Image Anal Stereol 2007;26:63-71 Original Research Paper LIMIT DISTRIBUTIONS OF SOME STEREOLOGICAL ESTIMATORS IN WICKSELL’S CORPUSCLE PROBLEM Lothar Heinrich University of Augsburg, Institute of Mathematics, 86135 Augsburg, Germany e-mail: heinrich@math.uni-augsburg.de (Accepted April 19, 2007) ABSTRACT 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 ared = 3,s = 2ors=1andd = 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. Keywords: asymptotic normality, Brillinger-mixing point processes, shot-noise processes, «-stable distribution functions.. INTRODUCTION In 1925 the Swedish statistician Sven D. Wicksell (1890-1939) studied the following problem which belongs meanwhile to the classical toolbox of stereologists. Suppose that a system of three-dimensional random spheres {B3(Xi,Ri) : i > 1} with midpoints {Xi : i > 1} forming a homogeneous point field in R3, and with identically distributed radii {Ri : i > 1} having an unkown common distribution function (briefly df) F3(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 cross-section 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 f2(r),r> 0, depending onF3 as follows: f2(r)=r [d F 3PL ( f(1-F3(p))dp) . (1) 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 F3 . 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 Mu¨cklich, 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 {[Xi,Ri] : i > 1} constitutes a stationary marked point process in R3. 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 Rd equipped with the Euclidean norm || • |d and the corresponding Borel a -field ,^(Rd). Let ^d = {[Xi,Ri] : i > 1} be a stationary, independently marked point process in Rd with generic non-negative mark R0 having the df Fd(r),r > 0. The intensity measure Ad(-) of *¥d is then given by Ad(B x (0,r]) = Xdvd(B)Fd(r), where vd denotes the d-volume and Xd = E#{^*d n [0, 1]d} denotes the intensity of the corresponding stationary non-marked point process ^ = {Xi : i > 1} (see Stoyan et al., 1995 for details). In order to impose an appropriate (mixing) condition on % we need the kth-order cumulant measures %(•) defined on the Borel c7-field ,^(Rdk) of any k > 2 (see, e.g., Heinrich and Schmidt, 1985 for a precise definition). The stationary of ^ enables us to define an associated (signed) measure—the reduced 63 Heinrich L: Limit distributions of stereological estimators kth-order cumulant measure—# by disintegration w.r.t. vd, i.e., (red) on d(k-1) B(R ) Yk{ x Bi) = XdY{ked)C*{Bi-x))vd{dx) i=1 i=1 To facilitate the interpretation of the below Conditions 1-3 we point out that, for disjoint bounded BhB2 G âg(Rd), the second-order cumulant measure y2(-) of the Cartesian product Bx x B2 is just given by the covariance of the numbers #{*¥% n Bi} and #{*¥% n B2}. The rate of decay of yi(B1 x B2) to zero, if the distance of Bx and B2 grows unboundedly, expresses some kind of asymptotic independence between distant parts of the point process ¥£. For a complete description of such weak dependences we have to put restrictions on all higher-order (reduced) cumulant measures of *¥*d. Finally, note that $ed) (•) = 0 for any k > 2 is necessary and sufficient for *F* to be a stationary Poisson point process. Further, let Bd(x,r) denote the closed sphere in Rd with radius r > 0 centered at x and let us put 0. Wicksell’s corpuscle problem in its hyper-stereological version can be described as follows: The system of d-spheres Zd = {Bd{XhRi) : i > 1} is intersected by the s-flat Hs = {x= (xh...,xd) G Rd : xs+1 = ...=xd = 0} (which can be identified with R s). We assume that the collection of non-empty s-spheres Es := Zdf]Hs = {Bs(Xi,Ri) : i > 1} in the linear subspace Hs can be observed (all radii and midpoints are visible, without considering overlappings and edge-effects) in an expanding sampling window Wn (s) := nW(s), where W(s) is a fixed convex set in R s with unit s-volume, i.e., vs(Ws )) =_1,and n runs through N = {1,2,...}. Note that Bs(Xi,Ri) / 0/ iff Ri := (Rf - WXiÌ-s)1/2 > 0. Here and in what follows, write x (resp. x) to indicate the projection of x G Rd onto Hs) (resp. onto the orthogonal complement of Hs which can be identified with Rd-s). The system of non-empty s-spheres Bs{XhRi) is completely described by the stationary marked point process ¥s = {[Xi,Ri] : i_> 1} in R s with intensity measure As (A x (0,r]) = Xsvs(A)Fs(r), where Fs denotes the df of the ‘typical radius’ R0. In the next section we restate the well-known explicit expressions of the df Fs and the intensity Xs in terms of Fd and Xd 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 Fd which are obtained from a single observation of all s- spheres whose midpoints lie in Wn s ). 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 non-normal 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 ==> and -^ we denote weak convergence (i.e., n—»°o n—yoo convergence in distribution) and convergence in probability P, respectively. 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 Wd is Brillinger-mixing. This special mixing condition implies that numbers of points of ^ = {Xi : 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 % (Heinrich, 1994), as well as for Poisson cluster processes ^ (Heinrich, 1988). However, it seems that the Poisson assumption can hardly be dropped in our Theorems 5 and 6 to derive «-stable limits (with a = (d- s)/2) for the fluctuation of the corresponding empirical df s of Fd 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.uni-augsburg.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 R? = Rj - WXiÌ-s > 0 the intensity measures Xs and Ad are connected by the identity K(Ax(a,b))= I l(xGA)x l(a23, 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 Tin iy1 Ri2-r2 is an unbiased estimation of Xd{1-Fd{r)) for any r > 0 and all n G N. On the other hand, the same calculation reveals that the variance of Un(r) is infinite (which has been first noticed in Franklin, 1981). We refer to the fact that, for fixed n G N, the empirical process Un(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 = Ri , however, between two such jumps Un(r) is strictly increasing. Such strange behaviour of this stereological estimator of Xd(1 -Fd{r)) gave rise to consider several modified and smoothed versions of Un(r) (see, e.g., Groeneboom and Jongbloed, 1995 for an isotonic estimation and its asymptotic analysis). ASYMPTOTIC RESULTS THE CASE s = d- 1 We first put together some mixing-type conditions for the point process ^ = {Xi : i > 1} of the midpoints ofthed-spheres. Condition 1 Assume that^ is Brillinger-mixing, i.e., f |7fred)(d(xi,...,xk_i)| 2. Condition 2 Assume that the reduced second-order cumulant measure /fed) (•) satisfies \y\ j(dx)| 0and ER0 < °°, then _______ Ì K2nd-\ d-\ [Un(r) -Xd(1 -Fd(r))j ==^N(0,o2(r)) , where N(0, a2) denotes a zero mean Gaussian random variable with variance o2.Forr>0, condition Eq. 3 is equivalent to fd-x{r) < °°, where the probability density function fd-l is given by Eq. 2 for s = d-1. Remark 1 Provided that Fd(0) = 0, Theorem 1 (for r = 0) yields a central limit theorem for the unbiased estimator f/U n(0) of the intensity Xd. Note that without assuming Brillinger-mixing -merely under Condition 3 - Un(r) turns out to be weakly consistent (as n ->¦ 00) for Xd{1 - Fd{r)). Hence, we get that Un(r) Un(0) 1-Fd(r) for any r>0. It should be noted that, in case ^ 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 Fd is a-Ho¨lder continuous for some a > 1/2 in [r,r+8], i.e., there exists a positive number HaS depending on a and 8 such that Fd{p)-Fd{r) < Hat5(p-r)a for r < p < r+ 8 and some 8 > 0 (see Golubev and Levit, 1998 for an analogous smoothness condition). The multivariate extension of Theorem 1 (by employing the well-known method of Cramer-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 —> 00. Theorem 2 Let the Conditions 1 and 2 and Eq. 3 for re{rl,...,rk},0 0 be satisfied. Then _______ I irlnd-X ,fn (r\ x Vg nd -Ä. U U1 - (1-Fd(r))\ =>N(0,s2(r)) , where s2(r) := (c72(r) + c72(0) (1 -Fd(r))2)/X2. There exists indeed a weakly consistent estimator of the asymptotic variance o2(r) (although its expectation does not exist) which is given by the following ‘overnormed’ random sum r) Ri2-r2 P 66 Image Anal Stereol 2007;26:63-71 Theorem 3 Under Condition 3 and ER0 < 8 it holds on {r)—^oL{r) for each r > 0 satisfying Eq. 3 As an immediate consequence of Eq. 2 for s = d-1, the ratios 2reg(r)/fàd -i)n are weakly consistent estimators of fd-x{r) for each r > 0 satisfying Eq. 3, where (Ad-l) n 1 nd-1 E1Xi GWn (d-l) i>1 is an unbiased and weakly consistent estimator of Ad-i = 2 Ad ERo (see Van Es and Hoogendoorn, 1990 or Golubev and Levit, 1998, for alternative kernel-type estimators of fd -(r)). Combining Theorem 1 with Theorem 3 together with Slutski’s theorem provides Corollary 2 Let the Conditions 1 and 2, ER0 < °° and Eq. 3 for some fixed r>0be satisfied. Then ._____________ / n2nd-1 Va2(r)lognd-i Un(r)-Xd(1-Fd(r)))^N(0, n—>oo Remark 3 By means of Corollary 2 ( applied to r = 0 provided Fd(0) = 0) we are able to construct an asymptotically exact confidence interval for the unknown intensity Xd of the midpoints of d-spheres. In order to find an asymptotic confidence interval for 1-Fd(r) we combine Corollary 1, Theorem 2 and Slutski’s theorem and obtain Corollary 3 Assume that the Conditions 1 and 2, ER0 < oo, Fd(0) = 0, and Eq. 3 forr = 0 and some fixed r>0are satisfied. Then n / sn(r) y where nd-\ lognd - U n 0) n[ ' -(1-Fd(r))J =>N(0,1) 1 /^ r)(f/U n(0))2 + a2(0)(f/U n(r))2. In other words, for any 0 < a < 1 and large enough observation window Wn ( d -l), the interval [b-(a,r),b+(a,r)} contains the value 1 - Fd(r) approximately with probability 1 - a, where b£(a,r) Un(r sn(r -jn 0 ±za/2^* logvd-i(Wn ( (d-i) Vd-i(W n (d-l) Here, za/i denotes the (1 - a/2)-quantile of the N(0, 1)-distribution. A further immediate consequence of Theorem 2 and Slutski’s theorem is Corollary 4 Let the assumptions of Theorem 2 and ER0 <°°be satisfied. Then U, Un(rj)-h(1-Fd(rj))) Wrj) n—^oo where the random variable %\ is %2-distributed with k degrees of freedom. The latter result can be used to test the goodness-of-fit of certain hypothesised radius df Fd (if Xd is known). THE CASE s = d- 2 For fixed n G N, define the empirical process VJr) jnd -lognd - i^Xi GWn (d-2) 1(Ri > r) Ri2-r2 which has an infinite mean for any r > 0. Nevertheless, Vn(r) is weakly consistent for Xd(1 - Fd(r)) under slight additional assumptions. Theorem 4 Under Condition 3 and ER2, < °° it holds Vn{r)-^Xd{1-Fd{r)) for any r>0, n—^oo and therefore, together with Fd(0) = 0, V ˆ nr) P V ˆ n 0)n- 1-Fd(r) for any r>0 Theorem 5 Let *¥d = {Xi : i > 1} be a stationary Poisson process with intensity Xd. If, in addition, \log(p2-r2)\dFd(p) 0 with Fd(r) < 1, then lognd-2 ( ,V n^r\ -1 ) -log(VXd(1 -Fd(r))) f;log(p2-r2)dFd(p) 1-Fdir) -1 + ^^S1 _ _ _ 67 Heinrich L: Limit distributions of stereological estimators where y := limn^00(1 + 1/2 + • • • + 1/n - logn) ~ 0.5772 denotes the Euler-Mascheroni constant and the random variable S1 possesses an astable dfwith characteristic exponent a = 1 and skewness parameter ß = 1 having the Fourier-Stieltjes transform Eexp{it Si} = exp|-2|t| -itlog|t|\ , forteR1. Remark 4 Nolan (1997) provides tables and numerical procedures for calculating the density of Si (and other stable densities). This gives at least in principle the possibility for testing the null hypothesis HQ : Fd = FJ0), Xd = X (0) THECASEd-s>2 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. Vn(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. Let p :=d-s and define %p\r) 1 _,„,_ nx \{Ri>r) ns ppZKXiZWP) {R 2 -r2)p/2 Theorem 6 Let *¥*d = {Xi : i > 1} be a stationary Poisson process with intensity Xd and ERpQ~2 < °°. Then, for any fixed r>0 with Fd(r) <1,it holds Y^(r) (cpXdJ(p2-r2)(p-2)/2dFd(p)) =S 2p) cos(pp) 2/p and the where cp = if(Xi,Xj,Ri) with different ‘response functions’ f|Wx Rd-s x (0,°°) ^ [0,oo), see Heinrich and Schmidt (1985) and references therein. However, only Un(r) has a finite first moment. In fact, applying Campbell’s theorem gives EUn(r) = Xd (1 -Fdir)) and further that E(Un(r))m < °° for 1< m < 2, but E(U n (r))2 = °°. In order to prove Theorem 1 we have to replace the terms (R2 -r2yxl2 (which are responsible for the large fluctuations of the sum) by truncated terms. More precisely, for any e > 0, we introduce the ‘truncated’ shot-noise process U n p ( r 1 = 1 E l(Xi G Wn (d-l) R 2 r2 1 Ri 2 -r 2 > max{£R i-r} £lnd~vlognd~v and the nonnegative random integer LB2 2max{£,R i-r2} id—1 lognd—1 £znd First step: For any Borel set Kl1 we have the identity {Un,e(r) G B} n {Nn,e(r) = 0} = {Un(r) G B} n {Nnß{r) = 0} which in turn implies the estimate P(Un,e(r) GB) - P(Un(r) GB) 1) < ENn,e(r). (5) Applying Campbell’s theorem to the shot-noise process Nn,e(r) we obtain after a short calculation using Eq. 3'and ERo < °° that ENn4r)=2Xdnd-1 n— 0 2 2 max{£, p2-r2} ld i eznd vlogn l^p2-x2-r2 dl dxdFd(p)—>0 for any e > 0. Second step: Using once more Eq. 3 and the formula /ol (1 - w2)-l/2dw = n/2 we may show that ________ I K2nd-\ ylog n^ log n^(EUn,e(r)-Xd(1-Fd(r)))-t0 _ 68 Image Anal Stereol 2007;26:63-71 and Condition 2 enables us to prove that n l im %¦ nd-1 lognd-1 VarUn,e(r)) Vr 2+e < h Xr))-a\r) ^pï^ri for any e > 0 In the third step we make use of Condition 1 and show that the cumulants of order m > 3 (abbreviated by the symbol Cumm) of {%2 nd~x / lognd^1 )l/2 U ˆ n ,e(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 n l im( jj.2 n -1 . m/2 nl?im8 lognd-1 Cumm{Un ,e(r)} < e(m-2)/2 Cm a2 for m > 3 where the constant Cm depends on the total variations of the signed measures yr ek d(-) ,k = 2,...,m. This last estimate confirms the asymptotic normality of the truncated shot-noise process Un,e(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.uni-augsburg.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 5 > 0 be arbitrarily small, but fixed and e > 0 be chosen small enough (in fact, e = en can be thought of as a positive sufficiently slowly decreasing null sequence). Define in analogy to É/U n,e(r) the truncated process Vn KXiGWn = d-2d-2 E Jin log n iy1 x ,(Ri_r > max&g elnd zlo (d-2) R?-r2 r2} e2nd 2lognd 2 and let Mn,e(r) denote the above random integer Nn,e(r) with'd-2insteadofd-1. Since the ‘truncation inequality’ (Eq. 5) remains valid for the shot-noise process Vn(r) with Mne(r) instead of Nne(r), it follows together with Chebychev’s inequality that P(\Vn(r)-Xd(1-Fd(r))\ > 8) < P(Mn,e(r) > 1) + P( \Vn,e{r) - Xd{1 - Fd(r)) \>S)< EMn,e(r) Var(Vn ,e(r)) (EVn ,e(r) — Xd{1 — Fd + §2 + §2 The following relations can be proved for any e > 0 : EMne(r)) 0 (since ER20 < 8) lim\EVn,e(r)-Xd(1-Fd(r)) n-X" and \ d 0 for any Borel-measurable, complex-valued function v(-)onRdx[0,°o) satisfying oo / / \v(x,p)-1\dFd(p)dx Rd 0 Choosing r} jTnd~1(p1 xi r1) yields the following expression for the logarithm of the characteristic function Eexp{it lognd-2 V n (r)}: n2 r2 Xdnnd-2 exp lnd_2 -1dydFd(p) exp{itz}-1 r 0 OO oo h I r (nnd-2{p2-r2)) * dzdFd(p). (6) 69 Heinrich L: Limit distributions of stereological estimators The inner integral in Eq. 6 can be approximated by elementary functions with explicit remainder term in the following way : Poisson process % = {Xi : i > 1} : j exp{ itz}-1d z A n |t|-itlog|t| +it (1-r-logA) 2 2\-1 + — (1+A|t|)e, 2 v y ' where A = an(p2 -r2) , an = nnd-2, and 0 denotes some complex number satisfying |0| < 1. Next, splitting the outer integral in Eq. 6 into two integrals over (rn(e),°°) and (r,rn(e)] with rn(e) = Vr + ^an)-1, we arrive at log Eexp{itlognd"2Vn(r)} = Xd{1-Fd{rn{e))) |t|-itlog|t| + it (1-y+logan £Xd + itXd log(p2-r2)dFd(p) + 2 dt2(1 + e|t|)d rn(e) rn{e) + 2Xd6an (p2-r2)dFd(p) with |0|<1. Since, in view of Eq. 4, the term in the last line vanishes as n -? °° for any e > 0 and also lognd-2(Fd(rn(e))-Fd(r)) d-2 r nP < logn , / |log(p2 -r 2 )|dFd(p)^0, it follows from the foregoing equation (after replacing t by t/Xd ( 1 - Fd (r) ) and some further rearrangements) that logEexpitlognd-2r1 V r 1 logEexp{it Si} + it it Ad(1-Fd(r)) Xr°°log(p2-r2)dFd(p) 1-Fdir + itlog('7rAd(1-Fd(r))ì+it 1-y which is nothing else but the assertion of Theorem 5. To prove Theorem 6 we make use of the subsequent representation of L{np)(t) := logEexp{itYp\r)} which can be derived in analogy to Eq. 6 by using the generating functional of the oo oo (p) , exp{itz}-1 Ln (t)=XdCOp z 1+2/p---- r (ns(p2-r2))~p/2 x (p2-r2-z-2/ pn-s)"1+p/2dzdFd(p) The following formula goes back to L. Euler and can be found in any ‘Table of Integrals’ for 0 < a < 1 : exp{itz}-1 r(1-a) (an\ dz =-----------cos (— I 7i+« a 2 na x |t|a-1 + isgn(Otan2 where r(1 - a) = /0°°e-xxrad x. 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