Nicolas Hengartner

Eric Matzner-Løber

Laurent Rouvière

Thomas Burr

Multiplicative Bias Corrected Nonparametric Smoothers

Keywords: Nonparametric regression, bias reduction, local linear estimate

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Introduction

In nonparametric regression, the bias-variance tradeoff of linear smoothers such as kernel-based regression smoothers, wavelet based smoother or spline smoothers, is generally governed by a user-supplied parameter. This parameter is often called the bandwidth. As an example, assuming that the regression function m is twice continuously differentiable, the local linear 1 smoother with bandwidth h and kernel K has conditional bias h 2 2 m ′′ (x) u 2 K(u) du + o p (h 2 ) and conditional variance

1 nh σ 2 (x) f (x) K 2 (u) du + o p 1 nh
where f stands for the density of the (one-dimensional) explanatory variable X and σ 2 (x) is the conditional variance of the response variable given X = x. See for example the book of [START_REF] Fan | Local Polynomial Modeling and Its Application, Theory and Methodologies[END_REF]. Since the bias increases with the second order derivative of the regression function, the local linear smoother tends to under-estimate in the peaks and over-estimate in the valleys of the regression function. See for example [START_REF] Simonoff | Smoothing Methods in Statistics[END_REF], [START_REF] Wand | Kernel Smoothing[END_REF], [START_REF] Scott | Multivariate Density Estimation: Theory, Practice, and Visualization[END_REF]. This behavior in peaks and valleys causes some trouble in some practical applications, such as for the estimation of energy spectrum.

The decay of radioactive isotopes often generates gamma particles whose energy can be measured using specialized detectors. Typically, these detectors count the number of particles in various energy bins over short time intervals such as one to ten minutes. This enables estimation of the energy distribution of the emitted particles, which is called the energy spectrum. For low or medium resolution detectors, the spectrum is typically composed of multiple broad peaks whose location and area characterize the radio-isotope.

That is we not only want the locations of the peaks, but also the shape and amplitudes in peak regions.

Because the actual bin counts are noisy, and the energy spectrum is fairly smooth, it has been proposed to estimate the energy spectrum using nonparametric smoothing techniques [START_REF] Sullivan | Wavelet analysis of sodium iodide spectra[END_REF], [START_REF] Gang | A nonlinear wavelet method for data smoothing of low-level gamma-ray spectra[END_REF]).

However, the bias of the nonparametric smoothers degrades isotope identification performance for any algorithm that includes peak area or ratios of areas [START_REF] Casson | Nuclear Reachback Reference Manual[END_REF]) and motivates studying methods to reduce bias.

All nonparametric smoothing methods are generally biased. There are many approaches to reducing the bias, but most of them do so at the cost of an increase in the variance of the estimator. For example, one may chose to undersmooth the energy spectrum. Undersmoothing will reduce the bias but will have a tendency of generating spurious peaks. One can also use higher order smoothers, such as local polynomial smoother with a polynomial of order larger than one. While again this will lead to a smaller bias, the smoother will have a larger variance. Another approach is to start with a pilot smoother and to estimate its bias by smoothing the residuals (Di Marzio and [START_REF] Marzio | On boosting kernel regression[END_REF]). Subtracting the estimated bias from the smoother produces a regression smoother with smaller bias and larger variance. For the estimation of an energy spectrum, the additive bias correction and the higher order smoothers have the unfortunate side effect of possibly generating a nonpositive estimate.

An attractive alternative to the linear bias correction is the multiplicative bias correction pioneered by [START_REF] Linton | A multiplicative bias reduction method for nonparametric regression[END_REF]. Because the multiplicative correction does not alter the sign of the regression function, this type of correction is particularly well suited for adjusting non-negative regression functions. [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF] showed that if the true regression function has four continuous derivatives, then the multiplicative bias reduction is operationally equivalent to using an order four kernel. And while this does remove the bias, it also increases the variance.

Many authors have extended the work of [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF]. Glad [1998a,b] proposes to use a parametrically guided local linear smoother and Nadaraya-Watson smoother by starting with a parametric pilot. This approach is extended to a more general framework which includes both multiplicative and additive bias correction by [START_REF] Martins-Filho | A class of improved parametrically guided nonparametric regression estimators[END_REF] (see also [START_REF] Mishra | Semiparametric estimator of time series conditional variance[END_REF] for an extension to time series conditional variance estimation).

For multiplicative bias correction in density estimation, we refer the reader to the recent works of [START_REF] Gustafsson | Local transformation kernel density estimation of loss distributions[END_REF], [START_REF] Hagmann | Local multiplicative bias correction for asymmetric kernel density estimators[END_REF] and Hirukawa [2010].

Although the bias-variance tradeoff for nonparametric smoothers is always present in finite samples, it is possible to construct smoothers whose asymptotic bias converges to zero while keeping the same asymptotic variance.

Hengartner and Matzner-Løber [2009] has exhibited a nonparametric density estimator based on multiplicative bias correction with that property, and have shown in simulations that their estimator also enjoyed good finite sample properties. [START_REF] Burr | Smoothing low resolution gamma spectra[END_REF] present such an estimator for nonparametric regression to estimate energy spectra. They illustrate the benefits of this approach on real and simulated spectra. The goal of this paper is to study the asymptotic properties of this estimator. It is worth pointing out that these properties have already been studied by [START_REF] Linton | A multiplicative bias reduction method for nonparametric regression[END_REF] for fixed design and further by [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF]. We emphasize that there are two major differences between our work and that of [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF].

• First, we do not add regularity assumptions on the target regression function. In particular, we do not assume that the regression function has four continuous derivatives as in [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF].

• Second, we show that the multiplicative bias reduction procedure performs a bias reduction with no cost to the asymptotic variance. It is exactly the same as the asymptotic variance of the local linear estimate.

We provide another asymptotic behavior under less restrictive assumptions than in [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF]. Moreover our results and proofs are completely different from the above referenced works.

This paper is organized as follows. Section 2 introduces the notation and defines the estimator. Section 3 gives the asymptotic behavior of the proposed estimator. A brief simulation study on finite sample comparison is presented in Section 4. The interested reader is referred to the Appendix where we have gathered the technical proofs.

Preliminaries

2.1 Notation.

Let (X 1 , Y 1 ), . . . , (X n , Y n ) be n independent copies of the pair of random variables (X, Y ) with values in R × R. We suppose that the explanatory variable X has probability density f and model the dependence of the univariate response variable Y to the explanatory variable X through the nonparametric regression model

Y = m(X) + ε. (1)
We assume that the regression function m(•) is smooth and that the disturbance ε is a mean zero random variable with finite variance σ 2 that is independent of the covariate X. Consider the linear smoothers for the regression function m(x) which we can write as

m(x) = n j=1 ω j (x; h)Y j ,
where the weight function ω j (x; h) depends on a tuning parameter h, that we think of as the bandwidth.

If the weight functions are such that n j=1 ω j (x; h) = 1 and n j=1 ω j (x; h) 2 = (nh) -1 τ 2 , and if the disturbances satisfy the Lindberg-Feller condition, then the linear smoother obeys the central limit theorem

√ nh m(x) - n j=1 w j (x; h)m(X j ) -→ N (0, τ 2 ). ( 2
)
We can use (2) to construct asymptotic pointwise confidence intervals for the unknown regression function m(x). But unless the limit of the scaled bias

b(x) = lim n-→∞ √ nh n j=1 w j (x; h)m(X j ) -m(x) ,
which we call the asymptotic bias, is zero, the confidence interval

m(x) -Z 1-α/2 √ nhτ, m(x) + Z 1-α/2 √ nhτ
will not cover asymptotically the true regression function m(x) at the nominal 1α level. The construction of valid pointwise 1α confidence intervals for regression smoothers is another motivation for developing estimators with zero asymptotic bias.

Multiplicative bias reduction

Here we present a framework for multiplicative bias reduction. Given a pilot smoother

mn (x) = n j=1 ω j (x; h 0 )Y j ,
the ratio

V j = Y j mn (X j )
is a noisy estimate of m(X j )/ mn (X j ), the inverse relative estimation error of the smoother mn at each of the observations. Smoothing V j by

α n (x) = n j=1 ω j (x; h 1 )V j
yields an estimate for the inverse of the relative estimation error which can be used as a multiplicative correction of the pilot smoother. This leads to the (nonlinear) smoother

m n (x) = α n (x) mn (x).
(3)

The estimator (3) was studied for fixed design by [START_REF] Linton | A multiplicative bias reduction method for nonparametric regression[END_REF] and further studied by [START_REF] Jones | A simple and effective bias reduction method for kernel density estimation[END_REF]. In both cases, they assumed that the regression function had four continuous derivatives, and show an improvement in the convergence rate of the corrected estimator. Glad [1998a,b] proposed to use a parametrically guided local linear smoother and Nadaraya-Watson smoother by starting with a parametric pilot. She shows that the resulting estimates improve on the local polynomial estimate as soon as the pilot captures some of the features of the regression function.

3 Theoretical Analysis of Multiplicative Bias

Reduction

In this section, we show that the multiplicative smoother has smaller bias with essentially no cost to the variance, assuming only two derivatives of the regression function. While the derivation of our results are for local linear smoothers, the technique used in the proofs can be easily adapted for other linear smoothers, and the conclusions remain essentially unchanged.

Assumptions

We make the following assumptions:

1. The regression function is bounded and strictly positive, that is, b ≥ m(x) ≥ a > 0 for all x.

2. The regression function is twice continuously differentiable everywhere.

3. The density of the covariate is strictly positive on the interior of its support in the sense that f (x) ≥ b(K) > 0 over every compact K contained in the support of f .

4. ε has finite fourth moments and has a symmetric distribution around zero.

5. Given a symmetric probability density K(•), consider the weights ω j (x; h) associated to the local linear smoother. That is, denote by K h (•) = K(•/h)/h the scaled kernel by the bandwidth h and define for k = 0, 1, 2, 3 the sums

S k (x) ≡ S k (x; h) = n j=1 (X j -x) k K h (X j -x).
Then

ω j (x; h) = S 2 (x; h) -(X j -x)S 1 (x; h) S 2 (x; h)S 0 (x; h) -S 2 1 (x; h) K h (X j -x).
We set

ω 0j (x) = ω j (x; h 0 ) and ω 1j (x) = ω j (x; h 1 ).
6. The bandwidths h 0 and h 1 are such that

h 0 → 0, h 1 → 0, nh 0 → ∞, nh 3 1 → ∞, h 1 h 0 → 0 as n → ∞.
The positivity assumption (assumption 1) on m(x) is classical when we perform a multiplicative bias correction. It allows to avoid that the terms Y j / mn (X j ) blows up. Of course, the regression function might cross the x-axis. For such a situation, Glad [1998a] proposes to shift all response data Y i a distance a, so that the new regression function m(x) + a does not any more intersect with the x-axis. Such a method can also be performed here.

Assumptions 2-4 are standard to obtain rate of convergence for nonparametric estimators. Assumption 5 means that we conduct the theory for the local linear estimate. The results can be generalized to other linear smoothers.

Assumption 6 is not restrictive since it is satisfied for a wide range of values of h 0 and h 1 .

A technical aside

The proof of the main results rests on establishing a stochastic approximation of estimator (3) in which each term can be directly analyzed.

Proposition 3.1. We have

m n (x) = µ n (x) + n j=1 ω 1j (x)A j (x) + n j=1 ω 1j (x)B j (x) + n j=1 ω 1j (x)ξ j ,
where µ n (x), conditionally on X 1 , . . . , X n is a deterministic function, A j , B j and ξ j are random variables. Under condition nh 0 → ∞, the remainder ξ j converges to 0 in probability and we have

m n (x) = µ n (x) + n j=1 ω 1j (x)A j (x) + n j=1 ω 1j (x)B j (x) + O P 1 nh 0 .
Remark 3.1. A technical difficulty arises because even though ξ j may be small in probability, its expectation may not be small. We resolve this prob-lem by showing that we only needs to modify ξ j on a set of vanishingly small probability to guarantee that its expectation is also small. Definition 3.1. Given a sequence of real numbers a n , say that a sequence of random variables

ξ n = o p (a n ) if for all fixed t > 0, lim sup n-→∞ P[|ξ n | > ta n ] = 0.
We will need the following Lemma.

Lemma 3.1. If ξ n = o p (a n ), then there exists a sequence of random variables

ξ ⋆ n such that lim sup n-→∞ P[ξ ⋆ n = ξ n ] = 0 and E[ξ ⋆ n ] = o(a n ).
We shall use the following notation

E ⋆ [ξ n ] = E[ξ ⋆ n ].

Main results

We deduce from Proposition 3.1 and Lemma 3.1 the following Theorem.

Theorem 3.1. Under the assumptions ( 1)-( 6), the estimator m n satisfies:

E ⋆ ( m n (x)|X 1 , . . . , X n ) = µ n (x) + O p 1 n √ h 0 h 1 + O p 1 nh 0 and V ⋆ ( m n (x)|X 1 , . . . , X n ) = σ 2 n j=1 w 2 1j (x) + O p 1 nh 0 + o p 1 nh 1 .
We deduce from Theorem 3.1 that if the bandwidth h 0 of the pilot estimator converges to zero much slower than h 1 , then m n has exactly the same asymptotic variance as the local linear smoother of the original data with bandwidth h 1 . However, for finite samples, the two step local linear smoother can have a slightly larger variance depending on the choice of h 0 . For the bias term, a limited Taylor expansion of µ n (x) leads to the following result.

Theorem 3.2. Under the assumptions ( 1)-( 6), the estimator m n satisfies:

E ⋆ ( m n (x)|X 1 , . . . , X n ) = m(x) + o p (h 2 1 ).
It is worth pointing out that we only suppose that the regression function is twice continuously differentiable. We do not add smoothness assumptions. For a study of the local linear estimate in the presence of jumps in the derivative, we refer the reader to [START_REF] Desmet | Local linear fitting and improved estimation near peaks[END_REF]. Combining Theorem 3.1 and Theorem 3.2, we conclude that the multiplicative adjustment performs a bias reduction on the pilot estimator without increasing the asymptotic variance. The asymptotic behavior of the bandwidths h 0 and h 1 is constrained by assumption 6. However, it is easily seen that this assumption is satisfied for a large set of values of h 0 and h 1 . For example, the choice

h 1 = c 1 n -1/5 and h 0 = c 0 n -α for 0 < α < 1/5 leads to E ⋆ ( m n (x)|X 1 , . . . , X n ) -m(x) = o p (n -2/5
) and

V ⋆ ( m n (x)|X 1 , . . . , X n ) = O p n -4/5 .
Remark 3.2. Estimators with bandwidths of order O(n -α ) for 0 < α < 1/5 are oversmoothing the true regression function, and as a result, they have biases that are of larger order of magnitude than their standard deviations.

We conclude that the multiplicative adjustment performs a bias reduction on the pilot estimator.

Numerical examples

While the amount of the bias reduction depends on the curvature of the regression function, a decrease is expected (asymptotically) everywhere, and this, at no cost to the variance. The simulation study in this section shows that this asymptotic behavior emerges already at modest sample sizes.

Local study

To illustrate numerically the possible reduction in the bias and associate increase of the variance achieved by the multiplicative bias correction, consider estimating the regression function m(x) = 5 + 3|x| 5/2 + x 2 + 4 cos(10x) at x = 0 (see Figure 1). The local linear smoother tends to under-estimate the regression function at their maximum, and hence, this example will provide a good example. Furthermore, because the second derivative of this regression function is continuous but not differentiable at the origin, the results previously obtained by [START_REF] Linton | A multiplicative bias reduction method for nonparametric regression[END_REF] do not apply.

The data are simulated according to the model

Y i = m(X i ) + ε i , i = 1, . . . , 100,
where ε i are independent N (0, 1) variables. We first consider the local linear estimate with a Gaussian kernel function and we study its performances over For the new estimate, the theory recommends to start with an oversmooth pilot estimate. In this regard, we take h 0 = 0.1 and study the performance of the multiplicative bias corrected estimate for h 1 ∈ H 1 = [0.005, 0.12]. To explore the stability of our two stages estimator with respect to h 0 , we also consider the choice h 0 = 0.02.

For such a choice, the pilot estimate clearly undersmoothes the regression function.

Bias and variance of each estimate are calculated at x = 0. To do this, we compute the value of each estimate at x = 0 for 200 samples (X i , Y i ), i = 1, . . . , 100. The same design X i , i = 1, . . . , 100 is used for each sample. Finally, due to the bias reduction, we note that our procedure also reduces the optimal mean square error (see Table 1). and the multiplicative bias corrected estimate (MBCE) with h 0 = 0.1 at point

x = 0.
We conclude our local study with a comparison between our estimate and the estimate proposed by Glad [1998a]. To do this, we compute the multiplicative bias corrected estimate using three parametric starts:

• first the guide is chosen correctly and belong to the true parametric family: m1 n (x) = β0 + β1 |x| 5/2 + β2 x 2 + β3 cos(10x);

• second, we consider a linear parametric guide (which is obviously wrong):

m2 n (x) = β0 + β1 x;
• finally, we use a more reasonable guide, not correct, but that can reflect some a priori idea on the regression function

m3 n (x) = β0 + β1 x + β2 x 2 + . . . + β8 x 8 .
All the estimates βj stands for the classical least square estimates.

The multiplicative bias correction is performed on these parametric starts using the local linear estimate. As expected, we first observe that the performance clearly depends on the choice of the parametric start. Table 1 and table 2 show that (in term of MSE) the estimate studied in this paper is better than the corrected estimated with parametric start m2 n and m3 n . Unsurprisingly, the best performance are obtained with the parametric guide m1 n (which belongs to the true model). In practice, when one has no or few a priori information on the target regression function, the method proposed in the present paper is preferable.

Global study

This paper does not conduct any theory to select the two bandwidths h 0 and h 1 in an optimal way. If automatic procedures are needed, they can be obtained by adjusting traditional automatic selection procedures for the classical nonparametric estimators (see [START_REF] Burr | Smoothing low resolution gamma spectra[END_REF]). In this part, we propose to use leave-one-out cross validation to choose both h 0 and h 1 .

We then compare the performance of the selected estimate with the local polynomial estimate in term of integrated square error. [START_REF] Hurvich | Smoothing parameter selection in nonparametric regression using and improved akaike information criterion[END_REF] report a comprehensive numerical study that compares standard smoothing methods on various test functions. Here, we take the same setting to compare the local linear estimate with its multiplicative bias corrected smoother. In each of the examples, we take the Gaussian kernel

K(x) = exp(-x 2 /2)/ √ 2π.
We use the following regression functions (see Figure 3):

(1)

m 1 (x) = sin(5πx) (2) m 2 (x) = sin(15πx) (3) m 3 (x) = 1 -48x + 218x 2 -315x 3 + 145x 4 (4) m 4 (x) = 0.3 exp [-64(x -.25) 2 ] + 0.7 exp [-256(x -.75
) 2 ] and we take a Gaussian error distribution with standard deviation σ j = 0.25 m j 2 for j = 1, . . . , 4.

We use a cross validation device to select both h 0 and h 1 . This selection procedure involves solving minimization problem that necessitate a search over a finite grid H of bandwidths h 0 and h 1 . Formally, given H, we choose ĥ0 and ĥ1 such as

( ĥ0 , ĥ1 ) = argmin (h 0 ,h 1 )∈H×H 1 n n i=1 (Y i -m i n (X i )) 2 .
Here m i n stands for the corrected local polynomial estimate after deleted the ith observation. To assess the quality of the selected estimate, we compare its performances with the local polynomial estimate for which the bandwidth is again selected by leave-one-out cross validation. The performance of an estimator m is measured by the integrated square error Table 3 presents the median over 100 replications of

ISE( m) = 1 0 (m(x) -m(x)) 2 dx,
• the selected bandwidths;

• the integrated square error;

• the integrated square error of the local linear estimate divided by the integrated square error of the corrected estimate (R ISE ).

Figure 4 displays the boxplots of the integrated square error for each estimate. We obtain significant ISE reduction for the four models. As predicted by Theorem 3.1, the data-driven procedure selects h 0 bigger than h: the pilot estimate is oversmoothing the true regression function. Of course, selecting both h 0 and h 1 is time consuming and can appear as the price to be paid to improve the local linear smoother.

LLE MBCE h ISE (×100) h 0 h 1 ISE (×100) R ISE m 1 0.
Figure 5 presents, for the regression function m 1 with n = 100 and 100 iterations, different estimators on a grid of points. In lines is the true regression function which is unknown. For every point on a fixed grid, we plot, side by side, the mean over 100 replications of our estimator at that point (left side)

and on the right side of that point the mean over 100 replications of the local polynomial estimator. Leave-one-out cross validation is applied to select the bandwidths h 0 and h 1 for our estimator and the bandwidth h for the local polynomial estimator. We add also the interquartile interval in order to see the fluctuations of the different estimators. In this example, our estimator reduces the bias by increasing the peak and decreasing the valleys. Moreover, the interquartile intervals look similar for both estimator, as predicted by the theory.

Proofs

This section is devoted the technical proofs.

Proof of Proposition 3.1

Write the bias corrected estimator

m n (x) = n j=1 ω 1j (x) mn (x) mn (X j ) Y j = n j=1 ω 1j (x)R j (x)Y j ,
and let us approximate the quantity R j (x). Define

mn (x) = n j=1 ω 0j (x)m(X j ) = E ( mn (x)|X 1 , . . . , X n ) ,
and observe that R j (x) = mn (x) mn (X j ) mn (x) mn (X j ) × 1 + mn (x)mn (x) mn (x) × 1 + mn (X j )mn (X j ) mn (X j )

-1 = mn (x) mn (X j ) × [1 + ∆ n (x)] × 1 1 + ∆ n (X j ) , where ∆ n (x) = mn (x) -mn (x) mn (x) = l≤n ω 0l (x)ε l l≤n ω 0l (x)m(X l ) . Write now R j (x) as R j (x) = mn (x) mn (X j ) [1 + ∆ n (x) -∆ n (X j ) + r j (x, X j )]
where r j (x, X j ) is a random variable converging to 0 to be define latter on.

Given the last expression and model (1), estimator (3) could be written as

m n (x) = n j=1 ω 1j (x)R j (x)Y j = n j=1 ω 1j (x) mn (x) mn (X j ) m(X j ) + n j=1 ω 1j (x) mn (x) mn (X j ) [ε j + m(X j ) (∆ n (x) -∆ n (X j ))] + n j=1 ω 1j (x) mn (x) mn (X j ) (∆ n (x) -∆ n (X j )) ε j + n j=1 ω 1j (x) mn (x) mn (X j ) r j (x, X j )Y j =µ n (x) + n j=1 ω 1j (x)A j (x) + n j=1 ω 1j (x)B j (x) + n j=1 ω 1j (x)ξ j .
which is the first part of the proposition. Under assumption set forth in Section 3.1, the pilot smoother mn converges to the true regression function m(x). [START_REF] Bickel | On some global measures of the deviations of density function estimates[END_REF] show that this convergence is uniform over compact sets K contained in the support of the density of the covariate

X. As a result sup x∈K | mn (x) -mn (x)| ≤ 1 2 . So a limited expansion of (1 + u) -1 yields for x ∈ K R j (x) = mn (x) mn (X j ) 1 + ∆ n (x) -∆ n (X j ) + O p |∆ n (x)∆ n (X j )| + ∆ 2 n (X j ) , thus ξ j = O p |∆ n (x)∆ n (X j )| + ∆ 2 n (X j ) .
Under the stated regularity assumptions, we deduce that

ξ j = O p 1 nh 0 .
leading to the announced result. Proposition (3.1) is proved.

Proof of lemma (3.1)

By definition lim sup

n-→∞ P[|ξ n | > ta n ] = 0
for all t > 0, so that a triangular array argument shows that there exists an increasing sequence m = m(k) such that

P |ξ n | > a n k ≤ 1 k for all n ≥ m(k). For m(k) ≤ n ≤ m(k + 1) -1, define ξ ⋆ n =    ξ n if |ξ n | < k -1 a n 0 otherwise. It follows from the construction of ξ ⋆ n that for n ∈ (m(k), m(k + 1) -1), P[ξ n = ξ * n ] = P[|ξ n | > k -1 a n ] ≤ 1 k ,
which converges to zero as n goes to infinity. Finally set k(n) = sup{k :

m(k) ≤ n}, we obtain E[|ξ| ⋆ n ] ≤ a n k(n) = o(a n ).

Proof of Theorem (3.1)

Recall that

m n (x) = µ n (x) + n j=1 ω 1j (x)A j (x) + n j=1 ω 1j (x)B j (x) + O P 1 nh 0 .
Focus on the conditional bias, we get

E(µ n (x)|X 1 , . . . , X n ) = µ n (x) E(A j (x)|X 1 , . . . , X n ) = 0 E(B j (x)|X 1 , . . . , X n ) = mn (x) mn (X j ) σ 2 ω 0j (x) mn (x) - ω 0j (X j ) mn (X j ) . Since n j=1 ω 1j (x)ω 0j (x) ≤ n j=1 ω 1j (x) 2 n j=1 ω 0j (x) 2 = O p 1 n √ h 0 h 1 , we deduce that E n j=1 ω 1j (x)B j (x) X 1 , . . . , X n = O p 1 n √ h 0 h 1 .
This proves the first part of the Theorem. Using the fact that the residuals have four finite moments and have a symmetric distribution around 0, a moment's thought shows that

V(Y j [∆ n (x) -∆ n (X j )] |X 1 , . . . , X n ) = O p 1 nh 0 and Cov(Y j , Y j [∆ n (x) -∆ n (X j )] |X 1 , . . . , X n ) = O p 1 nh 0 . Hence V ⋆ ( m n (x)|X 1 , . . . , X n ) = V n j=1 ω 1j (x) mn (x) mn (X j ) Y j X 1 , . . . , X n + O p 1 nh 0 .
Observe that the first term on the right hand side of this equality can be seen as the variance of the two stages estimator with a deterministic pilot estimator. It follows from Glad [1998a] that

V n j=1
ω 1j (x) mn (x) mn (X j ) Y j X 1 , . . . , X n = σ 2 n j=1 ω 2 1j (x) + o p 1 nh 1 , which proves the theorem.

Proof of theorem (3.2)

Recall that µ n (x) = j≤n ω 1j (x) mn (x) mn (X j ) m(X j ).

We consider the limited Taylor expansion of the ratio m(X j ) mn (X j ) = m(x) mn (x) + (X jx) m(x) mn (x) (X jx) 2 ω 1j (x) = S 2 2 (x; h 1 ) -S 3 (x; h 1 )S 1 (x; h 1 ) S 2 (x; h 1 )S 0 (x; h 1 ) -S 2 1 (x; h 1 )

.

For random designs, we can further approximate (see, e.g., [START_REF] Wand | Kernel Smoothing[END_REF]) Putting all pieces together, we obtain

S k (x, h 1 ) =    h k σ k K f (x
E ⋆ ( m n (x)|X 1 , . . . , X n ) -m(x) = o p (h 2 1 ) + O p 1 n √ h 0 h 1 + O p 1 nh 0 . Since nh 3 1 -→ ∞ and h 1 h 0 -→ 0,
we conclude that the bias is of order o p (h 2 1 ).

Figure 1 :

 1 Figure 1: The regression function to be estimated.

Figure 2 :

 2 Figure 2: Mean square error (dotted line), squared bias (solid line) and variance (dashed line) of the local linear estimate (left) and multiplicative bias corrected estimate with h 0 = 0.1 (center) and h 0 = 0.02 (right) at point x = 0.

Figure 3 :

 3 Figure 3: Regression functions to be estimated.

Figure 4 :

 4 Figure 4: Boxplot of the integrated square error over the 100 replications.

Figure 5 :

 5 Figure 5: The solid curve represents the true regression function, our estimator is in dashed line and local linear smoother is dotted.

For

  j ) Y j (1 + [∆ n (x) -∆ n (X j )]) + O p 1 nh 0 .

  x) 2 ω 1j (x)(1 + o p (1)) .It is easy to verify that Σ 0 (x; h 1 ) =

Table 1 :

 1 Optimal mean square error (MSE) for the local linear estimate (LLE)

		MSE Bias 2 Variance
	LLE	0.134 0.038	0.096
	MBCE 0.072 0.003	0.069

Table 2 :

 2 The performance of the resulting estimates is measured over a grid of bandwidths H 2 = [0.005; 0.4]. Bias and variance of each estimate are still estimated at x = 0. We keep the same setting as above and all the results are averaged over the same 200 replications. We display in Table2the optimal MSE calculated over the grid H 2 . Optimal mean square error for the multiplicative bias corrected estimates with parametric starts mj

		MSE Bias 2 Variance
	start m1 n	0.060 0.000	0.060
	start m2 n	0.134 0.038	0.096
	start m3 n	0.095 0.021	0.074

n , j = 1, 2, 3.

Table 3 :

 3 Median over 100 replications of the selected bandwidths and of the integrated square error of the selected estimates. LLE and MBCE stands for local linear estimate and multiplicative bias corrected estimate.

	023	0.920	0.041 0.032	0.727	1.191
	m 2 0.011	5.967	0.027 0.012	4.968	1.205
	m 3 0.029	2.063	0.071 0.054	1.139	1.648
	m 4 0.018	0.087	0.033 0.023	0.076	1.147

  ) + o p (h k ) for k even h k+1 σ k+1 K f ′ (x) + o p (h k+1 ) for k odd,

	where σ k K = u k K(u) du. Therefore
		Σ 2 (x; h 1 ) = h 2 1	u 2 K(u) du + o p (h 2 1 )
					= σ 2 K h 2 1 + o p (h 2 1 ),
	so that we can write µ n (x) as
	µ n (x) = mn (x)		m(x) mn (x)	+	σ 2 K h 2 1 2	m(x) mn (x)	′′	+ o p (h 2 1 )
		=m(x) +	σ 2 K h 2 1 2	mn (x)	m(x) mn (x)	′′	+ o p (h 2 1 ).
	Moreover
	m(x) mn (x)	′′	=	m2 n (x)m ′′ (x) m3 n (x)	-2	mn (x) m′ n (x)m ′ (x) m3 n (x)
					-	m(x) mn (x) m′′ n (x) m3 n (x)	+ 2	m(x)( m′ n (x)) 2 m3 n (x)
	and applying the usual approximations, we conclude that
					m(x)
					mn (x)

′′

= o p (1).


Silvina Montrul
email: montrul@uiuc.edu

Second language acquisition welcomes the heritage language learner: opportunities of a new field

Keywords: second language acquisition, heritage speakers, bilingualism, L1 loss, incomplete acquisition, language maintenance, language identity

With increasing recent changes in immigration and demographic patterns, the face of the traditional second language (L2) classroom in Spanish and the less commonly taught languages has changed dramatically in the Western World. In addition to serving typical L2 learners, these classes have recently had to accommodate an increasing number of heritage language learners. But just who are these heritage language learners? What linguistic knowledge do they bring to the classroom? And how do they differ from typical L2 learners? In this review article, I evaluate three recent books that address these questions and introduce issues relevant to the emerging field of heritage language acquisition. I argue that the heritage speaker population is very relevant for second language acquisition research because it allows us to address fundamental theoretical debates in our field from a different perspective, including debates on the role of age, input and environment, language transfer, linguistic mechanisms and the type of linguistic knowledge acquired before and after a critical period. I conclude by identifying promising avenues for research with heritage speakers and learners.

I Introduction

In recent years, the face of traditional foreign language classes in the USA, Canada, Australia and Western Europe has been changing dramatically, most noticeably in the teaching of Spanish, Russian, East Asian languages (Korean, Chinese, Japanese) and the less commonly taught languages (Hindi, Turkish, Arabic, among others). Until recently, these classes were designed for, and typically populated by, learners with no previous background in the target language. Today, however, many of these classes have been opening their doors to children of immigrants. Many of these individuals have acquired these languages as a first language (L1) to some degree -either monolingually or simultaneously with the majority language -but, for a variety of reasons, their heritage language became their secondary/second language. Naturally, having typical second language (L2) learners and heritage language learners with different linguistic profiles and experiences in the same classrooms poses serious challenges to both L2 teachers and learners alike. At present, many language practitioners feel justifiably disoriented when it comes to addressing the linguistic and cultural needs of these two types of learners.

Understanding the profiles and pedagogical needs of heritage language learners is what the emerging field of Heritage Language Acquisition and education is about. Pressing research questions currently driving this field include:

• Who are heritage language speakers? • What linguistic knowledge do they bring from childhood? €

• How can we help these learners develop, maintain and expand knowledge of the heritage language so they can use the language in a variety of formal and informal settings?

• And, perhaps most urgently, what and how should these learners be taught in the classroom?

In this article, I take a look at three recent books on minority language speakers that directly address these questions. In addition, I consider the question of how and why the heritage speaker and the heritage language learner are theoretically so relevant to the now well-established field of L2 acquisition.

II The books

The three books share several features. They all discuss at length the linguistic and cultural variables that characterize heritage language speakers and learners, providing a good sense of the heterogeneity of this population. They all highlight how national language policies and issues of language identity impinge on the linguistic fate of this population. In Valdés et al. (2006), Joshua Fishman (p. 1) defines heritage languages as the 'colonial, indigenous, and immigrant languages other than English of the United States'. Heritage languages are languages associated with the ethno-cultural heritage of particular minority populations, which for many historical and political reasons have been devalued. Minority-language-speaking children become bilingual at a young age because they have to become members of, and interact with, mainstream society. And even when the linguistic community for the heritage language has a sizable population, as with Spanish in the USA, this in itself does not guarantee a positive role for the heritage language in the lives of these individuals.

Existing definitions of heritage speakers vary widely. Some are too broad, including children of expatriate Americans, for example, growing up speaking Japanese in Japan, or even internationally adopted children whose adoptive parents do not speak the language of their child's country of origin. On the other hand, other definitions are too narrow, referring to individuals with high proficiency in the heritage language. Valdés (2000) defines heritage speakers in terms of what they can do linguistically; others stress cultural roots, including individuals who do not actually speak or understand the heritage language. While 'heritage speakers' are bilingual individuals who speak the heritage language to some degree, 'heritage language learners' are heritage speakers who seek to learn, re-learn, maintain or expand knowledge of their heritage language in the classroom (Kondo-Brown, 2006: 1).

Despite different opinions and perspectives, agreeing to some extent on who a heritage speaker is and what a heritage language learner needs in the classroom is at the heart of heritage language education. Answers to these questions are the first important steps in the creation and implementation of heritage language programs, curricula and assessment aimed at helping language minority students maintain their language and fulfil their linguistic and personal needs.

With respect to linguistic knowledge and patterns of language use, most heritage language speakers and learners, especially those born and schooled in their host country, have perceivable gaps. Typically, a heritage language acquired in childhood may not develop further once schooling in the majority language begins. As a result of incomplete acquisition, many heritage language speakers and learners may exhibit fossilization and/or language attrition in the heritage language. Beyond grammatical competence, many heritage language speakers and learners do not develop the full spectrum of sociolinguistic registers or the level of cognitive and academic literacy commanded by monolingual native speakers [START_REF] Cummins | Four misconceptions about language proficiency in bilingual education[END_REF][START_REF] Krashen | Language shyness and heritage language development[END_REF][START_REF] Valdés | Chicano Spanish: the problem of the 'underdeveloped' code in bilingual repertoires[END_REF]. Even when they may speak the language very fluently and with native-like pronunciation, many heritage speakers and learners lack command of late-acquired aspects of language, including forms of address, grammatical and discourse devices, and other aspects of meaning and pragmatics. Aware of their weaknesses, many heritage speakers experience language shyness, tending to avoid interaction with native speakers to minimize embarrassment and feelings of linguistic inadequacy [START_REF] Krashen | Language shyness and heritage language development[END_REF].

Without a doubt, the most severe outcome for heritage language speakers and learners is language loss. The three books place heavy emphasis on the education of heritage language learners whose role is to reverse the trend toward loss, and to salvage heritage languages from their gradual extinction. But more than anything, the three books seek to raise awareness about the importance of heritage languages for the cultural and national health of the nations involved. In their collective voices, and in different tones, the three volumes seek to expand the teaching of heritage languages in different academic institutions and to find more precise answers to the complex questions about which programs, teaching methods, and pedagogical materials are best suited for the language maintenance of these learners.

1 Valdés et al., 2006: Spanish in California Developing minority language resources is a case study of Spanish in California. Of all the heritage languages spoken in the USA today, Spanish has the longest history of teaching initiatives for heritage language learners in public schools and universities. Yet despite the size and continued growth of the Spanish-speaking community, Valdés et al. document ongoing language shift to English, which affects not only children but also highly educated Spanish-speaking professionals who need to use Spanish on a daily basis. Valdés et al. contextualize the results of a large-scale research project describing current instructional practices at the secondary and post-secondary levels in the teaching of Spanish to heritage language learners. The book consists of nine chapters, most of them written by Joshua Fishman and Guadalupe Valdés, with Rebecca Chávez and William Pérez collaborating on the chapters describing empirical research. In the first two chapters, Fishman places the root of American language ideology and current policies in their broader historical contexts. In Chapter 1, Fishman examines how the 'failed' history of second/foreign language teaching is closely related to American language ideology, and argues that Spanish and other heritage languages, constitute an untapped resource to promote foreign language competence.

Chapter 2 traces the history of minority language teaching in the USA, including indigenous, colonial and contemporary immigrant heritage languages. While many heritage languages will come and go owing to geopolitical and economic changes, Fishman foresees that Spanish will continue to grow in the USA, and advocates helping Spanish speakers preserve their language across generations.

In Chapter 3, Valdés takes us through time with a vivid and highly informative account of the Spanish presence in California since the time of the Spanish Conquest to the present. Valdés highlights how already in the 19th century Spanish speakers were silenced and oppressed by English speakers at the governmental level, and how, to this day, many anti-Spanish policies permeate California's public life. Valdés argues that Americans see Spanish as a threat to the English language and American culture, on the one hand, and as an obstacle to the successful assimilation of Latino immigrants, on the other. According to Valdés, the hostile treatment of the Spanish language and its speakers militates against language maintenance in this population, despite the continuous influx of Spanish speakers.

Chapter 4 presents a survey of 200 educated Latinos from the education, social services, law, medical and government sectors in California, a third of them with advanced degrees. The survey examined beliefs, attitudes and perceived needs about Spanish in California. The results showed that the Latino professionals rarely or only sometimes use Spanish at work, but recommended that Spanish classes focus on cultural connections, content areas and professions.

The next four chapters discuss current practices in foreign language and heritage language teaching of Spanish and present the results of two studies conducted with secondary schools and colleges and universities. In Chapter 5, Valdés surveys the teaching of Spanish in California and underscores the challenges faced by heritage language practitioners who want to develop language resources. Valdés reiterates the position voiced several times elsewhere [START_REF] Valdés | The teaching of Spanish to bilingual Spanish-speaking students: outstanding issues and unanswered questions[END_REF][START_REF]Introduction. Spanish for native speakers, volume I. AATSP Professional Development Series Handbook for teachers K-16[END_REF]: namely, that the varieties of ethnic languages spoken by heritage speakers cannot be compared fairly against the educated monolingual norm; and, second, that proficiency measures developed for foreign language learners do not adequately capture heritage language learners' levels of ability. In Chapters 6 and 7 we learn that a common feature of all the programs for which survey results were available is their emphasis on the standard academic dialect through success in course examinations, reading and writing. Common to all educational institutions surveyed was the instructors' general poor understanding of their students as bilingual individuals. In general, non-Latino instructors described the language proficiency of heritage language students in disapproving terms ('They communicate pretty well orally, not always correctly … have serious problems with reading and writing'; p. 217), often confusing lack of knowledge of a grammatical meta-language with linguistic limitations ('They should be able to analyze language, and know parts of grammar, tenses of verbs. Generally they use them, but cannot identify them'; p. 219). Most disturbing to Valdés et al. is that college instructors regard the educated Spanish of upper class Spanish-speaking monolinguals as the ideal and the primary goal of heritage language instruction. Learning to use the standard language well was assumed 'to lead directly to growth in pride and self-esteem for Latino students' (p. 233). In Chapter 8, Valdés engages in severe criticism of the available heritage language teaching practices, claiming they frequently involve ad hoc adaptations of foreign language teaching approaches applied blindly and inappropriately to heritage language learners (p. 235). Valdés advocates a research agenda for heritage language development (p. 242), with the objectives of:

• developing language evaluation assessment procedures that can identify key differences among heritage learners;

• investigating the implicit systems of different types of heritage learners in their non-dominant L1s;

• determining the degree of system restructuring needed for heritage speakers at different levels of proficiency to carry out particular functions in particular settings using appropriate linguistic forms; and

• identifying whether pedagogies used to bring about restructuring of the interlanguages of L2 learners can also be effective.

Fishman closes the book by explicitly encouraging the Spanish-speaking community to become more proactive in political discussions. This population, more than any other in the USA, has the unique opportunity to spearhead the reversal of transgenerational language loss. In general, Valdés et al. offer a very informative and detailed account of the sociopolitical situation of Spanish in California. Particularly interesting is the rich historical context they provide to help readers grasp the complexity of US language ideology and policy as it relates to current anti-Spanish sentiment in certain segments of the American population. While the specific goals of the research project described are laudable, it is not entirely clear that they have been met. In the end, the results of the two surveys -what Latino professionals think about language maintenance, and what educational institutions actually do to promote maintenance -are not clearly integrated in the conclusion. Throughout the book, current practices for teaching Spanish to heritage speakers at the secondary and post-secondary level are denounced for being too traditional in their focus on academic goals (reading and writing).

There are also contradictions. On several occasions throughout the text, Valdés claims that foreign language instruction has little to offer heritage language instruction, but in discussing Silva-Corvalán's (2003) study about incomplete mastery of the Spanish inflectional system, to mention just one example, Valdés writes:

In the case of incomplete acquisition, the instructional problem to be solved might involve, for example, the full acquisition of tense, aspect and mood in the L1. Instructional approaches might therefore include second language methodologies used in the teaching of both oral and written language to L2 learners. (p. 247) On the next page, however, she writes:

Without evidence to the contrary, one could not conclude that direct forms or formfocused instruction or other typical pedagogies used in L2 instruction would be particularly beneficial in the process of reacquisition or reversal of attrition. (p. 248) Although many of the observations about the political atmosphere in California are correct, the general rhetoric of protest throughout this volume leaves us with the uneasy feeling that the goals of heritage language education will be hard to meet. At the same time, the book succeeds admirably in raising awareness about the importance and urgency of maintaining Spanish and other heritage languages in the USA.

Kondo-Brown, 2006: East Asian Immigrants in North America

In Heritage language development, Kimi Kondo-Brown (2006) broadens the perspective presented in Valdés et al. by describing school-age heritage language learners of East Asian descent living in both the USA and Canada. This volume examines how policy-makers, school administrators, teachers and immigrant parents can work together to create optimal learning contexts for the acquisition and maintenance of Chinese, Japanese and Korean at home, in school and in the community. Kondo-Brown also urges more work promoting heritage language development and maintenance in both younger and older learners.

If there is one significantly different aspect between the two books, it is the tone. Valdés et al. reads like a political manifesto, exuding exasperation both with American language ideologies and policies and with many professionals' self-professed ignorance and prejudice toward Spanish heritage language learners. By contrast, Kondo-Brown's approach is less confrontational. She starts, for example, by reporting the positive personal consequences of bilingualism for language minority children highlighted in early Canadian research. Like Valdés et al., Kondo-Brown also discusses, but does not overtly condemn, the paradoxical nature of American -and to a lesser extent Canadian -language policies. Kondo-Brown suggests, for instance, that American policies encourage immigrant children to become English speakers at the expense of losing their heritage language, and that many of these learners come to the university to re-learn their family language and complete a foreign language requirement. Although Canadian language policies seem more favourable toward heritage language development than their American counterparts, Kondo-Brown suggests there is still strong opposition to teaching heritage languages in Canadian public schools (Cummins, 1993).

A major salient cultural difference between the Spanish and East Asian populations portrayed in the two books is the active role of East Asian families and communities in the language development and education of their children. Many school-age East Asian heritage language learners study their language in community-and church-based schools for Korean and Chinese and the hoshuukoo or Japanese supplementary schools. These are the main, critical, vehicles for promoting heritage language maintenance in this population during the elementary and high school years. And at the college level, many traditional East Asian language, literature and culture departments also offer courses serving mostly heritage language learners.

All chapters, except for the last one, present original empirical quantitative and qualitative studies describing aspects of heritage language learners' profiles and linguistic abilities. The book is structured into three parts, each consisting of three chapters. Part 1 (Chapters 2-4) focuses on bilingualism in the family and explores factors and practices affecting heritage language development in American and Canadian families. Chapter 2 (Guofang Li) is a case study of two Chinese families residing in Canada. The study reveals the ambivalence that immigrant parents feel: while they strongly support heritage language maintenance, they fear that the heritage language will impede full acquisition of English. Chapter 3 (Mitsuyo Sakamoto) examines similar issues in Japanese families in Toronto using a life history design.

Chapter 4 (Eunjin Park) highlights the fundamental role that grandparents play in six Korean families, especially in helping children acquire the complex forms of address and grammatical expressions of honorifics. When immigrant children do not interact with older interlocutors, they miss the chance to learn how social hierarchies and politeness rules are encoded in the language. Korean honorifics are a major pedagogical concern in the teaching of Korean as a heritage language.

The next three chapters discuss the role of organized heritage language instruction through educational institutions. Chapter 5 (Kiyomi Chinen and Richard Tucker) present a quantitative study of identity formation in second generation Japanese Americans in 7th-11th grade attending a California-based hoshuukoo, finding that not only does hoshuukoo help these learners develop literacy skills, but it also fosters a positive sense of ethnic group membership. In Chapter 6, Sarah Shin examines the negative effects that current test-driven practices in American elementary schools have on heritage language maintenance. Chapter 7 (Asako Hayashi) examines how different bilingual education programs in the USA and Japan promote heritage language maintenance and biliteracy. Hayashi found that degree of Japanese and English proficiency was related to frequency of language use inside and outside school.

The last three chapters focus on heritage language use and proficiency in school-age heritage language learners. In Chapter 8, Eun Joo Kim reports on a study of factors predicting proficiency in Korean heritage language learners. Accuracy on a Korean proficiency test was positively related to age of immigration, schooling in the language, and proficiency scores. Chapter 9 (Evelyn Yee-fun Man) focuses on strength of language contact and perception of ethnolinguistic vitality in Chinese heritage language children ages 8-18 living in Canada, finding that the children's attitudes, beliefs and perceptions of the Chinese community in Toronto are closely related to extent of use of Chinese. In Chapter 10, Kimi Kondo-Brown summarizes the findings from all these studies and others to suggest directions for further research. Kondo-Brown (2006) provides an excellent descriptive account of the East Asian immigrant children and their parents. It is very complete in discussing the role of three main agents in successful heritage language acquisition: the family, the school and the community. Although the East Asian community faces some of the same challenges and prejudices of the Spanish-speaking community in the USA, Kondo-Brown does not portray East Asian immigrants as victims of the language loss process instigated by language policies and educators. Rather, East Asian families are depicted as the main agents seeking support from school and the community to transmit their language and culture to their children, and thereby retarding intergenerational language loss. And they are quite successful. What distinguishes this volume from the other two is its discussion of heritage languages beyond the USA. For example, Joseph Lo Bianco shows how policy initiatives implemented through adult literacy programs contribute to heritage language retention by drawing examples from the linguistic situation in Australia and in Scotland, which has been undergoing revival of Gaelic since 1997. Although the Canadian context is mentioned in Kondo-Brown, Patricia Duff offers in Chapter 4 a more nuanced picture of heritage language education in Canada. Tracing the historical development of language policies favouring heritage languages in Canada, Duff claims that despite Canada's more pluralistic and accepting approach to multiculturalism and heritage languages (compared to US practice), the statistics also show a high incidence of heritage language loss among second generation Canadians.

But for those interested in second language acquisition and linguistic issues, Part II is undoubtedly the meat of this book, since it gives a sense of the linguistic repertoire of heritage language learners. Furthermore, some chapters directly compare the linguistic profiles and abilities of L2 learners to heritage speakers. While Spanish and East Asian heritage speakers take the bulk of the volume, the inclusion of two chapters on Russian heritage speakers in this part is particularly welcome.

A pioneer in the study of American Russian and of heritage language systems as linguistic systems in their own right, Maria Polinsky presents in Chapter 8 a small-scale study of two adult heritage speakers and two child heritage speakers, along with two full Russian speakers (child and adult) as controls. The study focuses on the narrative oral production of these speakers and quantifies several grammatical aspects, including number of words, pauses, clauses, embedded clauses, case markers, aspect and tense. Polinsky shows that adult heritage speakers with incomplete knowledge of the language are not just frozen children: their grammatical systems continue to restructure even with impoverished input. In the closing chapter, Terry Au summarizes the results of her research program comparing Spanish and Korean heritage speakers with very low proficiency in those languages to L2 learners of similar low proficiency. She finds that while heritage language learners have advantages over L2 learners in phonology and pronunciation, they do not differ from them in their accuracy of morphosyntax.

On the other side of the proficiency spectrum, in Chapter 11 Claudia Parodi presents both the linguistic characteristics of Chicano Spanish and the profile of advanced heritage language learners who speak that variety. Many of the political and affective issues discussed more broadly in Valdés et al. really come to life in this excellent account of the linguistic repertoire of a Spanish heritage speaker. Furthermore, Parodi provides specific suggestions for how heritage language instruction should differ from L2 instruction. In Chapter 11 Debra Friedman and Olga Kagan describe the academic writing abilities of Russian speakers who moved to the USA between late childhood and adolescence.

Even though these learners received schooling in Russia, they still have notable gaps in their academic knowledge of Russian. In their smallscale study, Friedman and Kagan compare the production of relative clauses and temporal cohesion devices used in written compositions by educated Russian native speakers, Russian heritage speakers and advanced learners of Russian as a foreign language. Friedman and Kagan found revealing differences between the three groups. The heritage language learners did not use as many relative clauses and temporal cohesion devices as the native speakers, while also showing more lexical knowledge and a variety of syntactic resources not evidenced in the L2 learners' narratives. Similar advantages for heritage language learners over L2 learners in vocabulary, collocations and other aspects of grammatical knowledge are reported by Kanno et al. in their study of advanced learners of Japanese, especially those learners who received supplementary schooling (hoshuukoo) during childhood. All chapters in this section -including Masako Douglas's study of Japanese heritage speakers -show that, in general, heritage language learners have more developed oral than writing skills as suggested by tests that require linguistic manipulation (e.g. cloze, grammaticality judgments, etc.).

While interesting and valuable, the small-scale studies in this section involve very few subjects, and can only be interpreted as preliminary linguistic descriptions. Ideally, the studies discussed by Polinsky, Friedman and Kagan, and Kanno et al. should be expanded into larger scale experimental studies so that their findings can be generalized beyond the individual cases they report, permitting solid implications to be drawn for the classroom.

III The relevance of heritage language speakers/learners for second language acquisition

How are heritage language speakers/learners relevant to the field of L2 acquisition and linguistic theory more generally? In Valdés et al. (2006) and elsewhere [START_REF] Valdés | The teaching of Spanish to bilingual Spanish-speaking students: outstanding issues and unanswered questions[END_REF][START_REF] Kondo-Brown | Differences in language skills: heritage language learner subgroups and foreign language learners[END_REF], Valdés repeatedly asserts that efforts at teaching heritage languages are blind ad hoc adaptations of foreign language teaching, and that heritage language research and teaching has so far proceeded without a solid theoretical foundation, implying that no progress in the field can be made without a theory of heritage language acquisition (see also Lynch, 2003). Specifically, in Chapter 8 she states:

Second language acquisition theories, as well as traditions now guiding traditional foreign language instruction have little to say about these students and what they should be taught. Existing research on incipient or developing bilingualism in foreign or second languages is of little relevance to teachers of heritage students. Moreover, views about second language (L2) developmental sequences and second language proficiency hierarchies can contribute little to the understanding of the instructional needs of this population. (Valdés et al., 2006: 119) Even if heritage language learners are in many ways different from typical L2 learners with no previous knowledge of the language, contrary to Valdés, I see many parallels between heritage language and second language acquisition and teaching worth exploring. My understanding of this quote from Valdés is that little is known about the linguistic skills of heritage language speakers, and about how these resemble or do not resemble those of L2 learners. Naturally, without proper understanding of how similar or different these two types of learners are, it is difficult to tell at this point whether the exact same methods applied to L2 learners in the classroom should also be applied to heritage language learners. This, of course, is an open question because other than anecdotal and descriptive accounts of differences, we have no empirical data measuring the effectiveness of L2 methods for heritage language learners.

Where I also disagree with Valdés is on the validity of existing learning theories to unravel heritage language acquisition and development. Heritage speakers are bilingual individuals who share features with L1 acquirers -such as early exposure to the language in a naturalistic setting during a critical period for the development of language -and with adult L2 learners, including restricted access to input, transfer errors and incomplete acquisition, among others. We do not have to reinvent the wheel here: existing theories of L1 and L2 acquisition and bilingualism can be extended to explain and make predictions about the nature of linguistic knowledge of heritage language learners at different ages of development. In fact, I have argued elsewhere (Montrul, 2005;[START_REF] Polinsky | Incomplete acquisition: American Russian[END_REF][START_REF] Montrul | Gender agreement in adult second language learners and Spanish heritage speakers: the effects of age and context of acquisition[END_REF] that currently available theories of second language acquisition are very relevant to the investigation and explanation of the nature of linguistic knowledge in these two types of learners. And, seen from another angle, heritage language learners are very relevant to second language acquisition and linguistics, since they afford these fields a unique opportunity to evaluate:

• current claims about the complex nature of L1 and L2 competence acquired before and after the so-called critical period;

• the type of knowledge (implicit, explicit) acquired as a result of input and experience;

• the competence-performance dichotomy in formal linguistics;

• the specific innate and environmental ingredients for developing stable native speaker competence;

• the role of transfer and

• the limits and possibilities of bilingual competence in adulthood.

For example, one bone of contention in generative second language research is the explanation for observed differences in the outcome of L1 acquisition and adult L2 acquisition. L2 learners tend to fossilize whereas L1 learners acquire the L1 grammar completely. A likely cause of L2 fossilization is input: L1 learners are exposed to continuous and abundant naturalistic input and have opportunities to use their language. L2 learners, by contrast, receive variable L2 input in type (instructed, naturalistic), amount (more or less frequent) and quality (native vs. non-native), and may not use the L2 as much as L1 learners.

The notion that there is a biologically determined propitious time in early childhood during which being exposed to language is crucial for developing linguistic skills has figured prominently in explanations of why adult L2 learners rarely reach the level of linguistic ability of native speakers (see, among others, [START_REF] Bley-Vroman | The logical problem of foreign language learning[END_REF][START_REF] Long | Maturational constraints on language development[END_REF][START_REF] Schachter | On the issue of completeness in second language acquisition[END_REF][START_REF] Johnson | Critical period effects on universal principles of language: the status of subjacency on the acquisition of a second language[END_REF][START_REF] Meisel | The acquisition of the syntax of negation in French and German: contrasting first and second language development[END_REF][START_REF] Dekeyser | Implicit and explicit learning[END_REF][START_REF] Hyltenstam | Maturational constraints in SLA[END_REF][START_REF] Paradis | A neurolinguistic theory of bilingualism[END_REF][START_REF] Hawkins | Interpretation of English multiple whquestions by Japanese speakers: a missing uninterpretable feature account[END_REF][START_REF] Tsimpli | The interpretability hypothesis: evidence from wh-interrogatives in second language acquisition[END_REF]. Other researchers downplay the effects of age, maintaining that access to Universal Grammar is not subject to a critical period [START_REF] Epstein | Second language acquisition, theoretical and experimental issues in contemporary research[END_REF][START_REF] White | How native is near-native? The issue of ultimate attainment in adult second language acquisition[END_REF]. Since heritage language speakers and learners have some characteristics of L1 and some of L2 acquirers, they represent an interesting testing case to tease apart competing positions like these. For example, both L2 learners and heritage speakers exhibit incomplete knowledge of the target language, but they differ with respect to experience (use) and age of acquisition.1 If age of acquisition is an important factor in bilingual outcomes, as theories of L2 acquisition assume, then it is possible that despite incomplete knowledge later in life, heritage speakers may still possess more native-like knowledge of the target language than L2 learners with similar command of the language, by virtue of the fact that the linguistic or/and cognitive mechanisms assumed to operate for language acquisition in childhood may be different from those deployed in adulthood. Alternatively, if the same learning mechanisms are accessible to the child and adult learner, L2 learners and heritage speakers should not differ.

The studies reported by Au in Brinton et al. (2008) found no advantages for heritage learners on morphosyntax, although morphosyntax was very broadly defined. By contrast, the other chapters in Part II of Brinton et al. as well as my own work on this area suggest that this initial impression may be premature, and that age and exposure also provide an advantage for L2 learners in different aspects of syntax, lexical semantics and inflectional morphology. For example, Montrul (2005) showed that low proficiency heritage speakers made the syntactic distinction between unaccusative and unergative verbs, while L2 learners did not. Montrul (2006) showed how intermediate proficiency heritage speakers had all aspects of the null subject parameter in place whereas the L2 learner counterparts have not yet reset the entire parameter. These studies suggest that aspects of syntax that are acquired very early in childhood are in place and retained by heritage speakers, while L2 learners acquire them more slowly. But when it comes to morphological errors, the picture is less clear. [START_REF] Montrul | Gender agreement in adult second language learners and Spanish heritage speakers: the effects of age and context of acquisition[END_REF] showed that both L2 learners and heritage speakers make errors with gender agreement, but heritage speakers make overall fewer errors. Yet, gender errors are manifested differently: heritage speakers made more errors than L2 learners in the written tasks, whereas the L2 learners were less accurate on an oral task, which was assumed to minimize metalinguistic knowledge. Thus, the picture that is emerging thus far is that heritage speakers appear to have an advantage over L2 learners in some aspects of morphology and syntax as well, not just in phonology.

Related to age is also the issue of language transfer. Recent literature on language acquisition and loss has shown that language transfer occurs both in the L1-to-L2 direction (a typical characteristic of L2 acquisition) and in the L2-to-L1 direction, in the case of L1 loss under intensive contact with an L2 [START_REF] Cook | The effects of the second language on the first[END_REF]. In both cases, the dominant language encroaches into the structure of the less dominant language in systematic ways. The few documented cases of L1 attrition in adults at the level of phonetics [START_REF] Major | Losing English as a first language[END_REF], lexical semantics [START_REF] Balcom | Crosslinguistic influence of L2 English on middle constructions in L1 French[END_REF] and the syntax-discourse interface [START_REF] Gürel | Linguistic characteristics of second language acquisition and first language attrition: Turkish overt vs. null pronouns[END_REF] can be easily characterized as L2 effects. Heritage speakers are cases of attrition and incomplete acquisition in childhood, the effects of which are more adverse than in adulthood [START_REF] Montrul | Gender agreement in adult second language learners and Spanish heritage speakers: the effects of age and context of acquisition[END_REF]. It is an open question, therefore, as to whether the transfer effects in these two situations are comparable: is L1 influence in adult L2 learners (late bilinguals) similar to L2 influence in early bilinguals? In short, although there is some basic research that systematically compares L2 learners and heritage speakers -as well as research focusing on the systematic nature of incompletely acquired L1 grammatical systems in children and adults [START_REF] Silva-Corvalán | Language contact and change: Spanish in Los Angeles[END_REF]2003;[START_REF] Håkansson | Syntax and morphology in language attrition: a study of five bilingual expatriate Swedes[END_REF][START_REF] Song | The learning and teaching of Korean in community schools[END_REF][START_REF] Anderson | Noun phrase gender agreement in language attrition, Preliminary results[END_REF][START_REF] Montrul | Incomplete acquisition and attrition of Spanish tense/aspect distinctions in adult bilinguals[END_REF][START_REF] Paradis | A neurolinguistic theory of bilingualism[END_REF][START_REF] Kondo-Brown | Differences in language skills: heritage language learner subgroups and foreign language learners[END_REF][START_REF] Polinsky | Incomplete acquisition: American Russian[END_REF]2008;[START_REF] Bolonyai | In)vulnerable agreement in incomplete bilingual L1 learners[END_REF] -this trend should continue. The findings from these research programs have much bearing on our understanding of the role of age on language acquisition and loss more generally, and they are crucial to inform linguistic theory and the heritage language teaching profession.

In conclusion, the study of heritage language learners has boomed in the last few years, as vividly illustrated in these three new books. Despite obvious differences between L2 learners and heritage speakers, there are important language learning issues that affect the two populations. Linguistic approaches to L2 acquisition have often been criticized for not having pedagogical implications. I believe that linguistic theory applied to L2 acquisition and adult early bilinguals remains a crucial tool for generating research questions and for guiding the design of linguistically-informed instruments to identify systematic and measurable differences and similarities between these two types of language learners. Once we acquire a better sense of the linguistic knowledge heritage language and L2 learners have or lack -and this is communicated to the teaching profession -practitioners will be in a much better position to advocate for separate programs for heritage language learners, or to address both learners' linguistic and pedagogical needs, especially when they find themselves sharing the same L2 class.

3

  Brinton et al., 2008: Some links with second language acquisition Heritage language education: a new field emerging includes 20 chapters written by prominent scholars in language policy, linguistics and education, and covers some of the same ground asValdés et al. and Kondo-Brown: issues related to policy, identity and types of programs, with heavy emphasis on Spanish and East Asian heritage speakers. Nancy Hornberger and Shuhan Wang open the first chapter with questions about precisely defining identity. The rest of the book is organized into three parts. Part I 'Heritage speakers: demographics, policy, and identity,' includes Chapters 2-7; Part II, 'Heritage speaker profiles and needs analysis,' contains Chapters 8-12; and Part III, 'Program development and evaluation,' includes Chapters 13-19. The book closes with Chapter 20, 'Salvaging a childhood language', by Terry Au.

I agree with an anonymous Second Language Research reviewer that age here is not just biological age but a macro-variable that also subsumes maturational state, cognitive development, degree of L1 proficiency, amount of L1 and L2 use, etc.

Acknowledgements

I am grateful to Margaret Thomas for giving me the opportunity to write this review and for her insightful editorial suggestions. I also thank an anonymous Second Language Research reviewer for feedback and suggestions.