LIMIT DISTRIBUTIONS OF SOME STEREOLOGICAL ESTIMATORS IN WICKSELL ’ S CORPUSCLE

Suppose that a homogeneous system of spherical particles (d-spheres) with independent identically distributed radii is contained in some opaque d-dimensional body, and one is interested to estimate the common radius distribution. The only information one can get is by making a cross-section of that body with an s-flat (1 ≤ s ≤ d − 1) and measuring the radii of the s-spheres and their midpoints. The analytical solution of (the hyper-stereological version of) Wicksell’s corpuscle problem is used to construct an empirical radius distribution of the d-spheres. In this paper we study the asymptotic behaviour of this empirical radius distribution for s = d − 1 and s = d − 2 under the assumption that the s-dimensional intersection volume becomes unboundedly large and the point process of the midpoints of the d-spheres is Brillinger-mixing. Of course, in stereological practice the only relevant cases are d = 3 , s = 2 or s = 1 and d = 2 , s = 1. Among others we generalize and extend some results obtained in Franklin (1981) and Groeneboom and Jongbloed (1995) under the Poisson assumption for the special case d = 3 , s = 2.


INTRODUCTION
In 1925 the Swedish statistician Sven D. Wicksell (1890Wicksell ( -1939) ) studied the following problem which belongs meanwhile to the classical toolbox of stereologists.Suppose that a system of threedimensional random spheres {B 3 (X i , R i ) : i ≥ 1} with midpoints {X i : i ≥ 1} forming a homogeneous point field in R 3 , and with identically distributed radii {R i : i ≥ 1} having an unkown common distribution function (briefly df) F 3 (r) , r ≥ 0 , is embedded in an opaque medium.Since the medium is opaque, one cannot observe the sphere radii directly.What can be observed is a bounded part of a planar crosssection through the medium, showing circular sections of some spheres.It has been shown in Wicksell (1925) that the observable circle radii have a common probability density function f 2 (r) , r ≥ 0 , depending on F 3 as follows: There exists a vast and widely scattered literature dealing with the numerical and statistical inversion of Eq. 1, i.e., the approximative determination of the df F 3 .Actually the solution of this problem is the essential point in numerous applications of Wicksell's corpuslce problem in various fields such as material science, biology, medicine and so on (see, e.g., Stoyan et al., 1995;Ohser and Mücklich, 2000 and references therein).
In Mecke and Stoyan (1980) the reader can find a rigorous derivation of Eq. 1 based on the assumption that the sequence of pairs {[X i , R i ] : i ≥ 1} constitutes a stationary marked point process in R 3 .The given proof extends straightforwardly to higher dimensions.For our purposes we presuppose in addition that the radii of distinct spheres are independent of each other and that they are also independent of the locations of the sphere centres.For the sake of generality we consider Wicksell's corpuslce problem in the d-dimensional Euclidean space R d equipped with the Euclidean norm • d and the corresponding Borel σ −field B(R d ).Stoyan et al., 1995 for details).In order to impose an appropriate (mixing) condition on Ψ * d we need the kth-order cumulant measures γ k (•) defined on the Borel σ −field B(R dk ) of any k ≥ 2 (see, e.g., Heinrich and Schmidt, 1985 for a precise definition).The stationarity of Ψ * d enables us to define an associated (signed) measure-the reduced HEINRICH L: Limit distributions of stereological estimators kth-order cumulant measure-γ To facilitate the interpretation of the below Conditions 1-3 we point out that, for disjoint bounded Wicksell's corpuscle problem in its hyperstereological version can be described as follows: The system of d-spheres which can be identified with R s ).We assume that the collection of non-empty s-spheres the linear subspace H s can be observed (all radii and midpoints are visible, without considering overlappings and edge-effects) in an expanding sampling window W (s) n := nW (s) , where W (s) is a fixed convex set in R s with unit s-volume, i.e., ν s (W (s) ) = 1 , and n runs through Here and in what follows, write x (resp.x) to indicate the projection of x ∈ R d onto H s ) (resp.onto the orthogonal complement of H s which can be identified with R d−s ).The system of non-empty s-spheres B s (X i , R i ) is completely described by the stationary marked point process , where F s denotes the df of the 'typical radius' R 0 .
In the next section we restate the well-known explicit expressions of the df F s and the intensity λ s in terms of F d and λ d together with the corresponding inversion formulae.After that we present our results on the asymptotic behaviour (as n → ∞) of appropriate empirical counterparts of the radius df F d which are obtained from a single observation of all sspheres whose midpoints lie in W (s) n .In particular, we state asymptotic normality (Theorem 1) and weak consistency (Theorem 4) in the cases s = d − 1 and s = d − 2, respectively.Using the terminology developed for limit theorems for sums of independent random variables we are in the situation of a nonnormal domain of attraction of the Gaussian resp.degenerate law (see Ibragimov and Linnik, 1971).In other words, we are faced with (weakly dependent) random variables having infinite variance resp.infinite expectation, but nevertheless, after suitable centering and overnorming, their sums satisfy a central limit theorem resp.a weak law of large numbers.By The Poisson framework, as presupposed in Franklin (1981), Groeneboom and Jongbloed (1995) and Golubev and Levit (1998), is replaced in the present paper by the assumption that the point process Ψ * d is Brillinger-mixing.This special mixing condition implies that numbers of points of Ψ * d = {X i : i ≥ 1} in distant regions become asymptotically uncorrelated.For a precise formulation of this condition the existence of all higher-order moment measures is needed.It should be mentioned that under milder moment assumptions similar asymptotic results can be obtained for absolutely regular point processes Ψ * d (Heinrich, 1994), as well as for Poisson cluster processes Ψ * d (Heinrich, 1988).However, it seems that the Poisson assumption can hardly be dropped in our Theorems 5 and 6 to derive α-stable limits (with α = (d − s)/2) for the fluctuation of the corresponding empirical df's of F d when d − s ≥ 2 .In the final section we put together the essential steps of the proofs of our results.All details of the proofs can be found in the paper http://www.math.uniaugsburg.de/stochastik/heinrich/papers/asymwick.pdf.

RELATIONSHIPS BETWEEN THE RADIUS DF'S
By means of Campbell's theorem (see, e.g., Stoyan et al., 1995) and the relation Letting r → 0, the previous formula yields and probability density function f s of R 0 (which always exists!) takes the form Here and throughout the integral ∞ r stretches over interval (r, ∞).To express the radius df F d in terms of the radius df F s for any s ∈ {1, . . ., d − 1} one has to solve the above Abel-type integral equation by unfolding.For doing this we distinguish between the cases d − s is even and d − s is odd, respectively.Put q = (d − s − 1)/2 , where x denotes the largest integer smaller than or equal to x and n!! = n(n − 2) • ...•4•2 or 3•1.Then the df F d can be expressed in terms of the probability density function f s in the following way: where in the last line q derivatives occur.
However, the statistical solution of the integral equation Eq. 2 leads to an inverse estimation problem which is rather unstable from both the computational and statistical view point (see Watson, 1971;Franklin, 1981;Van Es and Hoogendoorn, 1990;Groeneboom and Jongbloed, 1995;Stoyan et al., 1995;Mair et al., 2000 for further details).
In the most important case s = d − 1 it is rapidly verified by a straightforward application of Campbell's theorem (see, e.g., Stoyan et al., 1995) that for any r ≥ 0 and all n ∈ N .On the other hand, the same calculation reveals that the variance of U n (r) is infinite (which has been first noticed in Franklin, 1981).
We refer to the fact that, for fixed n ∈ N , the empirical process U n (r) regarded as random function of the argument r ≥ 0 is by no means monotonically decreasing.It possesses downward jumps at the random points r = R i , however, between two such jumps U n (r) is strictly increasing.Such strange behaviour of this stereological estimator of λ d (1 − F d (r)) gave rise to consider several modified and smoothed versions of U n (r) (see, e.g., Groeneboom and Jongbloed, 1995 for an isotonic estimation and its asymptotic analysis).

THE CASE s = d -1
We first put together some mixing-type conditions for the point process Condition 2 Assume that the reduced second-order cumulant measure γ for any bounded Borel set A ⊂ R 1 .
Condition 3 Assume that the reduced second-order cumulant measure γ (red) 2 (•) has finite total variation, i.e., A Poisson point process Ψ * d trivially satisfies the Conditions 1-3 since all of its higher-order cumulant measures disappear.Sufficient conditions for some other classes of point processes to be Brillingermixing are discussed in Heinrich (1988).For example, Poisson cluster processes are Brillinger-mixing iff the number of points in the typical cluster has moments of any order.Also, several types of dependently thinned Poisson processes such as Matérn's hard (and soft)core point processes possess this mixing property.In Ruelle (1988), conditions are derived that imply Brillinger-mixing of Gibbsian point processes with pair interactions.If Ψ * d is additionally isotropic with pair correlation function g(r) (see Stoyan et al., 1995), then Condition 2 is satisfied if This as well as Condition 3 are rather mild restrictions on the process Ψ * d .
Theorem 1 Let the Conditions 1 and 2 be satisfied.If for some fixed r ≥ 0 and ER 0 < ∞, then where N(0, σ 2 ) denotes a zero mean Gaussian random variable with variance σ 2 .For r > 0 , condition Eq. 3 is equivalent to f d−1 (r) < ∞ , where the probability density function f d−1 is given by Eq. 2 for s = d − 1 .
Remark 1 Provided that F d (0) = 0, Theorem 1 (for r = 0) yields a central limit theorem for the unbiased estimator U n (0) of the intensity λ d .
Note that without assuming Brillinger-mixingmerely under Condition 3 -U n (r) turns out to be weakly consistent (as n → ∞) for λ d (1 − F d (r)).Hence, we get that It should be noted that, in case Ψ * d is a stationary ergodic point process, the latter relation holds P−a.s.. Remark 2 For r > 0 , the assumption Eq. 3 is satisfied if the df F d is α-Hölder continuous for some α > 1/2 in [r, r + δ ], i.e., there exists a positive number H α,δ depending on α and δ such that for r ≤ ρ ≤ r + δ and some δ > 0 (see Golubev and Levit, 1998 for an analogous smoothness condition).
The multivariate extension of Theorem 1 (by employing the well-known method of Cramér-Wold) shows that the finite-dimensional distributions of the sequence of standardized empirical processes in Theorem 1 tend to those of a Gaussian 'white noise' process as n → ∞.
Theorem 2 Let the Conditions 1 and 2 and Eq. 3 for r ∈ {r 1 , . . ., where N k o, I k denotes a k-dimensional Gaussian random vector having zero mean components and a covariance matrix being equal to the unit matrix I k .
As an immediate consequence of Eq. 2 for s = d − 1 , the ratios 2 r σ 2 n (r)/( λ d−1 ) n are weakly consistent estimators of f d−1 (r) for each r ≥ 0 satisfying Eq. 3, where ) is an unbiased and weakly consistent estimator of λ d−1 = 2 λ d ER 0 (see Van Es and Hoogendoorn, 1990 or Golubev and Levit, 1998, for alternative kernel-type estimators of f d−1 (r) ).
Combining Theorem 1 with Theorem 3 together with Slutski's theorem provides Corollary 2 Let the Conditions 1 and 2, ER 0 < ∞ and Eq. 3 for some fixed r ≥ 0 be satisfied.Then Remark 3 By means of Corollary 2 ( applied to r = 0 provided F d (0) = 0) we are able to construct an asymptotically exact confidence interval for the unknown intensity λ d of the midpoints of d-spheres.
A further immediate consequence of Theorem 2 and Slutski's theorem is Corollary 4 Let the assumptions of Theorem 2 and ER 0 < ∞ be satisfied.Then where the random variable χ 2 k is χ 2 -distributed with k degrees of freedom.
The latter result can be used to test the goodnessof-fit of certain hypothesised radius df F d (if λ d is known).

THE CASE s = d -2
For fixed n ∈ N , define the empirical process which has an infinite mean for any r ≥ 0 .Nevertheless, V n (r) is weakly consistent for λ d (1 − F d (r)) under slight additional assumptions.

Theorem 4 Under Condition 3 and ER
for any r ≥ 0 , and therefore, together with F d (0) = 0 , for some fixed r ≥ 0 with F d (r) < 1 , then 5772 denotes the Euler-Mascheroni constant and the random variable S 1 possesses an α-stable df with characteristic exponent α = 1 and skewness parameter β = 1 having the Fourier-Stieltjes transform Remark 4 Nolan (1997) provides tables and numerical procedures for calculating the density of S 1 (and other stable densities).This gives at least in principle the possibility for testing the null hypothesis H 0 : Of course, the previous cases are of particular interest in stereological practice for d = 3, s = 2, d = 2, s = 1 and d = 3, s = 1.To be complete we also investigate the asymptotic behaviour of a simple generalization of U n (r) resp.V n (r) to the case d − s > 2 .The below result seems to be of interest for its own right (from the view point of pure asymptotics) and it gives insight how the instability increases when d − s becomes greater than two.

PROOFS OF THE THEOREMS
First observe that the empirical processes U n (r) , V n (r) , σ 2 n (r) , and Y (p) n (r) can be regarded as so-called shot-noise processes Heinrich and Schmidt (1985) and references therein.However, only U n (r) has a finite first moment.In fact, applying Campbell's theorem gives In order to prove Theorem 1 we have to replace the terms (R i 2 − r 2 ) −1/2 (which are responsible for the large fluctuations of the sum) by truncated terms.More precisely, for any ε > 0, we introduce the 'truncated' shot-noise process and the nonnegative random integer First step: For any Borel set B ⊆ R 1 we have the identity Applying Campbell's theorem to the shot-noise process N n,ε (r) we obtain after a short calculation using Eq. 3 and ER 0 < ∞ that for any ε > 0 .
Second step: Using once more Eq.3 and the formula Image Anal Stereol 2007;26:63-71 and Condition 2 enables us to prove that for any ε > 0 .
In the third step we make use of Condition 1 and show that the cumulants of order m ≥ 3 (abbreviated by the symbol Cum m ) of (π Ûn,ε (r) become arbitrarily small.More precisely, using some relationships and estimates for general shot-noise processes derived in Heinrich and Schmidt (1985) we arrive at the estimates where the constant C m depends on the total variations of the signed measures γ (k) red (•) , k = 2, . . ., m.This last estimate confirms the asymptotic normality of the truncated shot-noise process U n,ε (r) by applying the classical 'method of moments'.
Readers interested in detailed proofs of the Theorems 1-4 are referred to an extended version of the paper being available under http://www.math.uniaugsburg.de/stochastik/heinrich/papers/asymwick.pdf.
The proof of Theorem 3 is quite similar to that of Theorem 4. For this reason we will outline the essential proving steps only in case of Theorem 4.
Let δ > 0 be arbitrarily small, but fixed and ε > 0 be chosen small enough (in fact, ε = ε n can be thought of as a positive sufficiently slowly decreasing null sequence).Define in analogy to U n,ε (r) the truncated process and let M n,ε (r) denote the above random integer N n,ε (r) with d − 2 instead of d − 1.
Since the 'truncation inequality' (Eq.5) remains valid for the shot-noise process V n (r) with M n,ε (r) instead of N n,ε (r) , it follows together with Chebychev's inequality that P( V n (r) − λ d (1 − F d (r)) ≥ δ ) ≤ P(M n,ε (r) ≥ 1) The following relations can be proved for any ε > 0 : The proofs of the Theorems 5 and 6 rely on the exponential shape of the generating functional of the stationary, independently marked Poisson process Ψ d , which is as follows (see, e.g., Stoyan et al., 1995 for details): for any Borel-measurable, complex-valued 2 − r 2 ) yields the following expression for the logarithm of the characteristic function E exp{it log n d−2 V n (r)} : weak convergence (i.e., convergence in distribution) and convergence in probability P, respectively.

Image
Anal Stereol 2007;26:63-71    for any A ∈ B(R s ) and 0 ≤ a < b ≤ ∞ , where the indicator function 1(S) takes the values 0 or 1 depending on whether the set S is empty or not.Setting A = [0, 1] d and a = r , b = ∞ leads to the following Abel-type integral equation: Condition 3 and the rightcontinuity of F d completes the proof of Theorem 4.
d−2 (ρ 2 −r 2 ) −1 exp{it z} − 1 z 2 dz dF d (ρ) .(6) The rate of decay of γ 2 (B 1 × B 2 ) to zero, if the distance of B 1 and B 2 grows unboundedly, expresses some kind of asymptotic independence between distant parts of the point process Ψ * d .For a complete description of such weak dependences we have to put restrictions on all higher-order (reduced) cumulant measures of Ψ * d .Finally, note that γ d (x, r) denote the closed sphere in R d with radius r > 0 centered at x and let us put