Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise

Abstract

We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.

1. Introduction

A nonparametric regression (NPR) problem consists of estimating an unknown mean function that smoothly changes between observations at different design points. There are 𝑛 observations 𝑌_𝑖 of the form

𝑌_𝑖𝑖=𝜇(𝑛)+𝜉_𝑖for𝑖=1,…,𝑛, (1.1) where 𝜇 is the unknown smooth mean function on [0,1] and the errors 𝜉_𝑖 are observations from a zero-mean Gaussian process. For NPR problems that have a particular long memory structure to the covariance of the error terms, we will find a continuous Gaussian experiment approximation to the problem of estimating the mean.

Brown and Low [1] showed that the NPR experiment is asymptotically equivalent to the white-noise model where the mean function is observed in the presence of a Brownian motion process. This result paralleled the work by Nussbaum [2] in showing that asymptotic results in nonparametric function estimation problems can be simplified using approximations by the continuous white-noise experiments that Pinsker [3] studied. The original asymptotic equivalence results for NPR experiments were extended by Brown et al. [4] and Carter [5, 6] along with refinements in the approximations from Rohde [7] and Reiss [8].

All of these results assume that the errors 𝜉_𝑖 in (1.1) are all independent, and this assumption is critical in establishing the appropriateness of a white-noise model that also has independent increments. We want to consider the effect of correlation between the observations on these approximations. Presumably, if the correlation is weak, then the effect washes out asymptotically. However, we wish to consider cases where there is sufficient long-range correlation to affect the form of the approximation. In particular, we will show that the appropriate approximation is by a continuous Gaussian process experiment that is no longer white noise but is closer to a fractional Brownian motion.

Our approach is motivated by the work by Johnstone and Silverman [9] and Johnstone [10]. They investigated the wavelet decomposition of data of this type and used a fractional Brownian motion approximation in the limit:

𝑑𝑌(𝑡)=𝜇(𝑡)𝑑𝑡+𝑛^{−(𝛽+1)/2}𝑑𝐵_𝐾(𝑡)𝑡∈[0,1]. (1.2) They argued that the wavelet decomposition resulted in nearly independent coefficients which simplified the inference significantly. We will assume that the 𝐵_𝐾(𝑡) process is decorrelated by a wavelet decomposition and then show that this continuous model is asymptotically equivalent to the NPR experiment with the same covariance structure.

Theorem 1.1. The nonparametric regression experiment ℱ observes 𝑌_𝑖 as in (1.1) for an unknown mean function 𝜇 from a parameter set ℳ(𝑀,𝛼) defined in Section 1.2 and a known covariance structure as described in Section 1.3. This experiment is asymptotically equivalent to the experiment ℰ that observes 𝑑𝑌(𝑡)=𝜇(𝑡)𝑑𝑡+𝜎𝑛^{−(𝛽+1)/2}𝑑𝐵_𝐾(𝑡), (1.3) where 𝐵_𝐾(𝑡) is a Brownian motion with covariance kernel 𝐾.

This will be proven in two steps. First, Lemma 2.1 proves that the first 𝑛 wavelet coefficients in a decomposition of 𝑑𝑌(𝑡) are asymptotically sufficient in ℰ for estimating 𝜇. For the second step, Lemma 3.1 shows that a discrete wavelet transform of the observations from ℱ produces observations with nearly the same distribution as these asymptotically sufficient statistics.

Furthermore, in both experiments the lower-frequency terms in the wavelet decomposition are sufficient for estimating the means, allowing the higher-frequency terms to be used to give information about the variance process. This leads to Theorem 1.2, which proposes an experiment that allows some flexibility in the error structure.

Theorem 1.2. The NPR experiment ℱ observes the 𝑌_𝑖 as in (1.1), where the covariance structure depends on the parameters 𝛽 and 𝛾 and is such that the variance of the wavelet coefficients is 2^{𝛾+𝛽(𝑗+1)}.
The experiment ℰ observes the pair 𝛾𝛽𝛾𝛽∼𝒩(,log22(𝐱^⊤Λ𝐱)⁻¹), (1.4) (where 𝐱 and Λ are defined in Section 4) and then observes the continuous Gaussian process conditionally on 𝛾 and 𝛽: 𝑑𝑌(𝑡)=𝜇(𝑡)𝑑𝑡+𝑛⁻⁽¹⁺𝛽⁾2𝛾𝐾𝑑𝐵(𝑡), (1.5) where the covariance 𝐾 is such that 𝐾Var(𝐵(𝜓_𝑗𝑘))=2𝛽^(𝑗+1). The estimators 𝛽 and 𝛾 are the same as (𝛽,𝛾) but truncated so that −1≤𝛽≤0, and 𝛾≥−𝑐.
For 𝜇∈ℳ(𝑀,𝛼), −1<𝛽≤0, and 𝛾>−𝑐 for some constant 𝑐, the experiments ℱ and ℰ are asymptotically equivalent.

This theorem can be seen as an extension of Carter [6] Theorem 1.1 from a case where there is a single unknown variance for all the wavelet coefficients to a case where the variance changes as a log-linear function of the resolution level (or frequency).

Wang [11] addressed the issue of asymptotically sufficient statistics in the fractional Brownian motion process. In Section 3 of that article there is an argument that bounds the difference between minimax errors in an NPR experiment with correlated errors and an experiment that observes the mean in the presence of fractional Brownian motion error. This result extends the sort of approximation by Donoho and Johnstone [12] to correlated errors and is very much in the spirit of our Theorem 1.1 here. Our results differ from Wang [11] in that we have made a stronger assumption on the covariance structure of the errors in order to obtain the full asymptotic equivalence of the experiments as discussed in Section 1.1.

Lemma 2.1 is presented and proven in Section 2. Section 3 presents Lemma 3.1 and the proof of Theorem 1.1. The proof for Theorem 1.2 is in Section 4 with some relevant bounds in Sections 5 and 6.

1.1. Asymptotic Sufficiency

Instead of focusing on single estimation techniques, we will consider approximations of the entire statistical experiment. For large sample sizes, there is often a simpler statistical experiment that can approximate the problem at hand. One benefit of finding an approximating experiment is that it may have convenient sufficient statistics even when they are not available in the original experiment.

Our approximations will therefore be of experiments rather than particular distributions. A statistical experiment 𝒫 that observes data 𝑋 consists of a set of probability distributions {ℙ_𝜃} indexed by the parameter set 𝜃∈Θ. We wish to compare the information about 𝜃 in 𝒫 to another experiment 𝒬 that observes data 𝑌 from among the set of distributions {ℚ_𝜃} that are indexed by the same parameter 𝜃. Implicitly, we are concerned with two sequences of experiments 𝒫_𝑛 and 𝒬_𝑛 where 𝑛 roughly denotes the increasing sample size, but generally, we will leave off the subscript 𝑛. It will always be understood that the distributions depend on the “sample size.”

The NPR experiment will be approximated using Le Cam's notion of asymptotically equivalent experiments [13, 14] and asymptotically sufficient statistics [15]. Asymptotically equivalent experiments have corresponding inference procedures (such as estimators or tests) in each experiment that perform nearly as well. Specifically, if there is an estimator 𝜏(𝑋) in 𝒫 with risk ℙ_𝜃𝐿(𝜏(𝑋)), then, for bounded loss functions, there is an estimator 𝜎(𝑌) such that

sup_𝜃ℙ_𝜃𝐿(𝜏(𝑋))−ℚ_𝜃𝐿(𝜎(𝑌))→0 (1.6) as 𝑛→∞. These asymptotic equivalence results are stronger than the equivalence of minimax rates that is derived under a similar model by, for example, Wang [11]. Our results imply a correspondence over a range of bounded loss functions. Thus, the equivalence holds for a global 𝐿₂ error as well as local error measurements or other distances.

Asymptotic sufficiency is a stronger notion, where if 𝑇(𝑋) is a sufficient statistic for inference about 𝜃 in 𝒫, then 𝑇(𝑌) is asymptotically sufficient for 𝒬 when the total-variation distance between ℙ_𝜃 and ℚ_𝜃 is negligible. These asymptotically sufficient statistics generate experiments that are all asymptotically equivalent. In particular, 𝒫 and 𝒬 are asymptotically equivalent, and they are also asymptotically equivalent to the experiments generated by the distributions of 𝑇(𝑋) and 𝑇(𝑌). As a result, an estimator in 𝒫 should generally be of the form 𝜏(𝑇(𝑋)) and there is a corresponding estimator 𝜏(𝑇(𝑌)) that performs nearly as well in the 𝒬 experiment. There is a basic transitive property to the asymptotic equivalence that implies if 𝒫 is asymptotically equivalent to 𝒬, and 𝒬 is asymptotically equivalent to ℛ, then 𝒫 is asymptotically equivalent to ℛ.

Le Cam's asymptotic equivalence is characterized using the total-variation distance 𝛿(ℙ_𝜃,ℚ_𝜃) between the distributions. We will abuse this notation a bit by writing 𝛿(𝒫,𝒬)=sup_𝜃𝛿(ℙ_𝜃,ℚ_𝜃). It will often be more convenient to use the Kullback–Leibler divergence (𝐃(ℙ,ℚ)=ℙlog[𝑑ℙ/𝑑ℚ]) to bound the total variation distance:

𝛿(ℙ_𝜃,ℚ_𝜃)≤2𝐃(ℙ_𝜃,ℚ_𝜃)^1/2 (1.7) See [16]. The divergence is convenient for product experiments because 𝐃(∏ℙ_𝑖,∏_𝑖ℚ_𝑖)=∑_𝑖𝐃(ℙ_𝑖,ℚ_𝑖).

1.2. Wavelet Basis

We will use orthonormal wavelet bases to characterize the function space and to simplify the covariance structure of the errors.

Assuming we are considering periodic functions on the interval [0,1], we can construct periodic wavelet bases as by Daubechies [17, Chapter 9.3]. We start with a space 𝒱_𝑗 which consists of functions of the form

𝑓_𝑗(𝑡)=^2𝑗_𝑘=1𝑎_𝑗𝑘𝜙_𝑗𝑘(𝑡),𝑡∈[0,1], (1.8) where 𝜙_𝑗𝑘 is an orthonormal set of periodic functions generated via

𝜙_𝑗𝑘(𝑡)=2^𝑗/2𝜙(2^𝑗𝑡−𝑘). (1.9) We will work with a 𝜙 function having finite support [0,𝑁], and at the boundaries of the interval the 𝜙_𝑗𝑘(𝑡) are given the proper periodic extensions. This space generates wavelet functions 𝜓_𝑗𝑘 that span the difference between the 𝒱_𝑗 and 𝒱_𝑗−1 and can be written 𝜓_𝑗𝑘(𝑡)=2^𝑗/2𝜓(2^𝑗𝑡−𝑘) with the proper periodic adjustment at the boundary (e.g., 𝜓_𝑗,2𝑗(𝜖)=2^𝑗/2𝜓(2^𝑗𝜖) for a small 𝜖). This periodic adjustment has a small effect at the high resolution levels but is a larger factor for small values of 𝑗. In particular, the scaling function at level 0 is 𝜙₀(𝑡)=∑^𝑁_𝑘=0𝜙(𝑘+𝑡)=1.

The mean functions 𝜇(𝑡) will be assumed to be constructed from this wavelet basis:

𝜇(𝑡)=𝜃₀𝜙₀(𝑡)+²_𝑗≥0^𝑗_𝑘=1𝜃_𝑗𝑘𝜓_𝑗𝑘(𝑡). (1.10) We will restrict the mean functions to those that belong to a Hölder(𝛼) class of functions. Specifically, the class of periodic mean functions 𝜇(𝑡) is ℳ(𝑀,𝛼):

sup_{ℳ(𝑀,𝛼)}sup_𝑡,𝑠|𝜇(𝑡)−𝜇(𝑠)||𝑡−𝑠|^𝛼≤𝑀 (1.11) for some 1/2<𝛼<1 and 𝑀>0. This smoothness condition on the functions bounds the rate of growth of the higher-frequency terms in the orthonormal expansion. Originally from Meyer [18], in Daubechies [17, page 299 equation, equation (9.2.1)] gives the bound

max_𝑘|𝜃_𝑗𝑘|≤𝑀₂2^{−𝑗(𝛼+1/2)} (1.12) for a constant 𝑀₂ related to 𝑀. This bounds implies the useful bound on the squared error in the tail of the basis expansion:

_𝑗>𝑗∗2^𝑗2𝑗_𝑘=1𝜃²_𝑗𝑘≤𝑀²₂2^{−𝜀𝑗∗}(1−2^−𝜀)⁻¹, (1.13) where 𝜀=2𝛼−1.

1.3. Error Structure

These results rely on a specific structure to the covariance matrix of the errors in the NPR experiment. As by Johnstone [10], the fractional Brownian motion is the motivating example for our continuous Gaussian model. However, this model does not necessarily provide the independent coefficients that would simplify the inference. Instead, an error structure that has roughly some of the properties of the fractional Brownian motion will be considered.

Traditionally, the asymptotics of the NPR experiment have assumed independent noise. This white-noise model is especially convenient because all of the eigenvalues of the covariance operator are equal. Thus, any orthonormal basis generates a set of independent standard normal coefficients. With a more general covariance function, the eigenvalues are different and only particular decompositions lead to independent coefficients. Thus there is much less flexibility in the choice of basis, and this basis determines some of the structure of the covariance.

Following Johnstone [10], Johnstone and Silverman [9], Zhang and Waiter [19], Wang [11], and Cavalier [20] among others, we will assume a covariance structure that is whitened by a wavelet decomposition. When there is a long-range positive correlation between the observations, the wavelet decomposition tends to decorrelate the error process because the wavelet functions act like bandpass filters.

We will assume that there exists an orthonormal basis 𝜙₀ and 𝜓_𝑗𝑘 for 𝑗≥0 and 𝑘=0,…,2^𝑗−1 such that the decomposition of the error process generates independent normal coefficients. In other words, the error process is a zero-mean Gaussian process that is roughly

𝑑𝐵_𝐾(𝑡)=𝜉₀𝜙₀(𝑡)+_𝑗≥0𝑘𝜉_𝑗𝑘𝜓_𝑗𝑘(𝑡) (1.14) in the distributional sense where the 𝜉_𝑘𝑗 are independent normals. The Var(𝜉_𝑗𝑘) will be assumed to depend on 𝑗 and not 𝑘 as a sort of stationarity condition. In particular, we will assume that Var(𝜉₀)=𝜎² and then Var(𝜉_𝑗𝑘)=𝜎²2^𝛽(𝑗+1) for some 𝛽 in the interval (−1,0]. If 𝛽=0, then this is the white-noise process.

This is a convenient form for the error, but not completely unrealistic. Wavelet decompositions nearly whiten the fractional Brownian motion process. Wornell [21] argued that long-memory processes can be constructed via a wavelet basis with variances at resolution level 𝑗 shrinking like 2^−𝛾𝑗 for 0<𝛾<2. McCoy and Walden [22] showed that the discrete wavelet transform nearly decorrelates the noise in fractionally differenced white-noise processes. Alternatively, Wang [11] used a wavelet—vaguelette decomposition [23] to find a decomposition of the fractional Brownian motion that results in independent coefficients for a nearly orthonormal basis.

Section 7 demonstrates some properties of the specific Gaussian process generated by using the Haar basis as the wavelet basis. These properties are consistent with the sort of behavior that we want in the covariances of our observations. The correlation between observations decreases like 𝑑^−(𝛽+1) for 𝛽<0 where 𝑑 measures the distance between the locations of the coefficients.

A well-established method for estimating the parameter 𝛽 in these long-range dependent models is to fit a linear function to the log of an estimate of the variances of the coefficients at each resolution level. This idea goes back to at least Abry and Veitch [24] and is now a standard approach that has been improved upon in subsequent work; see Veitch and Abry [25], Stoev et al. [26], among others. This motivates the asymptotic sufficient statistics in Theorem 1.2 which are least squares estimates from the fitted line.

The assumptions in Theorem 1.1 on the covariance structure of the errors are strong and could limit the applicability of the result. However, if we allow the variances at different scales to have a range of linear relationship, we could then have a sufficiently rich class of error models. Theorem 1.2 allows for this somewhat larger class of models, and it seems likely that the changing magnitude of the variances over different resolutions level will have a greater effect on the distribution of the observed errors than the underlying basis.

2. Approximate Sufficiency in the Gaussian Sequence Experiment

The first step in the proof of Theorem 1.1 is to establish that a truncated wavelet decomposition is asymptotically sufficient for the continuous Gaussian experiment.

Lemma 2.1. The experiment ℰ is a Gaussian process experiment that observes 𝑑𝑌(𝑡)=𝜇(𝑡)𝑑𝑡+𝜎𝑛^{−(𝛽+1)/2}𝑑𝐵_𝐾(𝑡), (2.1) where 𝐵_𝐾 is a zero-mean continuous Gaussian process with covariance 𝐾. There are asymptotically sufficient statistics for estimating 𝜇∈ℳ(𝑀,𝛼): 𝑦₀,𝑦_𝑗𝑘∼𝒩(𝜃_𝑗𝑘,𝜎²𝑛^−𝛽−12^𝛽𝑗) (2.2) for 0≤𝑗≤𝑗^∗ as long as 𝑗^∗>log₂𝑛.2𝛼 (2.3)

In the Gaussian sequence experiment ℰ where only the mean 𝜇(𝑡) is to be estimated, the likelihood is

𝑑ℙ_𝜇𝑑ℙ₀(𝐲)=exp[2𝑦₀𝜃₀−𝜃²₀2𝜎²+𝜎⁻²𝑛^(1+𝛽)_𝑗≥0𝜎⁻²2^{−𝛽(𝑗+1)}_𝑘(𝜃_𝑗𝑘𝑦_𝑗𝑘−12𝜃²_𝑘𝑗)], (2.4) where ℙ_𝜇 is the distribution of 𝑌(𝑡) and ℙ₀ is the distribution of the version with mean 0 which would just be 𝜎𝑛^{−(𝛽+1)/2}𝐵_𝐾(𝑡).

We want to approximate this experiment ℰ by a similar experiment ℱ where the mean is projected onto the first 𝑗^∗ resolution levels, that is; 𝜇 is replaced by 𝜇:

𝜇(𝑡)=𝜃₀𝜙₀(𝑡)+^𝑗∗_𝑗=0𝑘𝜃_𝑗𝑘𝜓_𝑗𝑘(𝑡). (2.5) The likelihood becomes

𝑑ℚ_𝜇𝑑ℚ₀(𝐲)=exp[2𝑦₀𝜃₀−𝜃²₀2𝜎²+𝜎⁻²𝑛^{𝑗(1+𝛽)∗}_𝑗=02^−𝛽𝑗_𝑘(𝜃_𝑗𝑘𝑦_𝑗𝑘−12𝜃²_𝑘𝑗)]. (2.6) Therefore, this experiment ℱ has sufficient statistics 𝑦₀ and 𝑦_𝑗𝑘 for 0≤𝑗≤𝑗^∗. These observations are approximately sufficient in the ℰ experiment if the distance between the distributions in the two experiments is small.

By (5.3), the distance between these two sets of experiments is

𝛿(ℙ,ℚ)≤𝜎⁻¹𝑛^(𝛽+1)/2(_𝑗>𝐽2^−𝛽𝑗_𝑘𝜃²_𝑗𝑘)^1/2. (2.7) For the parameter space ℳ(𝑀,𝛼), (5.4) bounds the distance between the two experiments as

𝛿(ℙ,ℚ)≤𝜎⁻¹𝑛^(𝛽+1)/2𝑀2^−𝜀𝐽≤𝑀𝜎⁻¹2^{(𝐽−𝑗∗)(1+𝛽)/2−𝜀𝑗∗}, (2.8) which is negligible as 𝑛→∞ when the dimension of the sufficient statistics increases like

𝑗^∗>(1+𝛽𝛽+2𝛼)log₂𝑛 (2.9) for −1<𝛽<0. The worst case is when 𝛽=0, and thus we have proved Lemma 2.1.

3. Approximating the NPR Experiment

Theorem 1.1 can now be proven by approximating the sufficient statistics from Lemma 2.1 using the observations from the NPR experiment ℱ.

We suppose that we have 𝑛 observations from the NPR experiment as in (1.1) where the 𝜉_𝑖 are Gaussian random variables with a specified covariance function. Specifically, let 𝐖 be the 𝑛×𝑛 matrix that performs the discrete wavelet transform, and let 𝐖^⊤ be its inverse. The vector of random wavelet coefficients from Lemma 2.1, 𝐲=(𝑦₀,𝑦₀₀,𝑦₀₁,…,𝑦_{𝐽−1,2𝐽−1−1})^⊤ where 𝐽=log₂𝑛, can be transformed via the discrete wavelet transform to create 𝐲_𝐽=𝐖^⊤𝐲.

The expected value of this transformed vector is

𝔼𝑦_𝐽𝑖=𝜇_𝐽𝑖=2^𝐽/2𝜙(2^𝐽𝑡−𝑖)𝜇(𝑡)𝑑𝑡. (3.1) For a 𝜇(𝑡) function that is nearly constant around 𝑖/𝑛, 𝜇_𝐽𝑖≈2^−𝐽/2𝜇(𝑖/𝑛); so we can approximate 𝐲_𝐽 by (1/√𝑛)(𝑌₁,𝑌₂,…,𝑌_𝑁)^⊤.

In the original NPR experiment, the variances areVar(𝑌_𝑖)=Var(𝜉_𝑖)=𝐶𝜎² for a constant 𝐶 that depends on 𝛽 and the basis we will be using. The covariance matrix for these 𝑌_𝑖 will be assumed to be Σ=𝐖𝐃𝐖^⊤ where 𝐃 is the diagonal matrix of Var(𝑦_𝑗𝑘)=𝜎²_𝑛2^𝛽(𝑗+1). The variance of the 𝑦_𝐽𝑘 should be the same as that of 𝑌_𝑖𝑛^−1/2, and in the model described, Var(𝑦_𝐽𝑘)∝𝜎²_𝑛𝑛^𝛽. Therefore, 𝜎²_𝑛 should be set to 𝜎²𝑛^−1−𝛽.

Lemma 3.1. If the mean function 𝜇(𝑡) is in ℳ(𝑀,𝛼), then the total variation distance between the distribution ℙ_𝜇 of the vector (1/√𝑛)(𝑌₁,𝑌₂,…,𝑌_𝑁)^⊤ and the distribution ℙ_𝜇 of the vector 𝑦_𝐽 is sup_{𝜇∈ℳ(𝑀,𝛼)}𝛿(ℙ_𝜇,ℙ_𝜇)≤(𝑀𝑁^𝛼+1𝛼+1)𝑛^{1/2−𝛼+𝛽/2}. (3.2)

This lemma essentially goes back to the original work of Mallat [27] and parallels some of what was done by Brown and Low [1], Johnstone [10], Rohde [7], and Carter [5].

The NPR observations are such that the covariance matrix of 𝜉_𝑖 is also Σ, and therefore the total variation distance between the distributions is bounded in (5.2) by

1𝛿(ℙ,ℙ)≤√Δ2𝜋^1/2 (3.3) with

𝑖Δ=(𝜇(𝑛)−𝜇_𝐽𝑖)^⊤Σ⁻¹𝑖(𝜇(𝑛)−𝜇_𝐽𝑖). (3.4)

A standard calculation bounds the difference between the means when 𝜙(𝑡)<𝑀 with support on [0,𝑁] and the 𝜇(𝑡) are Hölder (𝛼) for 𝛼<1:

𝑖|𝜇(𝑛)−𝜇_𝐽𝑖|≤𝑛^{−1/2−𝛼}𝑀𝑁𝛼+1^𝛼+1. (3.5)

The covariance matrix is a positive definite matrix such that Σ⁻¹=𝐌𝐃⁻¹𝐌^⊤. The first column of the wavelet transform matrix is √𝑛𝟏 where 𝟏 is the vector of 1's. Therefore,

Δ≤(𝑀𝑁^𝛼+1𝛼+1)𝑛^−1−2𝛼𝑛𝑛^1+𝛽=(𝑀𝑁^𝛼+1𝛼+1)𝑛^{1−2𝛼+𝛽}, (3.6) which is negligible for large 𝑛 and 𝛼>1/2.

3.1. Proof of Theorem 1.1

The theorem follows from the fact that the observations 𝑦₀,{𝑦_𝑗𝑘} for 𝑗=0,…,𝐽−1 are asymptotically sufficient for the continuous process in (1.2). Then a linear function of these sufficient statistics 𝐲=𝐖^⊤𝐲 is still approximately sufficient. Thus, the experiment that seeks to draw inference about 𝜇 from the observations 𝑦_𝐽𝑖 is asymptotically equivalent to the experiment that observes (1.2) by Lemma 2.1.

Furthermore, by Lemma 3.1, the original NPR experiment that has the same covariance structure as 𝑦_𝐽𝑖 is asymptotically equivalent to that experiment and thus, by transitivity, to the experiment that observes the process (1.2) as well. This proves Theorem 1.1.

3.2. Remarks on the Covariance Structure

This result is restrictive in that it requires a specific known covariance structure. We are working under the assumption that the covariance matrix has eigenfunctions that correspond to a wavelet basis. This does not generally lead to a typical covariance structure. It does not even necessarily lead to a stationary Gaussian process; see the Haar basis example below.

The difficulty is that the requirement for having asymptotically equivalent experiments is quite strict, and the total variation distance between the processes with even small differences in the structure of the covariance is not negligible. For two multivariate Guassian distributions with the same means but where one covariance matrix is Σ and the other is 𝐷, a diagonal matrix with the same diagonal elements as Σ, the Kullback–Leibler divergence between the distributions is log|Σ|−log|𝐷|.

If the correlation between the highest level coefficients is Corr(𝜉_𝑗^∗_,𝑘,𝜉_𝑗^∗_,𝑘+1)=𝜌, then the contribution to the difference of the log determinants is on the order of 𝜌2^𝑗∗. The dimension of the problem is growing while the correlations are generally not going to 0 significantly quickly. For instance, in a typical wavelet basis decomposition of the true fractional Brownian motion Corr(𝜉_𝑗^∗_,𝑘,𝜉_𝑗^∗_,𝑘+1)=𝑐_𝛽 where 𝑐_𝛽 is a constant that depends on 𝛽 but not 𝑗^∗ or 𝑛.

Thus, the difference log|Σ|−log|𝐷| will not go to 0 as the sample size increases. Therefore, for the sort of long-range correlation structures that we are considering here, the eigenfunctions of the kernel 𝐾 need to be known or else the experiments will not be asymptotically equivalent.

4. Estimating the Covariance of the Increments

The key limitation of Theorem 1.1 is that it supposes that the covariance structure of the errors is known to the experimenter. To make the approximation more useful, it would help if the covariance structure was more flexible. A strategy similar to that used by Carter [6] can be used to estimate the variances of the coefficients.

By Carter [6], I showed that a model with a variance that changes slowly over time can still be approximated by the Gaussian process as long as all of the observations are independent. Our result here is that for correlated observations, if the variance is a linear function of the frequency, then a similar technique can be used to establish a set of asymptotically sufficient statistics.

Flexibility with regard to the covariance structure is added by allowing the magnitude of the Var(𝑦_𝑗𝑘) to depend on the resolution level 𝑗. The variances will be described by two parameters 𝛾 and 𝛽, which characterize the size of the error and the speed that it shrinks at higher resolution levels. These nuisance parameters can be estimated using part of the data, and then the inference can be carried out conditionally on the estimates.

Specifically, the experiment ℰ_𝑛 observes independent components 𝑦₀∼𝒩(𝜃₀,𝑛^−(1+𝛽)2^𝛾) and

𝑦_𝑗𝑘∼𝒩(𝜃_𝑗𝑘,𝑛^−(1+𝛽)2^{𝛾+(𝑗+1)𝛽})for0≤𝑗≤𝐽, (4.1) where the 𝑛^−(1+𝛽) factor is included to match up with the scaling functions at the 𝐽th resolution level. These observations form a new experiment with a parameter set that includes (𝜇,𝛾,𝛽) where 𝜇(𝑡)∈ℳ(𝑀,𝜀), −1<𝛽<0, and 𝛾 is bounded below by a constant −𝑐.

This experiment ℰ_𝑛 with the parametric variances is no longer an approximately sufficient statistic for the experiment that observes all of the 𝜃_𝑗𝑘. That experiment has too much information about the variance. If we observed the entire sequence at all resolution levels 𝑗, then 𝛾 and 𝛽 could be estimated exactly. We need to adopt another approximating experiment as in Carter [6] Many of the bounds in this section follow arguments from that paper.

4.1. Proof of Theorem 1.2

The theorem can be proven by applying Lemma 3.1 and then a version of Lemma 2.1 that uses only a small proportion of the low-frequency wavelet coefficients. The rest of the coefficients can be used to fix the parameters in the covariance of the observations.

The first step is to decompose the nonparametric regression into a set of wavelet coefficients. The 𝑛 NPR observations 𝑌_𝑖 can be transformed by dividing by √𝑛 and then performing the discrete wavelet transformation as in Lemma 3.1. The result is that a sequence of 𝑛 wavelet coefficients 𝑦₀ and 𝑦_𝑗𝑘 for 𝑗=0,…,𝐽−1 is equivalent to the original NPR observations with a total-variation distance between the distributions of

𝛿(ℙ_{𝜇,𝛾,𝛽},ℙ_𝜇𝛾𝛽)≤𝐶2^−𝛾𝑛^{1−2𝛼+𝛽}. (4.2) The supremum of this bound over all 𝛾>−𝑐 and 𝛽<0 is

𝛿(𝒫,𝒫)≤𝐶2^𝑐𝑛^1−2𝛼, (4.3) which will be negative for 𝛼>1/2.

The key strategy is to break the observations from this wavelet composition into pieces starting at level 𝑗^∗, where observations on 𝑗≤𝑗^∗ are assumed to be informative about the means, and the higher resolution levels are used to estimate the covariance structure.

For each resolution level with 𝑗>𝑗^∗, we generate the approximately sufficient statistics 𝑉_𝑗=∑_𝑘𝑦²_𝑗𝑘. Along with the 𝑦_𝑗𝑘 for 𝑗≤𝑗^∗, the collection of 𝑉_𝑗 is exactly sufficient if the means are 𝜃_𝑗𝑘=0 for 𝑗>𝑗^∗, because if there is no information about the means in the higher-frequency terms, then we have a piece of the experiment that is like a normal scale family. This new experiment ℰ_𝑣 is asymptotically equivalent to our ℰ_𝑛.

The error in approximating ℰ_𝑛 by ℰ_𝑣, where the means of the higher-frequencies coefficients are 0, is bounded by (5.3):

𝛿(ℰ_𝑣,ℰ_𝑛)≤(^𝐽−1_{𝑗=𝑗∗+1}𝑛^1+𝛽2^{−(𝛾+(𝑗+1)𝛽)}_𝑘𝜃²_𝑗𝑘)^1/2. (4.4) For 𝜃_𝑗𝑘 in ℳ(𝑀,𝜀) space, (5.4) bounds the distance as bound

𝛿(ℰ_𝑣,ℰ_𝑛)≤𝑀2^−𝛾/22^{(𝐽−𝑗∗−1)(1+𝛽)/2−𝛽/2−𝜀𝑗∗}≤𝑀2^𝑐/2+1/22^{𝐽/2−(1/2+𝜀)𝑗∗}, (4.5) which is negligible when 𝑗^∗>𝐽/(1+2𝜀).

This ℰ_𝑣 experiment has sufficient statistics 𝑦₀, 𝑦_𝑗𝑘 for 𝑗≤𝑗^∗, and

𝑉_𝑗∼Γ(2^𝑗,𝑛^−(1+𝛽)2^{𝛾+𝛽(𝑗+1)−𝑗})for𝑗=𝑗^∗+1,…,𝐽−1. (4.6) Furthermore, there are approximately sufficient statistics in this experiment (𝑦_𝑗𝑘,𝛾,𝛽) where 𝛾 and 𝛽 are the weighted least-squares estimates of 𝛾 and 𝛽 from the data log𝑉_𝑗. These are exactly sufficient statistics in the experiment ℰ_ℓ that observes the 𝑦₀ and 𝑦_𝑗𝑘 for the lower resolution levels 𝑗≤𝑗^∗ as before, in addition to the observations 2^𝑊𝑗 for 𝑗^∗<𝑗<𝐽 where

𝑊_𝑗∼𝒩(−(1+𝛽)𝐽+𝛾+𝛽(𝑗+1),2^−𝑗+1(log2)). (4.7)

The distance between ℰ_ℓ and ℰ_𝑣 depends on the distance between the distribution of the log of the Gamma variables and the normal approximation to this distribution. The calculation in of [6, Section 10.1] gives a bound on the Kullback–Leibler divergence of 𝐃(ℚ_𝑗,ℚ̆_𝑗)≤2^−𝑗 where ℚ_𝑗 is the distribution of 𝑉_𝑗, and ℚ̆_𝑗 is the distribution of 2^𝑊𝑗. Therefore, the total error between the two experiments is

𝛿(ℰ_ℓ,ℰ_𝑣)≤(^𝐽−1_{𝑗=𝑗∗+1}2^−𝑗)^1/2≤2^−𝑗∗/2. (4.8) Therefore, the observations in ℰ_ℓ are asymptotically sufficient for ℰ_𝑣 and thus also ℰ_𝑛 (as long as 𝑗^∗→∞ with 𝑛).

In the experiment ℰ_ℓ, the sufficient statistics for estimating 𝛾 and 𝛽 are the weighted least-squares estimators 𝛾 and 𝛽:

𝛾𝛽=(𝐱^⊤Λ𝐱)⁻¹𝐱^⊤Λ(𝐖+𝐽), (4.9) where Λ is the diagonal matrix with 2^𝑗 for 𝑗=𝑗^∗+1,…,𝐽−1 along its diagonal, 𝐱 is the design matrix with rows (1,𝑗−𝐽+1), and 𝐖+𝐽 is the column of observations 𝑊_𝑗−𝐽. The vector of estimators is normal with mean (^𝛾𝛽)^⊤ and covariance (1/2)(log2)(𝐱^⊤Λ𝐱)⁻¹.

Therefore, we can compare this experiment ℰ_ℓ to an experiment ℰ that observes the same (𝛾𝛽)^⊤, but the 𝑦₀ and 𝑦_𝑗𝑘 for 𝑗≤𝑗^∗ are replaced by Gaussian random variables 𝑦_𝑗𝑘 with variances (conditional on (𝛾,𝛽)) that are Var(𝑦_𝑗𝑘)=2𝛾_{+(𝑗−𝐽+1)}𝛽_−𝐽. The error in this approximation depends on the distance between the two sets of independent normal experiments with different variances. Letting ℙ_𝑗𝑘 be the distribution of 𝑦_𝑗𝑘 and ℙ_𝑗𝑘 the distribution of 𝑦_𝑗𝑘, the bound (6.12) in Section 6 gives

𝐃(ℚ_𝑗𝑘,ℚ_𝑗𝑘)≤(log𝑛)²𝑛−2^𝑗∗+𝑂((log𝑛)⁴(𝑛−2^𝑗∗+1)²),𝐃(ℚ₀,ℚ₀)≤(log𝑛)²𝑛−2^𝑗∗+𝑂((log𝑛)⁴(𝑛−2^𝑗∗+1)²). (4.10) There are 2^𝑗∗+1 independent normals 𝑦_𝑗𝑘 for 𝑗≤𝑗^∗ so that the total divergence is

𝛿(ℰ_ℓ,ℰ)²≤2(log𝑛)²2^{𝐽−𝑗∗}2−1+𝑂(^𝑗∗(log𝑛)⁴(𝑛−2^𝑗∗+1)²). (4.11) Therefore, the experiments ℰ_ℓ and ℰ are asymptotically equivalent for

𝑗^∗=𝐽−2log₂𝐽−𝜂_𝑛 (4.12) for some 𝜂_𝑛→∞.

We can improve this approximation by replacing the estimators 𝛽 and 𝛾 in ℰ by using

𝛽=−1,𝛽≤−1,𝛽,−1<𝛽<0,0,𝛽≥0, (4.13) and 𝛾=𝛾∨𝑐 to match up with the bounds on the parameter space. The new version of this experiment therefore observes (𝛾,𝛽) and the normal coordinates 𝑦_𝑗𝑘∼𝒩(𝜃_𝑗𝑘,𝑛¹⁺𝛽2𝛾2𝛽^(𝑗+1)) for 0≤𝑗≤𝑗^∗. The error between ℰ_ℓ and this new version of ℰ is smaller because |𝛾−𝛾+(𝛽−𝛽)(𝑗−𝐽)|≤|𝛾−𝛾+(𝛽−𝛽)(𝑗−𝐽)|, which makes the bound in (6.2) uniformly smaller.

Finally, we create a continuous Gaussian version of the ℰ experiment. This approximation ℰ observes all the 𝑦_𝑘𝑗 for 𝑘≥0 with means 𝜃_𝑗𝑘 and variances 𝑛⁻⁽¹⁺𝛽⁾2𝛾⁺𝛽^(𝑗+1). The ℰ are actually sufficient statistics for an experiment that observes (𝛾,𝛽) and 𝑦_𝑗𝑘 for 0≤𝑗≤𝑗^∗ and for 𝑗>𝑗^∗:

𝑦_𝑗𝑘∼𝒩(0,𝑛⁻⁽¹⁺𝛽⁾2𝛾⁺𝛽^(𝑗+1)). (4.14) The difference between the experiments ℰ and ℰ conditional on 𝛾 and 𝛽 is as in Section 2 and (5.4) less than 𝑀2⁻𝛾2^{(𝐽−𝑗∗)(1+}𝛽_)/2−𝛽_{/2−𝜀𝑗}^∗. The expectation of this bound when averaged over the possible values of (𝛾,𝛽) is a bound on the unconditional error. Furthermore, this expectation is less than the minimum over possible values of (𝛾,𝛽) (this is the real advantage that comes from going from (𝛾,𝛽) to (𝛾,𝛽)). Thus,

𝛿(ℰ,ℰ)≤2^𝑐−12^{(𝐽−𝑗∗)/2−𝜀𝑗∗}. (4.15) As before, this is asymptotically negligible when 𝑗^∗>𝐽/(1+2𝜀).

All that is left to do is to choose the level 𝑗^∗ at which to split the data so that the requirements from (4.5) and (4.15) that 𝑗^∗>𝐽/(1+2𝜀) and from (4.12) that 𝑗^∗=𝐽−2log₂𝐽−𝜂_𝑛 are all fulfilled. We could choose 𝜂_𝑛=𝜀𝐽/(1+2𝜀)−log₂𝐽 so that

𝑗^∗=𝐽+1+2𝜀𝜀𝐽1+2𝜀−log₂𝐽, (4.16) which is greater than 𝐽/(1+2𝜀) for 𝜀>0. This choice of 𝑗^∗ plugged into the bound in (4.5) gives us

𝛿(ℰ_𝑣,ℰ_𝑛)≤𝑁2^(𝑐+1)/2𝐽^1/2+𝜀2^{−𝜀𝐽/2}→0 (4.17) as 𝐽→∞ for 𝜀>0. At the same time, the bound in (4.11) becomes

𝛿(ℰ_ℓ,ℰ)²≤2(log2)²𝐽2^{𝐽𝜀/(1+2𝜀)}−𝐽⁻¹→0. (4.18) Thus, Theorem 1.2 is established.

5. Bounding the Total Variation Distance

We need a bound on the distance between two multivariate normal distributions with different means in order to bound the error in many of our approximations.

For shifted Gaussian processes, the total-variation distance between the distributions is

𝛿(ℙ_𝜇1,ℙ_𝜇2Δ)=1−2Φ(−^1/22), (5.1) where Δ=(𝜇₁−𝜇₂)^⊤Σ⁻¹(𝜇₁−𝜇₂). The expression in (5.1) for the total variation distance is concave for positive Δ, so a simple expansion gives

𝛿(ℙ_𝜇1,ℙ_𝜇21)≤√Δ2𝜋^1/2. (5.2)

For the Gaussian process with correlated components, we will assume that the variance of each wavelet coefficient is of the form Var(𝜓_𝑗𝑘)=𝜎²𝑛^−(1+𝛽)2^𝛽(𝑗+1) where the variance is calibrated so that Var(𝑌_𝑖)=𝜎²=𝑛Var(𝐵_𝐾(𝜙_𝐽,ℓ)). A bound on the error in the projection onto the span of 𝜓_𝑗𝑘 for 𝑗>𝑗^∗ comes from (5.2) which depends on

𝑛Δ=^1+𝛽𝜎²_𝑗>𝑗∗2^{−𝛽(𝑗+1)}_𝑘𝜃²_𝑗𝑘≤𝑛^1+𝛽𝜎²2^{−(1+𝛽)𝑗∗−𝛽}_𝑗>𝑗∗2^𝑗_𝑗𝜃²_𝑗𝑘≤𝑀²₂𝜎⁻²2^{(𝐽−𝑗∗)(1+𝛽)−𝜀𝑗∗}(2^𝛽−2^𝜀+𝛽)⁻¹, (5.3)(5.4) where the upper bound in (5.4) follows from 𝑛=2^𝐽, the definition of ℳ(𝑀,𝛼) and the bound in (1.13), and −1<𝛽<0. This error is negligible as 𝐽→∞ whenever

𝑗^∗>𝐽.2𝛼 (5.5)

6. Bounds from the Estimated Variances

In order to expand our asymptotically sufficient statistics out into a continuous Gaussian experiment, we need a bound on the total-variation distance between ℰ_𝑛 which, for 0≤𝑗≤𝑗^∗, observes a sequence of normals with variances 𝑛^−(1+𝛽)2^{𝛾+𝛽(𝑗+1)} and ℰ_𝑔, which observes a similar set of normals with variances 𝑛^−(1+𝛽)2𝛾₊𝛽_(𝑗+1).

For two normal distributions with the same means 𝜇, and variances 𝜎²₁ and 𝜎²₂, respectively, the Kullback–Leibler divergence is

𝐃(𝑁₁,𝑁₂1)=2[𝜎²₁𝜎²₂𝜎−1−log(²¹𝜎²₂)]. (6.1) Thus, for ℚ_𝑘𝑗 (the distribution of the 𝑦_𝑘𝑗) and ℚ_𝑘𝑗 (the distribution of the 𝑦_𝑘𝑗), the divergence between the conditional distributions given 𝛾 and 𝛽 is

𝐃(ℚ_𝑘𝑗,ℚ_𝑘𝑗∣1𝛾,𝛽)=22^𝛾−𝛾^{+(𝑗+1−𝐽)(𝛽−}𝛽₎−12−12[𝛾−𝛾+(𝑗+1−𝐽)(𝛽−𝛽)]log2,𝐃(ℚ₀,ℚ₀∣1𝛾,𝛽)=22^𝛾−𝛾−12−(𝛾−𝛾)log22. (6.2)(6.3) This divergence between conditional distributions can be used to bound the joint divergence:

𝐃(ℚ_𝑘𝑗,ℚ_𝑘𝑗)=𝔼𝐃(ℚ_𝑘𝑗,ℚ_𝑘𝑗∣𝛾,𝛽), (6.4) where the expectation is taken over the estimators 𝛾 and 𝛽.

To bound the expected value of the divergence in (6.2), we need the distribution of the estimators:

𝛾𝛽𝛾𝛽∼𝒩(,log22(𝐱^⊤Λ𝐱)⁻¹). (6.5) This implies that

𝔼exp[log2(𝛾−𝛾+(𝑗+1−𝐽)(𝛽−𝛽))]=exp[(log2)²2Var(𝛾+(𝑗+1−𝐽)𝛽)], (6.6) and therefore

𝔼𝐃(ℚ_𝑘𝑗,ℚ_𝑘𝑗∣𝛾,𝛽)=exp[(log2)²21Var(𝛾+(𝑗+1−𝐽)𝛽)]−2. (6.7)

Via elementary linear algebra calculations we get that for

𝜁=(^𝐽−1_{𝑖=𝑗∗+1}2^𝑖)(^𝐽−1_{𝑖=𝑗∗+1}(𝐽−𝑖−1)²2^𝑖)+(^𝐽−1_{𝑖=𝑗∗+1}(𝐽−𝑖−1)2^𝑖)²,Var(𝛾)=log22(^𝐽−1_{𝑖=𝑗∗+1}(𝐽−𝑖−1)²2^𝑖)𝜁⁻¹,Var(𝛽)=log22(^𝐽−1_{𝑖=𝑗∗+1}2^𝑖)𝜁⁻¹,Cov(𝛾,𝛽)=log22(^𝐽−1_{𝑖=𝑗∗+1}(𝐽−𝑖−1)2^𝑖)𝜁⁻¹. (6.8) As a result,

=Var(𝛾+(𝑗+1−𝐽)𝛽)log22×[_𝑖(𝐽−𝑖−1)²2^𝑖+(𝐽−𝑗−1)²_𝑖2^𝑖−2(𝐽−𝑗−1)_𝑖(𝐽−𝑖−1)2^𝑖]𝜁⁻¹. (6.9) To simplify this expression, let 𝑋 be a random variable with probability mass function proportional to 2^𝑖 for 𝑖=𝑗^∗+1,…,𝐽−1. Thus, the 𝜁 is equal to (∑2^𝑖)²[𝔼(𝐽−𝑋−1)²−[𝔼(𝐽−𝑋−1)]²]=(∑2^𝑖)²Var(𝑋). Similarly, the main factor in (6.9) is equal to

(2^𝑖)[𝔼(𝐽−𝑋−1)²+(𝐽−𝑗−1)²−2(𝐽−𝑗−1)𝔼(𝐽−𝑋−1)]=(2^𝑖)𝔼(𝑋−𝑗)². (6.10)

A simple bound of 0<𝑋−𝑗<𝐽 leads to 𝔼(𝑋−𝑗)²<𝐽². Furthermore, the variance of 𝑋 is decreasing in 𝑗^∗, and thus it is greater than 2/9 when 𝑗^∗<𝐽−2. Therefore,

Var(𝛾+(𝑗+1−𝐽)𝛽)≤9log24𝐽²∑2^𝑖≤𝐽²2^𝐽−1−2^𝑗∗ (6.11) because 9log2/4<2.

Thus,

𝐃(ℚ_𝑗𝑘,ℚ_𝑗𝑘1)=2exp[(log2)²21Var(𝛾+(𝑗+1−𝐽)𝛽)]−2≤12exp[(log2)²𝐽²2^𝐽−2^𝑗∗+11]−2≤(log2)²𝐽²2^𝐽−2^𝑗∗+1+𝐶𝐽⁴(2^𝐽−2^𝑗∗)²=(log𝑛)²𝑛−2^𝑗∗+1+𝑂((log𝑛)⁴(𝑛−2^𝑗∗+1)²). (6.12) Analogously, the expected divergence between ℚ₀ and ℚ_𝑗𝑘 is bounded by

𝔼𝐃(ℚ₀,ℚ₀∣1𝛾,𝛽)=2exp[−(log2)²21Var(𝛾)]−2=12exp[−(log2)²2((log2)𝔼(𝐽−𝑋−1)²2(∑2^𝑖1)Var(𝑋))]−2≤(log2)²𝐽²2^𝐽−2^𝑗∗+1+𝐶𝐽⁴(2^𝐽−2^𝑗∗)². (6.13)

If we add up these errors over the 2^𝑗∗ observations in the experiment, we get that the error in the approximation is less than 𝐶(log𝑛)²/(𝑛2^−𝑗∗−1), which is negligible for 𝑗^∗ sufficiently small.

7. Haar Basis Covariance

The Haar basis is a simple enough wavelet basis by which we can make some explicit calculations of the properties of the error distribution. We will show that the resulting errors 𝜉_𝑖 will have variances of approximately 𝑛^𝛽 as we expected, and the correlation between 𝜉_𝑖 and 𝜉_𝑗 will decrease at about a rate of |𝑖−𝑗|^−(1+𝛽).

The scaling functions for the Haar basis are constant on 2^𝑗 dyadic intervals at the resolution level 𝑗. The assumption is that we have a single scaling function coefficient with Var(𝑦₀)=1, and then every wavelet coefficient 𝑦_𝑗𝑘 is independent and has variance 2^𝛽(𝑗+1). Then the covariances can be calculated from the synthesis formula for the Haar basis.

The formula for synthesizing the scaling function coefficients 𝑦_𝐽𝑘 from the wavelet decomposition is

𝑦_𝐽𝑘=2^−𝐽/2𝑦₀+^𝐽−1_𝑗=0𝜁_{𝑗,𝐽,𝑘}2^{(𝑗−𝐽)/2}𝑦_𝑗𝑘^∗, (7.1) where 𝑘^∗ is the index such that 𝜓_𝑗𝑘^∗ has support that includes the support of 𝜙_𝐽𝑘. The 𝜁_{𝑗,𝐽,𝑘} is either 1 or −1 depending on whether 𝜙_𝐽𝑘 sits in the positive or negative half of the 𝜓_𝑗𝑘^∗ function.

Using the covariance structure described above, the variance of 𝑦_𝐽𝑘 is

Var(𝑦_𝐽𝑘)=2^−𝐽+^𝐽−1_𝑗=02^{(𝑗−𝐽)+𝛽(𝑗+1)}=2^−𝐽+2^𝛽−𝐽[2^(1+𝛽)𝐽−12^1+𝛽]−1=2^𝛽𝐽[12−2^−𝛽]−2^−𝐽[2^−𝛽−12−2^−𝛽] (7.2) for −1<𝛽<0. For 𝛽=0, the variance of each scaling function coefficient is 1 as in white noise. For 𝛽=−1, direct calculation leads to a variance of 2^−𝐽(1+𝐽/2).

To find the covariance between two variables 𝑦_𝐽𝑘1 and 𝑦_𝐽𝑘2, we need 𝑗^∗ which is the highest resolution level such that the support of 𝜓_𝑗∗𝑘^∗ includes the support of both scaling functions 𝜙_𝐽𝑘1 and 𝜙_𝐽𝑘2. The covariance is thus

Cov(𝑦_𝐽𝑘1,𝑦_𝐽𝑘2)=2^−𝐽+^𝑗∗−1_𝑗=02^{(𝑗−𝐽)+𝛽(𝑗+1)}−2^{𝑗∗−𝐽+𝛽(𝑗∗+1)}=2^−𝐽−2^{𝑗∗−𝐽+𝛽(𝑗∗+1)}+2^𝛽−𝐽[2^{(1+𝛽)𝑗∗}−12^1+𝛽]−1=2^{𝑗∗−𝐽+𝛽(𝑗∗+1)}[2−2^1+𝛽2^1+𝛽−1]−2^−𝐽[2^−𝛽−12−2^−𝛽]=2^𝛽𝐽(2^{−(1+𝛽)(𝐽−𝑗∗)})[1−2^𝛽1−2^−𝛽−1]−2^−𝐽[2^−𝛽−12−2^−𝛽]. (7.3) For large 𝐽 the correlation is on the order of 𝑑^−(1+𝛽) where 𝑑=2^{𝐽−𝑗∗} is a proxy for the distance between the observations. For 𝛽=0, all of these covariances are 0. For 𝛽=−1, the correlation is 𝑗^∗/(𝐽+2).

Acknowledgment

This work is supported by NSF Grant no. DMS-08-05481.