FURTHER RESULTS ON VARIANCES OF LOCAL STEREOLOGICAL ESTIMATORS

In the present paper the statistical properties of local stereological estimators of particle volume are studied. It is shown that the variance of the estimators can be decomposed into the variance due to the local stereological estimation procedure and the variance due to the variability in the particle population. It turns out that these two variance components can be estimated separately, from sectional data. We present further results on the variances that can be used to determine the variance by numerical integration for particular choices of particle shapes.


INTRODUCTION
One of the important unsolved problems in stereology concerns the stereological estimation of particle size distributions without specific assumptions about particle shape.It has been known for some time how to estimate stereologically the mean particle volume for particles of varying shape, cf.Jensen (1998).The resulting distribution of estimated particle volumes has been used as an estimate of the distribution of the true particle volumes.It is clearly important to be able to judge when such a procedure is justified.
The particular case of estimating the volumeweighted mean particle volume has recently been treated in Cabo et al. (2003).It is here shown that an estimator based on planar observation is one order of magnitude more efficient than the traditional one based on observation along lines.In the present paper, we concentrate on the ordinary (unweighted) particle volume distribution.It is shown that an estimator of mean particle volume based on planar observations is superior to one based on line observations, especially for elongated particles.The variance of the estimator can be decomposed into the variance due to the stereological estimation procedure and the variance due to the variability in the particle volumes.We will show how to estimate these variance components separately, from sectional data.If the variance due to the stereological estimation procedure is small compared to the variance due to the variability in the particle volumes, the distribution of estimated particle volumes can be regarded as an estimate of the distribution of the true particle volumes.

MARKED POINT PROCESSES
We define the particle model by means of marked point processes.For more details, we refer to Stoyan et al. (1995).Let Ψ m = {[X i ; Ξ i ]} be a marked point process such that X i is a point in R n and Ξ i belongs to the space M d of d-dimensional differentiable manifolds in R n with finite d-dimensional Hausdorff measure and with the reference point at the origin O.The point X i then serves as a reference point of X i + Ξ i , the i-th particle.
The marked point process Ψ m will be assumed to be stationary, i.e., Ψ m + x = {[X i + x; Ξ i ]} has the same distribution as Ψ m for every x ∈ R n .Stationarity of Ψ m implies stationarity of the unmarked point process Ψ = {X i }.Denote by λ its intensity and assume that 0 < λ < ∞.Let SO(n, L r ) be the subgroup of SO(n) consisting of rotations keeping an r-dimensional linear subspace L r fixed.Then, Ψ m is said to be invariant under rotations in The intensity measure of the marked point process is defined for where V = λ n is the volume and P m is the mark distribution.By Ξ 0 we denote a random manifold with distribution P m .If Ψ m is invariant under rotations in SO(n, L r ), then BΞ 0 has the same distribution as Ξ 0 for all B ∈ SO(n, L r ).

THE LOCAL STEREOLOGICAL ESTIMATORS
The local stereological estimators are based on information collected from section planes in R n through a reference point of the particle.In this section, we present the actual form of local stereological estimators of Hausdorff measure for a generic particle K ∈ M d .Then a section plane of dimension p is a p-dimensional linear subspace (for brevity called psubspace) of R n , p = 0, 1, . . ., n.For comprehensive exposition of local stereology, see Jensen (1998).
There are various forms of the local estimators, depending on the restriction put on the p-subspace.Denote by L n p,L r the set of p-subspaces containing a fixed r-subspace L r , 0 ≤ r < p ≤ n.Let λ d n be the ddimensional Hausdorff measure in R n and let us use the short notation dx d instead of λ d n (dx).Note that the ordinary Lebesgue measure is λ n n = λ n .For K ∈ M d , the local stereological estimator of λ d n (K), based on a p-subspace L p ∈ L n p,L r , d − n + p ≥ 0, has the form, cf.Jensen (1998), (5.24), where σ n = 2π n/2 /Γ(n/2) is the surface area of the unit sphere In a design-based setting, Eq. 1 is an unbiased estimator of λ d n (K).Thus, let µ n p,L r be the unique probability measure on L n p,L r , invariant under rotations from SO(n, L r ).In what follows, we will write dL n p,L r as short for µ n p,L r (dL p ).By an isotropic p-subspace in R n , containing the fixed r-subspace L r , we mean a random p-subspace with constant density with respect to µ n p,L r .Then, if K satisfies the regularity conditions stated in Jensen (1998), Proposition 5.4, and Lp is an isotropic p-subspace containing L r , the local estimator m Example 1.For K ∈ M 3 in R 3 there are three local stereological estimators of the volume V (K), (5) The estimators Eq. 3 and Eq. 4 are related by a socalled Rao-Blackwell procedure.We have For later reference, we also present the local estimator of λ where 1 ≤ r + 1 < p ≤ n, d − n + p ≥ 0 and ∇ q (y 1 , . . ., y q ) denotes the q-dimensional Hausdorff measure of the parallelepiped spanned by y 1 , . . ., y q .Assuming that K satisfies the regularity conditions from Jensen (1998), Theorem 5.6, m(n,d) Example 2. For d = n = 3, p = 2 and r = 0, the estimator of V (K) 2 has the form

THE VARIANCE OF LOCAL ESTIMATORS
We will now return to the model-based case.We let Ξ 0 be a generic random particle, distributed according to P m and denote by Em the expectation with respect to this distribution.Let L p(0) be a fixed p-subspace in R n , containing an r-subspace L r , 0 ≤ r < p ≤ n, d − n + p ≥ 0.
Below, we give explicit results for the second moment of m (n,d) p,L r Ξ 0 , L p(0) .For this purpose, the following proposition is very useful.
Proposition 1.Let P m be invariant under rotations in SO(n, L r ).The estimator m p,L r is fixed, Lp is an isotropic p-subspace containing L r and Ξ 0 and Lp are independent.
Proof.The result follows from the fact that for any non-negative measurable function h, The left-hand side can be rewritten using the invariance of P m under rotations in SO(n, L r ) and Jensen (1998), Lemma 8.4, where B ∈ SO(n, L r ).From invariant measure theory there exists an invariant probability measure α n When convenient we use the short notation m(Ξ 0 ) for m Using Proposition 1 and Eq. 2, we get Moreover, the relation Eq. 2 for a fixed Ξ 0 can be written as almost surely, and for the variance of m(Ξ 0 ), we get Generally, two random variables with the same expectation and variance don't have to be equal almost surely.But in our situation we can show that the equality of variances suffices.
Proposition 2. Let Ξ 0 be a typical manifold with distribution P m which is invariant under rotations in SO(n, L r ).Then Proof.The equality in Eq. 9 happens if Em var m(Ξ 0 , Lp ) | Ξ 0 = 0 which can be rewritten (using the independence of Ξ 0 and Lp ) as We would like to show this for all p-subspaces L p(0) .
Let us suppose that there exists a subspace L p(0) and a set A(L p( 0 0) ).But from the invariance of P m under rotations in SO(n, L r ) we obtain P m (A(L p )) = P m (A(L p(0) )) > 0 which means that (P m × µ n p,L r )(A) > 0 and this leads us to a contradiction.
If var m m(Ξ 0 ) = var m λ d n (Ξ 0 ), then the Hausdorff measure of the manifold is determined from the local section without error.Such local stereological estimators are exact, i.e., the variance of the estimator is created only by the randomness of particles.The simplest example of a particle with exact local volume estimator is a ball.Proposition 3. Let Ξ 0 be an n-dimensional ball in R n centred at O with probability one.Then Since the right-hand side is ω n R n σ p−r σ n−r , where Applying Proposition 1, Eq. 10 follows immediately.
In Jensen et al. (1999) the class of particles having an exact volume estimator (called quasi-spherical bodies) is studied.
By the similar reasoning as in the previous proof we can show that a sphere has exact surface area estimator.
Proposition 4. Let Ξ 0 be an (n − 1)-dimensional sphere in R n with centre O almost surely.Then sphere in R n and x ∈ K ∩ L p .Now the proof proceeds along the same lines as the proof of Proposition 3, the integration over a pdimensional ball is replaced by the integration over a (p − 1)-dimensional sphere.
Remark 1.The lower dimensional spheres are not necessarily quasi-spherical.For example, consider the case n = 3, d = 1, p = 2 and r = 0. Then the local estimator of λ 1 3 (Ξ 0 ) has the form where R is the radius of Ξ 0 and α is the angle between L 2(0) and the plane containing the circle Ξ 0 .
The local estimator Eq. 1 can be simplified if K is star-shaped at O, i.e., K ∩ L 1 is a line-segment for all be the n-chord function of the set K at O, cf.Gardner (1995), Definition 6.1.1.Furthermore, let ρn,K be the section function, cf.Gardner (1995), Chapter 7.
Proposition 5. Let K be a star-shaped set at O. Then m (11) Proof.Using the polar decomposition of Lebesgue measure we obtain In particular, for r = 0, the local stereological estimator is proportional to the section function Our aim is now to derive some explicit results for the second moment of m(Ξ 0 ).This will give an easy way of finding var m m(Ξ 0 ) (without simulation) for particular choices of shapes of Ξ 0 and will give insight into what kind of shapes of Ξ 0 result in an estimator with large variance.In what follows we always assume that Ξ 0 is invariant under rotations in SO(n, L r ).Proposition 6.Let Ξ 0 be a symmetric and star-shaped set at O. Then for the local estimator with d = n, p = 1 and r = 0 we have Proof.Since Ξ 0 is star-shaped at O, we see from Eq. 11 that the local estimator is proportional to the n-chord function, Using the symmetry of Ξ 0 (ρ Ξ 0 (ω) = ρ Ξ 0 (−ω)) and Proposition 1 we obtain the stated result.
Sometimes, it can be useful to have an alternative expression of Em m Proposition 7. Let Ξ 0 be symmetric and star-shaped set at O. Then for the local estimator with d = n, p = 1 and r = 0 we have Em m Proof.The formula in Proposition 6 can be rewritten as where in the last step we have used (Jensen, 1998), Proposition 4.1 with g(x) = x n .
Remark 2. The assumptions of the previous two propositions are not restrictive as they may appear.If Ξ 0 is not a symmetric and star-shaped set, we can define an equivalent symmetric star-shaped set star(Ξ 0 ), cf.Jensen (2000), by Proof.Under the regularity conditions of (Jensen, 1998), Theorem 5.6, we know that Using the generalized Blaschke-Petkantschin formula (Jensen, 1998), Theorem 5.6, with Notice that due to the assumed regularity conditions, ∇ 2 (π L ⊥ r x, π L ⊥ r y) = 0 on a set of (λ d n ×λ d n )-measure zero and the integral on the right-hand side is well-defined.The result now follows immediately from ∇ 2 (x, y) = x 2 y 2 − x, y 2 1/2 .

PAWLAS Z ET AL: Variances of local stereological estimators
Note that the second moment of m (n,d) p,L r Ξ 0 , L p(0) does not depend on the G-factor.For n-dimensional particles the formula for the variance given in Proposition 7 was expressed through integrals over S n−1 ∩ L ⊥ r in Jensen et al. (1999).Finally, we consider the special case of star-shaped particles and r = 0. Proposition 9. Let Ξ 0 be a star-shaped set at O with O ∈ Ξ 0 almost surely.Suppose that p ≥ 2. Then Proof.It is not difficult to see that the result follows immediately from Proposition 8 and the following formulation of polar decomposition of Lebesgue measure

EXAMPLES
In this section we use the results to find explicit expressions of the variance of local stereological estimators for specific particle shapes.

THE PLANAR CASE
For n = d = 2, p = 1 and r = 0, the variance can easily be determined, using Proposition 6 or Proposition 7 for various shapes of Ξ 0 .We give the formulas for var m m (2,2)

TRIAXIAL ELLIPSOIDS
We suppose that Ξ 0 is an ellipsoid centred at O and with semiaxes of lengths a, b and c.In R 3 there are three local stereological volume estimators, namely Eq. 3, Eq. 4 and Eq. 5.
For p = 1 the second moment of m ) can be written as (using Proposition 6 and spherical coordinates) or in the form where we used Proposition 7 and the transformation ).If we suppose that there exist constants a 0 , b 0 , c 0 and a non-negative random variable ρ such that a = ρa 0 , b = ρb 0 and c = ρc 0 (i.e., the typical particle shape is fixed, only size and direction are random), then the variance of the local estimator becomes where V 0 = 4π 3 a 0 b 0 c 0 is the volume of an ellipsoid with semiaxes a 0 , b 0 , c 0 and the constant κ can be determined from either Eq. 12 or Eq. 13 by means of numerical integration.For p = 2 and r = 0 we can proceed in similar way.From Proposition 9 we have Em m Note that there is a mistake in Jensen (2000), the constant 8 π 3 should be replaced by 1 8π 3 .For fixed Ξ 0 the double integral can again be computed numerically.The formula Eq. 14 still holds, the values of κ for several choices of ratios a 0 /b 0 and b 0 /c 0 are summarized in Table 1.We have also computed κ for Image Anal Stereol 2006;25:155-163 an intermediate estimator, usually called the nucleator, cf.Gundersen (1988), where ω 1 ∈ S 2 ∩ L 2 is an isotropic direction in an isotropic plane L 2 and ω 2 ∈ S 2 ∩L 2 is orthogonal to ω 1 .Our approach based on numerical integration enables more precise results than those obtained by simulation in Jensen (2000).Note that √ κ − 1 is the coefficient of error of the local volume estimator for a corresponding ellipsoid with semiaxes a 0 , b 0 and c 0 .
Table 1.The values of κ from Eq. 14 for three types of volume estimators and various shapes of ellipsoids.Higher values of κ mean higher variance caused by the local stereological estimation.For ball (κ = 1) we have an exact estimator with Note that the error is larger for prolate spheroids (b = c) than for the corresponding oblate spheroids (a = b).
In view of Eq. 6, it is not surprising that smaller values of error are obtained for the local estimator based on plane sections.
In the remainder of this subsection we consider the last local volume estimator Eq. 5. Obviously, it depends on the choice of the fixed line L 1 (usually called vertical axis) relative to the ellipsoid.We assume that the vertical axis has the same direction as one of the semiaxes of the ellipsoid (say the one of length c).Then the profile Ξ 0 ∩ L 2(0) is a planar ellipse with semiaxes of length A and c.Hence, the local estimator has the following form Let α be the angle between L 2(0) and the semiaxis of length a. Then A can be expressed as the function of a, b and α and the second moment of m(Ξ 0 ) is E m For fixed shape of Ξ 0 the constant κ in Eq. 14 does not depend on c 0 , For the values mentioned in Table 1 we get κ = 1 if

OTHER SPATIAL PARTICLES
A table similar to Table 1 can be determined for other choices of particle shape.
As an example, let Ξ 0 be obtained by scaling the prototype cuboid with edges of lenghts a 0 , b 0 , c 0 .It means that Ξ 0 is an isotropically oriented cuboid with edges of lengths ρa 0 , ρb 0 , ρc 0 , where ρ is a non-negative random variable.Then Eq. 14 holds with V 0 = a 0 b 0 c 0 and κ can be calculated numerically (see Table 2).The obtained values are slightly larger than for ellipsoids.
Table 2.The values of κ for various shapes of cuboids.If the particle distribution is invariant under the rotations keeping the vertical axis fixed and the direction of the vertical axis is parallel to the edge of length c of the cuboid, then As the next example consider a regular tetrahedron of random size.For the estimator based on line section κ = 1.20049 and for the estimator based on plane section κ = 1.01775.

HIGHER DIMENSIONS
If n > 3 we do not always have to use numerical integration in order to derive the explicit formula for the variance.For instance, if Ξ 0 is an ellipsoid in R 4 with semiaxes a 1 , a 2 , a 3 , a 4 , then Em m (4,4) This result can be derived from Proposition 7, using elliptical coordinates and lengthy but straightforward calculations.

ESTIMATION OF VARIANCES
It is interesting to find the estimators of the variances σ 2 m = var m m (n,d) p,L r Ξ 0 , L p(0) and σ 2 λ = var m λ d n (Ξ 0 ), separately, from sectional data.First we introduce ratio-unbiased estimators of µ Let W be a fixed bounded Borel set in R n with positive volume.We consider a sample of particles X i + Ξ i with X i in the sampling window W . Then the local estimator is determined from the central section (X i + Ξ i ) ∩ (X i + L p(0) ) for each sampled particle.In order to be in accordance with Eq. 1 we can think that the centred particle Ξ i is sectioned by a fixed psubspace L p( 0 In what follows we assume that the mark distribution P m of a stationary marked point process Ψ m is invariant under rotations in SO(n, L r ).It can be shown that (see Jensen (1998), Proposition 8.5) The proof is based on Campbell's formula for marked point processes (see Stoyan et al. (1995) is a ratio-unbiased estimator of Em λ d n (Ξ 0 ) 2 .We can also estimate Em m This estimator is again ratio-unbiased, as can be easily seen from Campbell's formula for marked point processes.
The problem is how to estimate As long as we restrict ourselves to the case of independently marked point process (i.e., the Ξ i are independent and identically distributed and independent of Ψ), and are unbiased estimators of σ 2 m = var m m(Ξ 0 ) and σ 2 λ = var m λ d n (Ξ 0 ), respectively.In the general case of dependent particles we propose following estimators of σ 2 m , or the estimator taking into account edge corrections where ∑ h means that the sum is taken over i and j such that X i ∈ W , X j ∈ W and X i − X j ≥ h.An appropriate choice of the parameter h ≥ 0 is some trade-off between small value (larger bias) and Anal Stereol 2006;25:155-163 large value (few observations, larger variance).Using Campbell's formula it is easy to show that σ 2 m is in both cases ratio-unbiased if m(X i + Ξ i ) and m(X j + Ξ j ) are independent whenever X i − X j ≥ h 0 .The estimate of σ 2 λ has then the form

Image
In applications, the distribution of estimated particle volumes (or other size parameters) has been used as an estimate of the true particle volume distribution.This procedure is justified if the variance due to the stereological estimation procedure is small compared to the variance due to the variability in the particle population.We can estimate both variances from central sections using Eq. 15 and Eq.16 or Eq.17, Eq. 18 and Eq.19.If the estimates are closed we can expect that the distribution of estimated sizes will be close to the true size distribution.The practical implications of this observation will be investigated elsewhere.
where 1 A (•) stands for the indicator function of the set A and B denotes the Borel σ -algebra.The stationarity PAWLAS Z ET AL: Variances of local stereological estimators of Ψ m implies the following decomposition y) dy 2 dx 2 and ∇ 2 (x, y) is twice the area of the triangle with vertices O, x and y.
thus Proposition 6 and Proposition 7 can be used for any Ξ 0 if Ξ 0 is replaced by star(Ξ 0 ) in the right-hand side of Propositions 6 and 7. Now we turn to the case p ≥ r + 2.