Fabienne Comte

Adeline Samson

Julien J Stirnemann

DECONVOLUTION ESTIMATION OF ONSET OF PREGNANCY WITH REPLICATE OBSERVATIONS

Keywords: Deconvolution, Density estimation, Nonparametric methods, Dating of pregnancy, Mean square risk, Replicate observations 1

In general, the precise date of onset of pregnancy is unknown and may only be estimated from ultrasound biometric measurements of the embryo. We want to estimate the density of the random variables corresponding to the interval between last menstrual period and true onset of pregnancy. The observations correspond to the variables of interest up to an additive noise. We suggest an estimation procedure based on deconvolution. It requires the knowledge of the density of the noise which is not available. But we have at our disposal another specific sample with replicate observations for twin pregnancies. This allows both to estimate the noise density and to improve the deconvolution step. Convergence rates of the final estimator are studied and compared to other settings. Our estimator involves a cut-off parameter for which we propose a cross-validation type procedure. Lastly, we estimate the target density in spontaneous pregnancies with an estimation of the noise obtained from replicate observations in twin pregnancies.

Introduction

In spontaneously conceived pregnancies, the date of pregnancy is unknown. Although pregnancies occur at around 14 days following last menstrual period (LMP), the fertile window of a woman may vary widely based upon hormonal studies (Wilcox et al. [2000]). These studies, however, provide day-specific probabilities of a fertile window within a female cycle in non-pregnant women and not the probability density of onset of pregnancy in pregnant women. Since the exact date of pregnancy is never precisely known in women conceiving spontaneously, the probability distribution function of onset of pregnancy within female cycles is unknown in the general population. This density, however, may have important implications both for clinical practice and physiology knowledge.

Ultrasound is the most widely used method for dating pregnancies in clinical practice. First trimester biometric measurements such as the crown-rump length (CRL) have been proven to perform better than LMP for dating pregnancies. Several formulas, derived from simple regression analysis have been developed for dating pregnancies (Sladkevicius et al. [2005]), the most widely used being the formula initially suggested by Robinson [1973]. Denoting by X the interval between LMP and true onset of pregnancy, and by Y the interval between LMP and ultrasound estimate, the purpose of this study is to estimate the density f of X. However, only the noisy observations (1) Y j = X j + ε j , j = 1, . . . , n are available. Here, the X j and the ε j for j = 1, . . . , n are independent identically distributed and the sequences (X j ) 1≤j≤n and (ε j ) 1≤j≤n are independent. Moreover, in this setting the density f ε of ε is unknown and the noise cannot be directly estimated from a preliminary sample of ε j . Since X is measured with an unknown error, the estimation of f may be seen as a deconvolution problem. Regarding the assumptions on the distribution of the error, several approaches have been studied in the literature. Numerous works have addressed this problem under the assumption of a known density for the error. These works comprise kernel methods (see Fan [1991], Liu and Taylor [1989], Stefanski and Carroll [1990], Hesse [1999], Delaigle and Gijbels [2004]) as well as wavelet methods (see Fan and Koo [2002], Pensky and Vidakovic [1999]). Minimax optimality of convergence rates have been studied by Fan [1991], Butucea [2004], Butucea and Tsybakov [2008a]. When a sample of the error is given, density estimation has been addressed by Diggle and Hall [1993] and Neumann [1997]. The latter considers the case of ordinary smooth densities for both desnities of the error and X, and provides minimax rates of convergence. More contributions by Johannes [2009] and Comte and Lacour [2011] propose different approaches with regard to bandwidth selection. A full scheme of estimation in this setting with data-driven bandwidth selection may be found in Comte and Lacour [2011].

In this article, we consider yet a different setting in which neither a known density nor a sample of noise are available. Rather, we consider the situation of replicate and noisy observations of the random variable X. Consider we have a sample of pregnancies with two replicate measurements of X:

(2)

Y -j1 = X -j + ε -j1 , Y -j2 = X -j + ε -j2 , j = 1, . . . , M with X -j , ε -j1 and ε -j2 , for j = 1, . . . , M , independent and identically distributed. The sequences (X -j ) 1≤j≤M , (ε -j1 ) 1≤j≤M and (ε -j2 ) 1≤j≤M are independent. These noisy observations could be replicate measurements of CRL of the same embryo or measurements of CRL in twin pregnancies. Therefore, we consider that two independent samples are available: the first, of size M , containing replicate observations and the second, of size n containing non-replicate observations. Density estimation by deconvolution with replicate observations has been studied by Delaigle et al. [2008], Li and Vuong [1998] and Meister and Neumann [2009]. Our approach suggests a similar estimator that is related to the truncated estimator of Neumann [1997]. The second sample with replicate observations allows both to estimate the noise density and to improve the deconvolution step. Our estimator involves a cut-off parameter for which we propose a cross-validation type procedure, following the procedure described in Comte and Lacour [2011].

The outline of this article is as follows. In Section 2 we define our estimator. We then majorate the L 2 risk based upon a new version of the fundamental lemma of Neumann [1997]. Convergence rates are compared to the settings of known noise density and observed noise samples. We discuss the relationship between M , n and the resulting convergence rates and show that the convergence rate is the same as that found with an assumed known noise density in several cases. In Section 4 we discuss model selection and the choice of an appropriate penalty. Simulations are conducted to illustrate the performance of our estimator together with a comparison with existing results in Section 5. Finally, in Section 6, we apply our method to real data and estimate the distribution f of onset of pregnancy within a female cycle using ultrasound measurements in twin pregnancies as replicate noisy observations.

Model and estimator

We denote f Y , f and f ε the densities of Y , X and ε. We denote by g * (x) = e itx g(x)dx, the Fourier transform of any integrable function g. The characteristic functions of each of the variables Y , X and ε are therefore denoted f * Y , f * and f * ε respectively. For a function g : R → R, we denote by g 2 = R g 2 (x)dx the L 2 norm. For two real numbers a and b, we denote a ∧ b = min(a, b). As a rule in this paper, unless otherwise specified, C and C will denote universal constants that may change from line to line.

The convolution problem may be written as

f Y (x) = f f ε (x) = f (x -t)f ε (t)
dt where denotes the convolution operator. Using the characteristic functions, we have

f * Y (u) = f * (u)f * ε (u).
In our setting we consider f ε unknown, therefore we must estimate it or at least the square of its characteristic function f * ε as we will see shortly. In the following, we consider the model described by the two independent samples (1) and (2). A noise sample is not available and we suggest that the estimation of f * ε relies upon replicate observations given in (2) instead.

The following preliminary assumption regarding the behavior of f ε will be considered fulfilled throughout the article.

Assumption(A1)

We assume ε is symmetric and that its characteristic function never vanishes.

Since assuming ε symmetric is equivalent to assuming f * ε real-valued, we can see that (A1) is equivalent to the assumption 2.2 in Delaigle et al. [2008]. Therefore, Assumption (A1) implies that ∀t ∈ R, f * ε (t) ∈ R * + . Under this reasonable assumption, we have:

E e it(ε -j1 -ε -j2 ) = |E e itε -j1 | 2 (A1) = E e itε -j1 2 = f * ε (t) 2 . Therefore, given that E e it(ε -j1 -ε -j2 ) = E e it(Y -j1 -Y -j2
) and under the hypothesis (A1), Delaigle et al. [2008] propose the following estimator of (

f * ε ) 1 M M j=1 cos t(Y -j1 -Y -j2 ) 1/2
As we will see shortly, our estimator of f * ε is used over a domain for which f * ε is strictly positive. Therefore we have the following estimation of (f * ε ) 2 without the absolute value:

(

3) (f * ε ) 2 (t) = 1 M M j=1 cos t(Y -j1 -Y -j2 )
We then define a truncated estimate of 1/f * ε :

(4)

1 f * ε (t) = 1 (f * ε ) 2 (t)≥M -1/2 (f * ε ) 2 (t) Fourier inversion of f * = f *
Y /f * ε yields the following estimator of f using a πm cut-off for integrability purpose :

(5) fm (x) = 1 2π πm -πm e -ixt f * Y (t) f * ε (t) dt, where f * Y (t) = 1 n + M   n j=1 e itY j + M j=1 e itY -j1   .
This estimator can also be viewed as a deconvolution kernel estimator with the sinc kernel and the bandwidth 1/(πm).

Remark 1. Data Y -j2 are not used in f * Y (t). Indeed, using the n+2M variables increases the variance of the estimator of f * Y (t) due to the dependency between Y -j1 and Y -j2 (see Appendix).

For further discussion, we recall the definitions of the estimators given in (3), (4) and (5) for the cases (i) with known density of noise, denoted with superscript KN and (ii) with an independent sample (ε -j ) of noise observations, denoted with superscript ON (see Comte and Lacour [2011]):

(i) In the case of a known density f ε of the noise, the estimate of f is:

f KN m (x) = 1 2π πm -πm e -ixt f * Y (t) f * ε (t) dt
where f * Y is defined as previously. (ii) In the case of an unknown density but with a sample of (ε -j ), the estimate of the characteristic function

f * ε is f ON * ε (t) = 1 M M j=1 e itε -j
and its truncated estimate

1 f ON * ε (t) = 1 | f ON * ε (t)|≥M -1/2 f ON * ε (t) .
The estimate of f in this case is

f ON m (x) = 1 2π πm -πm e -ixt f * Y (t) f ON * ε (t) dt,
where f * Y is defined as previously with M = 0. The additional sample with size M can not be used here to improve the estimation of f * Y since it concerns the noise only. These estimators differ from (5) in several ways:

(a) First, because of replications, we can only estimate (f * ε ) 2 and not directly f * ε as in case (ii).

(b) Subsequently, the levels of truncations required will also differ in (4) and (ii). This directly arises from the previous comment since the truncation will be in M -1/4 in (4), whereas it is in M -1/2 in the ON case (ii).

(c) Finally, compared to our setting and (ii) which share the fact that f ε is unknown, the case (i) does not require a truncation and the estimator is the classical expression of density estimation by deconvolution with a Fourier transform of compact support. The risk bound obtained in this case is considered as a benchmark of the best reachable bound.

The estimator fm (x) also differs from Delaigle et al. [2008] in several ways:

(a) A cut-off πm is used instead of a ridge parameter. This cut-off allows to consider super-smooth densities f ε and f , which is not the case with the ridge parameter. The cutoff use yields to restrict to the sinc kernel in order to optimize the bias of the estimator.

(b) As already mentioned earlier, we only use the estimation of f * ε where f * ε is nonnegative so the absolute value used by Delaigle et al. [2008] is not required. This substantially simplifies the theoretical study.

Upper bound of the L 2 risk

Let us define f m such that f * m = f * 1 [-πm,πm] (.). The function f m is the function which is in fact estimated by fm . Therefore, this implies a nonparametric bias measured by the distance between f and f m . Delaigle et al. [2008] study the pointwise risk of their estimator. In this paper, we propose to complete their theoretical study by considering the integrated risk of our estimator. We wish to bound the mean integrated squared error (MISE) defined as E( f -fm

2 ). We will first generalize Neumann's Lemma (Neumann [1997]) to the case of replicate measurements and use this result to deduce a risk bound.

3.1. General MISE bound. The extension of Neumann's lemma for replicate measurements is Lemma 1. Assume that (A1) holds, and let p be an integer, p ≥ 1. There exists a constant C p such that

E 1 f * ε (t) - 1 f * ε (t) 2p ≤ 2 |f * ε (t)| 2 ∧ 3M -1/2 |f * ε (t)| 4 ∧ C p M -1 |f * ε (t)| 6 p .
Proof is given in appendix.

Let us define, for

k ∈ N * , ∆ k (m) = πm -πm dt |f * ε (t)| k and ∆ (f ) k (m) = πm -πm |f * (t)| 2 |f * ε (t)| k dt as well as their estimates ∆k (m) = πm -πm dt | f * ε (t)| k and ∆(f) k (m) = πm -πm | f * Y (t)| 2 | f * ε (t)| k+2 dt since f * = f * Y f * ε .
Using the previous lemma we may deduce the following bound for the MISE:

Proposition 1. Assume that (A1) holds and fm is defined by (5). Then there exists a constant C such that

(6) E( f -fm 2 ) ≤ f -f m 2 + C ∆ 2 (m) n + M + ∆ (f ) 2 (m) √ M ∧ ∆ (f ) 4 (m) M .
Proof is given in appendix.

The precise decomposition is

E( f -fm 2 ) ≤ f -f m 2 + C ∆ 2 (m) n + M + ∆ 4 (m) (n + M ) √ M ∧ ∆ 6 (m) (n + M )M ∧ ∆ 2 (m) n + M + ∆ (f ) 2 (m) √ M ∧ ∆ (f ) 4 (m) M . (7)
This decomposition underlines the different terms involved in the bound of the integrated risk. We recognize in this last inequality (7) the bias ff m 2 and variance denoted Var(m

) := Q 1 (m) + Q 1 (m) ∧ Q 2 (m) + Q 3 (m) with Q 1 (m) := ∆ 2 (m)/(n + M ) Q 2 (m) := ∆ 4 (m)/((n + M ) √ M ) ∧ ∆ 6 (m)/((n + M )M ) Q 3 (m) := ∆ (f ) 2 (m)/ √ M ∧ ∆ (f ) 4 (m)/M.
We can also recognize Q 1 (m) as the variance term that arises alone when f * ε is assumed as known (see Comte et al. [2006]) and for sample size n + M . The following terms Q 2 (m) and Q 3 (m) are specific to our setting involving replicate observations and show the loss in the resulting rates. In the case of observed noise (see Comte and Lacour [2011]), the upper bound is Comte and Lacour [2011] easily obtain a variance term bounded by Q 1 (m). Therefore, compared to the case with observed noise or known density, the term Q 3 (m) resulting from the estimation of (f * ε ) 2 with replicate observations is a substantial step in complexity. Moreover, an interesting question would be to see if Q 2 (m) would not be smaller than Q 1 (m), at least in a empirical way. A theoretical and an empirical study of the behavior of each of these terms is presented later.

f -f m 2 + C∆ 2 (m)/n + (C + 2)∆ (f ) 2 (m)/M. Moreover, for M ≥ n,
3.2. Resulting rates. Let us consider the following classical assumptions regarding the behavior of f * ε :

Assumption (A2) There exist α ≥ 0, β > 0, γ ∈ R (γ > 0 if α = 0) and k 0 , k 1 > 0 such that k 0 (x 2 + 1) -α/2 exp(-β|x| γ ) ≤ |f * ε (x)| ≤ k 1 (x 2 + 1) -α/2 exp(-β|x| γ )
The noise distribution is called ordinary smooth if γ = 0 and supersmooth otherwise. A Gaussian noise is supersmooth with γ = 2 and a Laplace noise is ordinary smooth with γ = 0 and α = 2. Now, we know that, under (A2), the dominating variance term has the following order [see Comte et al., 2006, Comte andLacour, 2011]:

Q 1 (m) = ∆ 2 (m) n + M ≤ C m 2α+1-γ exp(2β(πm) γ ) n + M .
Such orders are non standard for variance terms in nonparametric estimation and in particular larger than orders m/n which are obtained for standard problems (e.g. density estimation without noise, corresponding to α = γ = 0, M = 0). If we want to give precise examples of the rates that can be obtained in the deconvolution context, we must also make assumptions on the rate of decrease of f * . Classically, we consider the following smoothness spaces for density f on R:

Assumption (A3) f ∈ A a,b,c (l) = {f ∈ L 1 ∩ L 2 , |f * (x)| 2 (x 2 + 1) a exp(2b|x| c )dx ≤ l} with c ≥ 0, b > 0, a ∈ R (a > 1/2 if c = 0), l > 0.
As previously, when c > 0, the function f is known as supersmooth, and as ordinary smooth otherwise. The spaces of ordinary smooth functions correspond to classical Sobolev classes, while supersmooth functions are infinitely differentiable (analytic function), and we have necessarily c ≤ 2. It includes for example Gaussian (c = 2) and Cauchy (c = 1) densities.

Then, under (A3), we have the following bias order:

(8) f -f m 2 ≤ Cm -2a exp(-2b(πm) c ).
Moreover, under both (A3) and (A2), we have

Q 3 (m) ≤ C m 2(α-a) + e (2β(πm) γ )-2b(πm) c ) + √ M ∧ m 2(2α-a) + e (4β(πm) γ )-2b(πm) c ) + M .
Therefore, we have the following results when both f and f ε are ordinary smooth.

Proposition 2. We consider assumptions (A1), (A2), (A3) and the ordinary smooth case for both f and f ε with c = γ = 0. The bound (6) then becomes

E( f -fm 2 ) ≤ C m -2a + m 2α+1 n + M + m 2(α-a) + √ M ∧ m 2(2α-a) + M .
If moreover M ≥ n and a ≥ α -1/2, then we have for 2a+2α+1) , and if

m opt = M 1/(2a+2α+1) , (9) E( f -fmopt 2 ) ≤ CM -2a/(2a+2α+1) If M = n ω with ω < 1, m opt = n 1/(
(10) 2α ≤ a ≤ ω 1 -ω (α + 1 2 ), then (11) E( f -fmopt 2 ) ≤ Cn -2a/(2a+2α+1) .
The first inequality is a consequence of ( 6), ( 8) and elementary bounds for the variance. Inequalities ( 9) and ( 11) allow us to recover the deconvolution rates as if the noise had known density for a sample size of observations of order M and n respectively. This rate is known to be the optimal one when the noise density is assumed to be known (see Fan [1991], Butucea [2004], Butucea and Tsybakov [2008a,b]). The first case was already mentioned in Delaigle et al. [2008]. The case M < n is new and interesting as it corresponds to our real data setting. Note that condition (10) reduces to a ≥ 2α when ω tends to 1. Another case is generally considered, where still c = 0 but γ > 0: Proposition 3. We consider assumptions (A1), (A2), (A3) and the ordinary smooth case for f while f ε is super smooth:

c = 0, γ > 0. Then if M ≥ n and m = m opt = π -1 (log(M )/(8β)) 1/γ , we have (12) E( f -fmopt 2 ) ≤ C[log(M )] -2a/γ
The rate in ( 12) is the optimal rate in this context. Unfortunately, it is logarithmic, but practical experiments show that, nevertheless, the procedure works well. It happens that, when the noise density is known, the choice of the optimal cutoff m opt is known also. In our case, an adaptive procedure would be required in this case as well, since β, γ are unknown. This enhances the interest of adaptive procedures in general.

Proposition 3 is often summarized as: "when the noise is super-smooth, the rate of deconvolution is logarithmic". This is not true since the regularity parameters of f are involved in the computation of the rate. Let us give two counterexamples. Example 1. Consider M = n, a = α = 0 and 2bπ = 2βπ = 1, c = γ = 1. We have

E( f -fm 2 ) ≤ C e -m + e m M + 1 √ M ∧ e m M .
Then the choice m opt = log(M )/2 yields

E( f -fmopt 2 ) ≤ CM -1/2
which is also still an optimal deconvolution rate in this case, and is obviously much better than logarithmic.

Example 2: Gaussian-Gaussian case. Consider M = n and a = α = 0 and 2b = 2β = 1, c = γ = 2. In that case, the risk bound can be written

E( f -fm 2 ) ≤ C e -(πm) 2 + m -1 e (πm) 2 M + 1 √ M ∧ e (πm) 2 M .
Then, choose

πm opt = 1 2 log(M ) + 1 4 log(log(M )) 1/2
, so that the risk is bounded by

E( f -fmopt 2 ) ≤ C(log(M )) -1/4 M -1/2
which is still a rate better than logarithmic for a sample size of order M .

We can give a more general result.

Proposition 4. Assume that assumptions (A1), (A2), (A3) are fulfilled with c > γ > 0.

If M ≥ n, then there exists a constant C such that

E( f -fm 2 ) ≤ C m -2a exp(-2b(πm) c ) + m 2α+1-γ e 2β(πm) γ M (1 + m 2α e 2β(πm) γ √ M ) .
Consequently, for any , 0 < < 1/2, the choice

πm 0 = ( log(M )/(2β)) 1/γ yields E( f -fm 0 2 ) ≤ C[log(M )] (2α+1-γ)/γ M -1+ .
The last inequality also presents a general context where a polynomial rate can be reached in the supersmooth case.

3.3. Discussion about the dominating variance term. We have shown that the optimal rate could be reached both for M ≤ n or M ≥ n, depending on the configuration of the parameters. Now, for cross validation choice of the cutoff parameter, it is useful to determine which term in the variance decomposition is dominating. In particular, we hypothesize that the main variance term defined in (6) may be simply Var

(m) = CQ 1 (m) = C∆ 2 (m)/(n + M ) where C is a numerical constant. An estimation of Var(m) is Var(m) = Q1 (m) + Q1 (m) ∧ Q2 (m) + Q3 (m) with Q1 (m) = ∆2 (m) n + M Q2 (m) = ∆4 (m) (n + M ) √ M ∧ ∆6 (m) (n + M )M Q3 (m) = ∆(f) 2 (m) √ M ∧ ∆(f) 4 (m) M
An example of empirical behavior of Q1 (m), Q2 (m) and Q3 (m) is depicted in Figure 1 for M n and M n and in the setting of ordinary smooth or supersmooth functions for f chosen as Laplace and Gaussian respectively, together with an ordinary smooth Laplace noise. Expectedly, Q1 (m) is larger than the other terms when M n. Interestingly, this seems to be also true when M n, at least empirically. This finding was invariable throughout simulations, thus making Var(m) C ∆2 (m)/(n + M ) an appropriate choice regardless of the respective values of n and M . The previous considerations will become prominent in the choice of a penalty in Section 4.

Cross-validation cut-off selector

The parameter m belongs to the set of admissible values

M n = {m = k/κ, k = 1, . . . , nκ , Var(m) ≤ C 0 }.
for fixed κ > 0 and C 0 , and z denoting the integer part of z. The general outline of the method used to select among all considered indexes M n is borrowed from Comte and Lacour [2011]. Our approach aims to select m ∈ M n based upon an adequate bias-variance compromise. First, notice that in our case f -

f m 2 = f 2 -f m 2 . Indeed, denoting •, • the scalar product, we have f -f m 2 = f 2 -2 f * , f * m + f m 2 = f 2 -2 f * (x)( f * 1 [-πm,πm] )(x)dx + f m 2 = f 2 -f m 2
The theoretical optimal choice of m is defined as:

m th = argmin m∈Mn f -f m 2 + Var(m) = argmin m∈Mn -f m 2 + Var(m)
The previous value of m th may only be estimated since ff m and Var(m) are both unknown. Using the estimator fm defined by (5), we will consider the following preliminary where pen(m) is a penalty term related to the variance that remains to be defined.

4.1.

Choice of an appropriate penalty. The penalty term is related to the variance Var(m) as defined by ( 6). When M ≥ n and f is sufficiently more regular than f ε , the optimal rates computations show that Var(m) C∆ 2 (m)/(n + M ). The empirical behavior of Q1 (m), Q2 (m) and Q3 (m) seems to suggest that Var(m) C ∆2 (m)/(n + M ) even when M n (see Figure 1). Therefore, based upon Theorem 1 in Comte and Lacour [2011], we define pen(m) as ( 14)

pen(m) = K 0 log ∆ 2 (m) log(m + 1) 2 ∆ 2 (m) n + M
where K 0 is a numerical constant. Note that the term log ∆ 2 (m) / log(m + 1) has the order of a constant when the noise is ordinary smooth and yields only a slight correction when the noise is supersmooth. This quantity has been introduced by Comte and Lacour [2011] to take into account negligible loss in the rates that sometimes (not systematically) occur in this last case.

4.2. Estimation procedure. We define pen(m) the estimator of pen(m) as:

pen(m) = K 1 log ∆2 (m) log(m + 1) 2 ∆2 (m) n + M .
Throughout numerical estimations we will consider K 1 = 1, after a set of simulation experiments to calibrate it. The computation of fm is performed following Comte and Lacour [2011] and Comte et al. [2006] by considering the following expression of the estimator (5) as an orthogonal projection. We denote ϕ(x) = sin(πx)/πx the sinus cardinal function. The orthonormal basis {ϕ m,j } j∈Z is thus defined by ϕ m,j (x) = √ mϕ(mxj), m ∈ N * , j ∈ Z and ϕ * m,j (x) = e -ixj/m 1 [-πm,πm] (x)/ √ m. We also recall from Comte and Lacour [2011] the estimated projection coefficients

âm,j = √ m 2 (-1) l 2 0 e ilπx f * Y f * ε (πm(x -1))dx.
The estimator (5) may then be expressed: Finally, by plugging ( 16) in ( 15) we obtain f m which is our final estimator. The choice of the constant κ will influence the quality of the final estimation since it governs the number of models that are proposed before selection. Choosing κ small will offer only a restricted number of models for the algorithm to choose from, whereas choosing κ large will allow a more refined estimation of mn . For the simulations in Section 5, we choose κ = 4 to keep the computing time reasonable. Conversely, for the real data application in Section 6, we choose κ = 30. 5. Simulation study 5.1. Design of simulation. Noise was given a Laplace or a Gaussian density with variance σ 2 as follows:

• Laplace noise.

f ε (x) = σ 2 e -σ|x| and f * ε (x) = σ 2 σ 2 + x 2 • Gaussian noise. f ε (x) = 1 √ 2π e -0.5x 2 /σ 2 and f * ε (x) = e -0.5σ 2 x 2
We compared our results to estimations under the assumption of a known noise density (see Comte et al. [2006], Comte and Lacour [2011] for the description of the estimation and penalties for Gaussian and Laplace noises). Following Comte et al. [2006], we considered the following four different densities:

(i) Mixed Gamma distribution: X = 1/ √ 5.48W with W ∼ 0.4Γ(5, 1) + 0.6Γ(13, 1) (ii) Cauchy distribution:

f (x) = (1/π)/(1 + x 2 ) (iii) Laplace distribution: f (x) = e - √ 2|x| / √ 2 (iv) Gaussian distribution: X ∼ N (0, 1)
Except the case of the Cauchy density, these densities are normalized with unit variance, thus allowing the ratio 1/σ 2 to represent the signal-to-noise ratio, denoted s2n. We considered signal to noise ratios of s2n = 5 and s2n = 10 in our simulations. To study the influence of the relationship between n and M , we considered several values of n and values of M = n and M = √ n. Additionally, we considered the density (v) defined by X ∼ 0.5N (-3, 1)+0.5N (2, 1) with a signal-to-noise ratio of 4 for comparison with Delaigle et al. [2008].

5.2.

Results. The values of the MISE risk multiplied by 100 for each density and simulation scenario and computed from 100 simulated data sets, are given in Table 1. As expected, the risk decreases as n or M increases. Similarly, when increasing the level of contamination of the data by reducing the signal-to-noise ratio, the risk increases. Compared to Gaussian noise, Laplace noise demonstrates overall lower risks whatever the other simulation parameters. Indeed, Gaussian noise is supersmooth whereas Laplace noise is ordinary smooth thus explaining the improvement in risk. Strikingly, in most cases the estimation of the square of the characteristic function of noise density f * ε has reduced the risk compared to the known density case. This phenomenon is counter-intuitive and we do not have a clear explanation. However, this has been noticed in Comte and Lacour [2011].

Table 2 presents the comparison of the penalized estimator and the estimator given by Delaigle et al. [2008] for the Gaussian mixture density (v). The integrated squared Table 1. Results of simulations presented as MISE×100. In each case the MISE was averaged over 100 estimations. The case "f ε known" corresponds to the estimator denoted f KN m with sample of size 2n (to be compared to the case M = n). error (ISE) is computed over 100 estimations and we present the results using the median and inter-quartile range (IQR). In all cases, the penalized estimator enjoys lower risks compared to those given by Delaigle et al. [2008].

In Figure 2, we present an estimation of f using the penalized estimator as in the simulated case (i) aforementioned. We considered the Gaussian mixture distribution (v) contaminated by Gaussian and Laplace noise with a signal-to-noise ratio of 4 with n = M = 200 and n = M = 500. The bimodal character of distribution (v) is well described by the estimation in both cases whereas the increase in precision for n = M = 500 is Table 2. Comparison of the ISE between the estimators of Delaigle et al. [2008] and the penalized estimator for the Gaussian mixture density (v). For the sake of comparison, the results are presented by the median × 100 (inter-quartile range × 100) of 100 estimations with M = n. mostly visible in the Laplace noise case which closely matches the theoretical density in that case.

Density estimation of onset of pregnancy

As defined previously, X denotes the interval between last menstrual period (LMP) and the true onset of pregnancy. We denote Y the interval between LMP and the onset of pregnancy estimated by the sonographic measurement of the crown-rump length (CRL) with Y = X +ε. Two separate independent samples are available: the first is an M -sample of spontaneous twin pregnancies, M = 86, each embryo with its own CRL measurement; the second is an n-sample of spontaneous singleton pregnancies, n = 1378, with Y j = X j + ε j . Each of these samples is a sample of the general unselected population and was obtained from the screening unit of the department of obstetrics and maternal-fetal medicine of the children's hospital Necker -Enfants Malades in Paris, France. Since the onset of pregnancy is identical for both twins, we thus have replicate noisy observations Y -j1 = X j + ε -j1 and Y -j2 = X -j + ε j 2 , -j = 1, . . . , M . We wish to estimate f which represents the distribution of probability of onset of pregnancy within a female cycle. As expected, the mode of the distribution is at around 13 days, meaning that the likelihood of onset of pregnancy is greatest at 13 days following the last menstrual period. However this distribution is positively skewed with a significant remaining probability of onset after 20 days. The risk was assessed by simulation in the setting of our data by considering X ∼ Γ(16, 1.2) and a Laplace ε ∼ Lap(0, 0.95) or Gaussian ε ∼ N (0, 1.2) noise. These densities were chosen empirically because they fitted our estimate (see Figure 3 (a) for the comparison of the empirical characteristic function of ε with Laplace and Gaussian characteristic functions). Under this simulation model, the risk MISE×100 was 0.038 and 0.034 for Laplace and Gaussian noise respectively over 100 estimations. We emphasize that the strong sideeffects which are observed on the estimated characteristic function in Figure 3 (a) can also be seen on simulated data (for a size sample 86) and mainly appear when going from the direct noise observation to the replicate case (where only differences of noise are observed).

Concluding remarks

We have presented an adaptive deconvolution estimator of a density when the noise density is unknown. Instead, a sample of noisy replicate observations is available. Although this estimator seems to perform nicely in simulation, it can exhibit poorer theoretical rates than in other settings. This expected loss is directly related to the use of replicate observations for the estimation of the characteristic function of noise density or more precisely the square of its module. Simulations show that the influence of the relative values of M and n is likely to be small. We also find that the gain in precision for increasing values of M may be small. Whereas this may be of little value in the field of engineering, it is of importance in biomedical applications or clinical research. Indeed, obtaining a sample of ε is often difficult or impossible in these applications, as well as a strong prior assumption regarding its density. However, replicate data may be found in clinical or biomedical applications. Nevertheless they are likely to be scarce since they involve multiple measurements/observations in one patient. In the case of dating pregnancy this is dealt with by using twin pregnancies instead. The estimation of a density of onset of pregnancy may find multiple clinical applications. The knowledge of the underlying variability of onset of pregnancy may help clinicians in the follow-up of pregnancies and mostly regarding growth monitoring by ultrasound and delivery since both these aspects rely upon an accurate estimation of onset of pregnancy. Furthermore, this density is of interest for the physiology of the female cycle, confirming with simple clinical data the variation in onset of pregnancy that could be expected from biological experiments (Wilcox et al. [2000]).

Acknowledgement

The authors thank Pr Jean-Christophe Thalabard for his advice, suggestions and support.

Appendix

V ar f * Y (t) = 1 (n + 2M ) 2   V ar   n j=1 e itY j   + V ar   M j=1 e itY -j1   + V ar   M j=1 e itY -j2   + 2cov   M j=1 e itY -j1 , M j=1 e itY -j2     = n + 4M (n + 2M ) 2 |f * ε (t)| 2 (1 -|f * X (t)| 2 ) The variance of f * Y (t) is equal to 1 n + M |f * ε (t)| 2 (1 -|f * X (t)| 2 ).
Easy calculations provide (n + 4M )/(n + 2M ) 2 ≥ 1/(n + M ), which end the proof of Remark 1.

.2. Proof of Lemma 1. The proof is given for p

= 1. Let us denote R(t) = 1 f * ε (t) - 1 f * ε (t)
.

Firt, write the decomposition

E(|R(t)| 2 ) = E 1 (f * ε ) 2 (t)<M -1/2 (f * ε ) 2 (t) + 1 (f * ε ) 2 (t) E   1 (f * ε ) 2 (t)≥M -1/2 ( (f * ε ) 2 (t) -(f * ε ) 2 (t)) 2 (f * ε ) 2 (t)( ( f * ε ) 2 (t) + f * ε (t)) 2  
On the one hand, we get

E(|R(t)| 2 ) ≤ 1 (f * ε ) 2 (t) + M (f * ε ) 2 (t) E[( (f * ε ) 2 (t) -(f * ε ) 2 (t)) 2 ] ≤ 2 (f * ε ) 2 (t) On the other hand, let us distinguish two cases. (i) If (f * ε ) 2 (t) ≤ 2M -1/2 , we have both 1/(f * ε ) 2 (t) ≤ 2M -1/2 /(f * ε ) 4 (t) and M -1/2 /(f * ε ) 4 (t) ≤ 2M -1 /(f * ε ) 6 (t) and E(|R(t)| 2 ) ≤ 1 (f * ε ) 2 (t) + M 1/2 (f * ε ) 4 (t) E[( (f * ε ) 2 (t) -(f * ε ) 2 (t)) 2 ] ≤ 1 (f * ε ) 2 (t) + M -1/2 (f * ε ) 4 (t) ≤ 3 M -1/2 (f * ε ) 4 (t) = 3( M -1/2 (f * ε ) 4 (t) ) ∧ ( M -1 (f * ε ) 6 (t) ) (ii) If (f * ε ) 2 > 2M -1/2
, using the Bernstein Inequality as in Neumann yields:

P(| (f * ε ) 2 (t)| < M -1/2 ) ≤ P(| (f * ε ) 2 (t) -(f * ε ) 2 (t)| > (f * ε ) 2 (t) -M -1/2 ) ≤ P(| (f * ε ) 2 (t) -(f * ε ) 2 (t)| > (f * ε ) 2 (t)/2) ≤ 2 exp(-M (f * ε ) 4 (t)/16) ≤ O M -1 (f * ε (t))
-4 p and with the same decomposition as above, this yields

E(|R(t)| 2 ) ≤ E 1 (f * ε ) 2 (t)<M -1/2 (f * ε ) 2 (t) + 1 (f * ε ) 4 (t) E   1 (f * ε ) 2 (t)≥M -1/2 ( (f * ε ) 2 (t) -(f * ε ) 2 (t)) 2 ( ( f * ε ) 2 (t) + f * ε (t)) 2   + 1 (f * ε ) 2 (t) E   1 (f * ε ) 2 (t)≥M -1/2 ( (f * ε ) 2 (t) -(f * ε ) 2 (t)) 2 ( ( f * ε ) 2 (t) + f * ε (t)) 2 1 f * ε ) 2 (t) - 1 (f * ε ) 2 (t)   ≤ 1 (f * ε ) 2 (t) P( (f * ε ) 2 (t) < M -1/2 ) + M -1 (f * ε ) 6 (t) + M -1/2 (f * ε ) 6 (t) E[( |f * ε ) 2 (t) -(f * ε ) 2 (t)| 3 ] ≤ M -1 (f * ε ) 6 (t) + M -1 (f * ε ) 6 (t) + M 1/2 (f * ε ) 6 (t) M -3/2 ≤ c M -1 (f * ε ) 6 (t) Thus, in that case where M -1/2 /(f * ε ) 4 (t) ≥ 2M -1 /(f * ε ) 6 (t), we get E(|R(t)| 2 ) ≤ M -1 |f * ε (t)| 6
. This ends the proof of the lemma.

.3. Proof of Proposition 1. Let us study the integrated mean square risk. By writing in the Fourier domain that

f * -f * m = (f * -f * m ) + (f * m -f * m ) = f * 1 [-πm,πm] c + (f * m -f * m )1 [-πm,πm] , we get, as f -f m 2 = (2π) -1 f * -f * m 2 = (2π) -1 ( f * 1 [-πm,πm] c 2 + (f * m -f * m )1 [-πm,πm] 2 ), that (17) f -fm 2 = f -f m 2 + f m -fm 2 .
Moreover, by applying the Parseval formula, we obtain

f m -fm 2 = 1 2π πm -πm f * Y (u) f * ε (u) - f * Y (u) f * ε (u) 2 du. It follows that (18) f m -fm 2 ≤ 1 π πm -πm | f * Y (u)| 2 |R(u)| 2 du + 1 π πm -πm | f * Y (u) -f * Y (u)| 2 |f * ε (u)| 2 du.
The last term of the right-hand-side of ( 18) is the usual term that is found when f * ε is known, and the first one is specific to the framework with estimated f * ε . We take the expectation of ( 18):

E( f m -fm 2 ) ≤ 2 π πm -πm E(| f * Y (u) -f * Y (u)| 2 |R(u)| 2 )du + 2 π πm -πm |f * Y (u)| 2 E(|R(u)| 2 )du + 1 π πm -πm (n + M ) -1 |f * ε (u)| 2 du. Applying Lemma 1 yields: E( f m -fm 2 ) ≤ 2 π πm -πm (E(| f * Y (u) -f * Y (u)| 4 )E(|R(u)| 4 )) 1/2 du + 2 π πm -πm |f * (u)| 2 |f * ε (u)| 2 E(|R(u)| 2 )du + 2 ∆ 2 (m) n + M ≤ 2 π πm -πm (n + M ) -1 1 |f * ε (u)| 2 du + 2C 1 π πm -πm |f * (u)| 2 |f * ε (u)| 2 M -1/2 |f * ε (u)| 4 ∧ M -1 |f * ε (u)| 6 du + 2 ∆ 2 (m) n + M (19)
By gathering ( 17) and ( 19), we obtain the result.

Figure 1 .

 1 Figure 1. Empirical behavior of Q1 (m), Q2 (m) and Q3 (m) as a function of m when M n and M n for f ordinary smooth and supersmooth, chosen as Laplace (left figures) and Gaussian (right figures) respectively. The influence of M and n is illustrated by M = 200, n = 2000 (top figures) and M = 2000, n = 200 (bottom figures).

  This expression of the estimator allows us to use Inverse Fast Fourier Transform (IFFT) Algorithms in the estimation process. Therefore, for numerical tractability we use only a finite sample of projection coefficients with fm = | |≤Kn âm, ϕ m, . Theoretical results assert that K n = n always suits (seeComte et al. [2006]) but we make the constant choice K n = 255.Since M n is unknown we consider an estimation of this domain,M n = {k/κ, k = 1, . . . , mn κ} by defining an integer mn such that mn = argmax m ∈ N,

f

  Mixed Gamma f ε known 0

Figure 2 .

 2 Figure 2. Estimations for n = M = 200 (dashed line) and n = M = 500 (dotted line) for the Gaussian mixture density (plain line) (v). (left): Laplace noise; (right): Gaussian noise. Two independent samples were used, of size n and M respectively.

  (left): estimation of the characteristic function of the noise in twin pregnancies (plain line). Laplace (dashed line) and Gaussian (dotted line) characteristic functions are plotted for comparison. (right): estimation of the density of onset of pregnancy.

Figure 3

 3 Figure 3 (b) presents the penalized estimator f m(.).As expected, the mode of the distribution is at around 13 days, meaning that the likelihood of onset of pregnancy is greatest at 13 days following the last menstrual period. However this distribution is positively skewed with a significant remaining probability of onset after 20 days. The risk was assessed by simulation in the setting of our data by considering X ∼ Γ(16, 1.2) and a Laplace ε ∼ Lap(0, 0.95) or Gaussian ε ∼ N (0, 1.2) noise. These densities were chosen empirically because they fitted our estimate (see Figure3(a) for the comparison of the empirical characteristic function of ε with Laplace and Gaussian characteristic functions). Under this simulation model, the risk MISE×100 was 0.038 and 0.034 for Laplace and

  : Proofs .1. Proof of Remark 1. Let us denote the estimator of f * Y based on the samples (Y j ), (Y -j1 ) and (Y -j2 ): dependency between (Y -j1 ) and (Y -j2 ), we have

C

  Butucea. Deconvolution of supersmooth densities with smooth noise. Canadian Journal of Statistics, 32(2):181-192, 2004. C Butucea and A B Tsybakov. Sharp optimality in density deconvolution with dominating bias. i. Theory of Probability and its Applications, 52(1):24, 2008a. C Butucea and A B Tsybakov. Sharp optimality in density deconvolution with dominating bias. II. Theory of Probability and its Applications, 52(2):237, 2008b. F. Comte and C. Lacour. Data-driven density estimation in the presence of additive noise with unknown distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4):601-627, September 2011. F Comte, Y Rozenholc, and M L Taupin. Penalized contrast estimator for adaptive density deconvolution. Canadian Journal of Statistics, 34(3):431-452, 2006. A Delaigle and I Gijbels. Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Annals of the Institute of Statistical Mathematics, 56(1):19-47, 2004. A Delaigle, P Hall, and A Meister. On deconvolution with repeated measurements. The Annals of Statistics, 36(2):665-685, 2008. P J Diggle and P Hall. A fourier approach to nonparametric deconvolution of a density estimate. Journal of the Royal Statistical Society. Series B (Methodological), 55(2): 523-531, January 1993. J Fan. On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics, 19(3):1257-1272, September 1991. J Fan and J Y Koo. Wavelet deconvolution. Information Theory, IEEE Transactions on, 48(3):734-747, 2002. C H Hesse. Data-driven deconvolution. Journal of Nonparametric Statistics, 10(4):343-373, 1999. J Johannes. Deconvolution with unknown error distribution. The Annals of Statistics, 37 (5):2301-2323, October 2009. T Li and Q Vuong. Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis, 65:139165, 1998. M C Liu and R L Taylor. A consistent nonparametric density estimator for the deconvolution problem. Canadian Journal of Statistics, 17(4):427-438, 1989. A. Meister and M. Neumann. Deconvolution from non-standard error densities under replicated measurements. Statistica Sinica, 20 (4):1609-1936, 2009. M H Neumann. On the effect of estimating the error density in nonparametric deconvolution. Journal of Nonparametric Statistics, 7(4):307, 1997. M Pensky and B Vidakovic. Adaptive wavelet estimator for nonparametric density deconvolution. Annals of Statistics, pages 2033-2053, 1999. H P Robinson. Sonar measurement of fetal crown-rump length as means of assessing maturity in first trimester of pregnancy. British Medical Journal, 4(5883):28-31, October 1973. P Sladkevicius, S Saltvedt, H Almstrm, M Kublickas, C Grunewald, and L Valentin. Ultrasound dating at 12-14 weeks of gestation. a prospective cross-validation of established dating formulae in in-vitro fertilized pregnancies. Ultrasound in Obstetrics and Gynecology: The Official Journal of the International Society of Ultrasound in Obstetrics and Gynecology, 26(5):504-511, 2005. L A Stefanski and R J Carroll. Deconvoluting kernel density estimators. Statistics, pages 169-184, 1990. A J Wilcox, D Dunson, and D D Baird. The timing of the "fertile window" in the menstrual cycle: day specific estimates from a prospective study. BMJ (Clinical Research Ed.), 321(7271):1259-1262, November 2000. MAP5, UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints Pères, 75006 Paris, France

En général, la date précise de début de grossesse est inconnue et est seulement estimée à partir de mesures de l'embryon par ultrasons. Nous voulons estimer la densité de la variable aléatoire correspondant à l'intervalle entre la date des dernières règles et le début réel de la grossesse. Les observations dont nous disposons correspondent aux variables d'intérêt mesurées avec un bruit additif. Nous proposons une procédure d'estimation basée sur des techniques de déconvolution. Cela requiert la connaissance de la loi du bruit mais elle n'est pas disponible. En revanche, nous disposons d'un échantillon d'observations répétées obtenues à partir de grossesses gémellaires. Cela nous permet à la fois d'estimer la loi du bruit et d'améliorer la procédure de déconvolution. La vitesse de convergence de l'estimateur résultant est étudiée et comparée à celles obtenues dans d'autres contextes. Notre estimateur contient un paramètre de troncature pour lequel nous proposons une sélection par validation croisée. Pour finir, nous étudions notre estimateur en simulation puis nous l'appliquons pour estimer la densité visée sur notre échantillon de grossesses spontanées avec estimation du bruit via des données répétées obtenues sur un échantillon de grossesses gémellaires.


Barbara Hendrischke (trad.), The Scripture on Great Peace. The Taiping jing and the Beginnings of Daoism

unconditionally using his edition is more and more questionable today, especially since the publication of Yu Liming's 兪唾汗 Taiping jing zhengdu ′l眠い矣 (Chengdu: Ba Shu shushe, 2001), a critical edition which solves most of the linguistic problems, even though it is not free from flaws.

After years of familiarity with the text, Hendrischke nonetheless offers an occasionally ambiguous vision of the basic material. 2 She writes, so as to justify her dropping the chapter numbering, that "each section typically deals with a single topic, and the sections assembled in one chapter often have little in common" (p. IX), then immediately adds, as a justification for her retaining the section numbering, that "the Taiping jing is a long scripture, and yet I have observed that its internal logic is remarkable" (p. X). To notice that each textual unit in the Taiping jing deals with "a single topic" somewhat misses the point, for only the shortest sections resist the temptation to digress (for mere lack of space). Quite often, the "topic"-either imposed by the master or stemming from a disciple's question, as Hendrischke explains at the end of her book (p. 348)-hardly constitutes more than a starting point and, as the dialogue develops, the speaker and his audience are lead to tackle a rather broad range of themes. (And the Taiping jing contains parts which are not dialogues.) The phenomenon called by our translator "random change of topic" (p. 45) is not unfamiliar to the seasoned Taiping jing reader. If sections seemingly unrelated sometimes occupy the same chapter, the contrary is also frequent; see, for example, sections 44-45 (chapter 36); 47-48 (chapter 37); 52-54 (chapter 40); 56-58 (chapter 42)… The extant canonical text results from a literary history pervaded by uncertainty but known to have been long and eventful; its literary structure is far more complex than what its apparently random or convenient organization may suggest at first sight, and the section numbering is no more and no less "irrelevant" than the chapter numbering. long introduction (pp. 1-66) preceding the translation is not intended for the specialist (see p. 54, n. 5) but rather for the student in Chinese studies or the lay reader. Divided into nine unequal paragraphs, it is a complete but rather conventional synthesis of what has already been published on the topic in several languages and by a great number of scholars. It is to her credit that Hendrischke straightaway casts doubt on the abusive label "Taoist," which has been attached to the Taiping jing since it was incorporated into the Taoist Depository centuries after the emergence of the Great Peace tradition (p. 3). The first paragraph (THE NOTION OF GREAT PEACE, pp. 4-13) situates the theme of "Great Peace" (Taiping) in the intellectual and literary context of China from the third century to the end of the Later Han 掲亢 dynasty (25-220), reminding us that the theme was originally not specific to any given school or current (p. 5). The second paragraph (THE PEACE THAT WILL SAVE THE WORLD, pp. 13-6) emphasizes the peculiar value given the notion in the Taiping jing, in contradistinction to contemporary usage. According to the translator, the ideology of the text challenged the very permanence of Han rule, and heralded the messianic movements which were to characterize the period of political division (3rd. to 6th. cent.) (p. 16). We may add, 2 In this regard, it will not be unnecessary to specify that the canonical text is divided into 57 "chapters" (juan 蝦), numbered from 35 to 119, which are subdivided into 129 "sections" (mostly called fa 基 or jue 窒), numbered from 41 to 213. (Here I adopt Hendrischke's translations to designate the textual units, in order to make references to the primary source easier.) This "master text" is incomplete. It is preceded by the Taiping jing chao ′l眠毘 (a sort of digest of the master text, in 10 chapters, numbered from 1 to 10 but not subdivided into sections) and followed by the Taiping jing fuwen xu ′l眠孺◇伊 (a 2-page preface) and the Taiping jing shengjun bizhi ′l眠命 м祝ポ (a 7-page collection of stanzas focused on a single theme). These four texts are anonymous and undated. nevertheless, that the views of the Taiping jing did absolutely not call into question the monarchical régime, which was seen as the ideal sociopolitical model because of its conformity to universal order; this is why the Taiping jing cannot be regarded as a "revolutionary" text, as Hendrischke rightly writes further on (p. 40).

The next two paragraphs (THE TAIPING MOVEMENT, pp. 16-24; THE MOVEMENT OF THE CELESTIAL MASTERS, pp. 24-30) sum up all the available data concerning the two major sociopolitical movements of the end of the Han era. The single flaw of these pages is the total assimilation of the A.D. 184 movement as the Yellow Turbans rebellion to a "Taiping (or Great Peace) movement". This assimilation might induce the unprepared reader to believe there is a clear historical connection, scientifically established, between this so-called "Great Peace movement" and the Scripture of Great Peace of which he or she is about to read translated parts. In fact, the expression taiping dao ′l洋 appears only once in the dynastic histories, in a quotation of a source written less than a century after the events took place but today lost. 3 This locus classicus happens to be our unique source concerning the religious activities of the group-which, interestingly, is not referred to using the usual pejorative nickname of official sources ("yellow turban bandits") in this quotation. On the other hand, Taoist sources never associate the phrase "taiping dao" (or taiping zhi \ dao) with the name of Zhang Jue 積援, the historical leader of the A.D. 184 insurgents. Moreover, linking Zhang Jue's name to a Great Peace text was first done by Fan Ye 購恠 (398-445) in his Hou Han shu, a work completed in 445, i.e. more than two and a half centuries after the rebellion; Fan Ye mentions a Taiping qingling shu ′l藻叮七, not a "Taiping jing". Even if we re-examine every facet of the problem, it is bound to remain insoluble unless, for instance, a first-hand Great Peace text dating back to the Han is discovered in archaeological context. Rather than constantly writing virtually the same things all over again, we should bear in mind that most of the relevant primary sources were produced several centuries after the events they report took place, and question the historicity of what has come to be accepted as established facts. Only through strict historical criticism will simplification and over-interpretation be kept at bay.

this respect, reading the fifth paragraph (THE TAIPING MISSIONARY PROJECT, pp. 30-1) will prove useful to any reader who failed to distinguish between Yellow Turbans, Celestial (or Heavenly) Masters ♀施 (the early Taoist Church), and the promoters of Great Peace. But is it legitimate to translate the word zui 椋 as "crime" in Celestial Master context and "sin" in the Taiping jing, and then use this difference to claim for Zhang Lu's rule a "theocratic" nature? (See also, p. 26, about Zhang's state in Hanzhong: "China's first, and for a long time its only, theocracy.") Shouldn't we rather emphasize the fact that the Chinese language resorts to a single term to cover both our Western concepts, including in current usage (zuiren 椋 means "culprit" as well as "sinner")? For this suggests, with other pieces evidence whose discussion would be inappropriate here, that it is the very nature of political power which is intrinsically religious in China; hence Zhang Lu's rule was no more and no less "theocratic" than the rule of any "Son of Heaven".

The sixth paragraph (HISTORICAL STAGES OF A SCRIPTURE ON GREAT PEACE, pp. 31-8) recapitulates the literary history of the texts named after Great Peace, and tries to relate them as much as possible to the ever-changing political context of the relevant period. As other specialists of the Taiping jing, Hendrischke relies on late, isolated occurrences to justify the usually retrospective identifications conveniently offered to fill in disturbing gaps. The following paragraph (THE ORIGIN OF THE TPJ, pp. 38-43) leaves these fragments of an uncertain literary history and turns to the text itself. In spite of her conviction that "the jing] as we have it today goes back to the sixth century," Hendrischke goes back to the Han era-the social and mental background against which the ideology of the text seems to have been formed, even if she admits that a thorough examination of the text enables one to restore but a "vague" picture of this cultural environment (p. 41).

The eighth paragraph (LANGUAGE AND STYLE, pp. 43-7) brilliantly shows the peculiarity of the text within Chinese literature and discusses the array of characteristics suggesting a vernacular origin: the occurrence of numerous disyllabic and trisyllabic compounds; the specific usage of particles; the length of the sentences; the verbosity and redundancy of expression. Contrary to what we are told (p. 44), however, the Taiping jing marks the plural when necessary, by prefixing the character zhu 實, for instance in the expression zhushen 實出, "the deities" or "the gods" (36 occurrences in Wang Ming's edition). Concerning the issue of composition, Hendrischke's views have changed. Although she used to be convinced of the rhetorical function (and, therefore, artificiality) of the dialogue style, she seems now inclined to accept the existence of a sort of proto-material made out of raw notes taken by an audience of pupils or disciples. Even though Hendrischke makes a rather good case for her theory (see pp. 89-90, n. 38, for relevant textual references), it is hardly compatible with the most basic principles of methodology. Great caution is required whenever, basing ourselves on style, we construct an interpretative model supposed to extend beyond the boundaries of literary form. The fact that the Taiping jing comes from a milieu of which almost nothing is known allows one to believe this milieu to have been radically different from the rest of the contemporary social fabric, but absolutely does not offer any proof that the main part of the extant text underwent only the slightest editorial alteration during the centuries separating the production of the earliest Great Peace writings from the integration of a Taiping jing into the Ming Taoist Canon. And, needless to say, the dialogue form as a literary style appeared in China before the formation of the scriptural tradition of Great Peace.

The last paragraph (THE SCRIPTURE'S MESSAGE OF SALVATION, pp. 47-54) presents a thematic selection from the "program" of the authors of the Taiping jing, particularly as regards their social views. One wonders if this development was really necessary, given that, first, it does not present anything new, and second, the translation of each section is preceded by a short introductory paragraph with the same purpose (see below).

Hendrischke's major achievement in this book is the superb English translation, without question a scholarly tour de force (pp. 67-342). She had earlier contributed the translation of two units from the Taiping jing to an anthology recently published (see Victor H. Mair, Nancy S. Steinhardt, and Paul R. Goldin [eds.], Hawai'i Reader in Traditional Chinese Culture, Honolulu: University of Hawai'i Press, 2005, pp. 225-30); this remarkable preliminary work is naturally included in the present volume (pp. Hendrischke chose to translate the first 25 sections of the text in reading sequence, from section 41 (chapter 35) to 66 (chapter 49), thus encompassing the first 14 chapters (out of 57) of the canonical text. Though her translation only covers some 20% of the canonical text (15% if the Taiping jing chao is to be included as relevant material), a huge step has been made towards a full English translation.

Each translated section opens with a paragraph introducing its topic and summarizing its content. Then the translation unfolds, followed by dense endnotes. Setting the dialogue alternately in normal style (the master's part) and in italics (the disciple's part) proves to be the right editorial choice, while the accuracy of the translation helps recreate the seeming spontaneity of the verbal exchanges: as a result, the dialogue comes to life. Regretfully, a warning in the opening pages of the book (p. 4) about the length of the notes-a caution seldom found in scholarly publications-now takes on its full meaning: the endnotes, numerous and often long, somewhat interfere with the reading. Admittedly a few lines are essential to explain any concept peculiar to the text or clearly point out any textual emendation; however, digressions filling several paragraphs or pages will end up confusing rather than enlightening the reader. Keeping to the 53 notes of the first translated chapter, such may be said of notes 9 (on the figure twelve thousand, pp. 22 (on the concept of central harmony, pp. 82-3), 29 (on the division of the world into 81 territories, pp. 86-8), and 47 (on infanticide, pp. 92-3); as for the last note (p. 94), which offers a translation of the section's end title, its content should simply be moved back to its corresponding location in the Chinese text, i.e. after the last line of the section. (On the problem of these "end titles," see also below.) may object to the English title "Scripture on Great Peace" (discussed by Hendrischke, pp. that it seems to imply that Great Peace is restricted to the thematic content of the scripture, whereas Great Peace actually is a cosmic revelation of which the is an integral part; hence "Scripture of Great Peace," or simply "Great Peace Scripture" which follows the original syntax, are both preferable renderings. Still concerning translation details, one may wonder why shen 出 is systematically rendered as "spirit" throughout the book, with awkward results like "spiritlike man" for shenren 出 (p. 207, etc.); while the "Controller of Fate" (siming L廻), also a shen, and also present in the human body, has earned the right to be called a "deity" (p. 90, n. 39). Finally, as regards the phrase jianling 熊∝ from the Dianlüe (quoted in Pei Songzhi's commentary to the Sanguo zhi), the translation "to control evil" (p. 27) seems grammatically incorrect; it would appear that the phrase refers to the illegitimacy of the concerned officials (according to Terry Kleeman, "Community and Daily Life in the Early Daoist Church," communication to the international conference "Rituals, pantheons and techniques: A history of Chinese religion before the Tang," Paris, December [14][15][16][17][18][19][20][21]2006).

The bibliography (pp. 373-91) and index (pp. are preceded by an appendix (THE COMPOSITION OF THE TPJ, pp. 343-72), in which Hendrischke addresses specialists. This closing essay was perhaps initially intended to introduce the translation, because it returns to most of the issues already dealt with in the opening part of the book-the peculiarity of the language (p. 346); the lack of internal organization of the canonical text, the fact that each section deals with a single topic "or at least with interconnected topics" (the admission would be welcome at the beginning of the book), and the reason for retaining only the section numbering (p. the dialogue form and its alleged origin in genuine notes taken during sessions of religious instruction (pp. 348-9), etc. Despite these duplications, this essay on the textual history and literary structure of the Taiping jing raises the right questions and offers well-documented and often convincing elements of answers.

Why does this partly redundant essay appear at the end of the book? Integrating it into the introduction and deleting all repetitions would have profitably reduced the critical apparatus-which is already overloaded with 961 endnotes filling one third (140) of the total pages-and allowed several extra sections to be included in the translation, which merely takes up 30% of the book pages.

Slightly more "technical" than the introduction, this appendix also lays itself more open to criticism. Significantly, Hendrischke's caution now gives birth to a number of sequential hypotheses, embodied at the discursive level in a recurrent conjectural vocabulary ("may assume," "must assume," "must expect," "seem," etc.). This tangle of hypotheses allows little room for historically established facts. An ingenuous sentence perfectly expresses the methodological problems faced by any modern approach to the text: "The external appearance of the [Taiping jing] conforms to what we can conjecture about its origin" (p. 346). Given that those conjectures are mainly based on the current appearance of the text, it is certainly no wonder that the extant text seems to "conform" to them. Wouldn't it rather be that the "external appearance" of the text dictates how we reflect upon its origin?

Hendrischke rightly draws our attention to the titles located at the end of some sections and chapters, she interprets as "one-line summaries". Generally ignored by scholars and dealt with in less than one page here (pp. 345-6), this material indeed requires a full study, which would allow us first to check if these end titles were as badly transmitted as are told, to the extent that they are now "often corrupt beyond any hope of emendation" (p. 345). Is textual corruption not always conveniently adduced whenever reading problems fail to be solved? Perhaps such a study would also allow us to verify if the initial titles may be safely attributed en bloc to Taoist editors of the sixth century, and the end titles (the so-called "summaries") to a different and necessarily earlier editorial intervention.

Equally delicate is the issue of the "textual layers" (or strata) first defined on a stylistic basis by Xiong Deji 佶嗤杉 (1913-87) in a foundational article ("Taiping jing de zuozhe he sixiang ji qi yu Huangjin he Tianshidao de guanxi" ′l眠疑Ж却会景併∞寡冉 鰭ヾ会♀施洋疑牴悦, Lishi yanjiu 恍Q巧駅, 4, 1962, pp. 8-25; on the limits of a strictly definition of these textual layers, see my arguments at the beginning of my paper in Asia Major, third series, 15, 2, 2002, pp. 1-5). Hendrischke makes the point that the stylistic variations are very often coupled with thematic differences (p. 348) then, once again, maintains that the dialogic material (her layer A) stems from original "note taking" (p. 349). But, when two extra layers (A' and B') are added to the basic three layers (A, B, and C), one begins to wonder if the concept of textual strata as a heuristic tool is really effective. Would a notion of thematic field not be more relevant? It is no accident that Hendrischke later on adopts a thematic approach to defend the validity of her division into strata, finally casting on her own method and confessing the comparative obscurity of her exposé (pp. 3). Barbara Hendrischke's superb translation of the first chapters of the Taiping jing should quickly become a must-read for every student and scholar involved in the history of early imperial China, and demanding readers will soon the book to take the place it deserves among the basic reference works of their academic institution or library. Even if it is not labeled "volume 1," The Scripture on Great Peace looks like the first of a series whose subsequent volumes should henceforth be eagerly awaited. It is hoped for that the talented translator will take up the challenge of completing the first full Western translation of the Taiping jing ever and ensure a publishing follow-up to this praiseworthy work.

Grégoire Espesset Kyoto University

The Dianlüe 科陀 (c270) by Yu Huan 津猶, cited by Pei Songzhi 刧間\ (372-451) in his commentary(completed

429) to the Sanguo zhi 髄易; taken up in the commentary written by Li Xian 姻屎 (651-84) and his collaborators to the Hou Han shu 掲亢七 between 675 (date of Li's accession to the status of Heir Apparent) and 680 (date of Li's imprisonment). Famously, Li Xian's commentary mentions and quotes the Taiping jing four times.