically equivalent and are noted a n b n if the ratio a n / b n is bounded away from zero and as n .
The sequence a n is called optimal rate of convergence , (or minimax rate of convergence ) on the class V for the L p - risk if a n R n ( V , p ) 1 / p . We say that an estimator m n of m attains the optimal rate of convergence if sup m V E m n - m p p R n ( V , p ) .

In order to fix the idea, we consider only the L 2 - risk in the remaining part of this section, thus p : = 2 .

In [link] , [link] , the authors found that the optimal rate of convergence attainable by an estimator when the underlying function belongs to the Sobolev class W q s is a n = n - s 2 s + 1 , hence R n ( V , 2 ) = n - 2 s 2 s + 1 . We saw in "Linear smoothing with wavelets" that linear wavelet estimators attain the optimal rate for s - Hölder function in case of the L 2 - risk (also called `IMSE'). For a Sobolev class W q s , the same result holds provided that q 2 . More precisely, we have the two following situations.

  1. If q 2 , we are in the so-called homogeneous zone. In this zone of spatial homogeneity, linear estimators can attain the optimal rate of convergence n - s / ( 2 s + 1 ) .
  2. If q < 2 , we are in the non-homogeneous zone, where linear estimators do not attain the optimal rate of convergence. Instead, we have:
    R n lin ( V , 2 ) / R n ( V , 2 ) , as n .

The second result is due to the spatial variability of functions in Sobolev spaces with small index q . Linear estimators are based on the idea of spatial homogeneity of the function and hence do perform poorly in the presence of non-homogeneous functions. In contrast, even if q < 2 , the SureShrink estimator attains the minimax rate [link] . The same type of results holds for more general Besov spaces, see for example [link] , Chapter 10.

Adaptivity of wavelet estimator

We just saw that a nonlinear wavelet estimator is able to estimate in an optimal way functions ofinhomogeneous regularity. However, it may not be sufficient to know that for m belonging to a given space, the estimator performs well. Indeed, in general we do not know which space the function belongs to. Hence it is ofgreat interest to consider a scale of function classes and to look for an estimator that attains simultaneously the best rates of convergence across the whole scale. For example, the L q - Sobolev scale is a set of Sobolev function classes W q s ( C ) indexed by the parameters s and C , see [link] for the definition of a Sobolev class. We now formalize the notion of an adaptive estimator.

Let A be a given set and let { F α , α A } be the scale of functional classes F α indexed by α A . Denote R n ( α , p ) the minimax risk over F α for the L p - loss:

R n ( α , p ) = inf m ^ n sup m F α E m ^ n - m p p .
The estimator m n * is called rate adaptive for the L p - loss and the scale of classes F α , α A if for any α A there exists c α > 0 such that
sup m F α E m n * - m p p c α R n ( α , p ) n 1 .

The estimator m n * is called adaptive up to a logarithmic factor for the L p - loss and the scale of classes F α , α A if for any α A there exist c α > 0 and γ = γ α > 0 such that

sup m F α E m n * - m p p c α ( log n ) γ R n ( α , p ) n 1 .

Thus, adaptive estimators have an optimal rate of convergence and behave as if they know in advance in which class the function to be estimated lies.

The VisuShrink procedure is adaptive up to a logarithmic factor for the L 2 - loss over every Besov, Hölder and Sobolev class that is contained in C [ 0 , 1 ] , see Theorem 1.2 in [link] . The SureShrink estimator does better: it is adaptive for the L 2 - loss, for a large scale of Besov, Hölder and Sobolev classes, see Theorem 1 in [link] .

Conclusion

In this chapter, we saw the basic properties of standard wavelet theory and explained how these are related to the construction of wavelet regression estimators.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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Anassong
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
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s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, An introduction to wavelet analysis. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10566/1.3
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