## 1. Introduction

Cavalieri sections are extensively used in stereology to estimate volumes, whereas Cavalieri slices may be used to estimate any other geometric measure, notably cell or particle number. The prediction of the variance of the corresponding estimators is therefore of interest. The problem is nontrivial because the observations are systematic, hence dependent in general. The state of the art has recently been reviewed in Cruz-Orive & García-Fiñana (2005) for Cavalieri sections, and in Cruz-Orive & Geiser (2004) for Cavalieri slices.

Suppose that the target is the volume *Q* of a bounded object. Fix a sampling axis and consider the area *f*(*x*) of the intersection between the object and a plane normal to the sampling axis at a point of abscissa *x*. Then, *Q* is the area under the graph of the function *f* : ℝ→ℝ^{+}. In general, *f* is assumed to be square integrable, and it is called the measurement function. The ‘trend’ of the variance of the Cavalieri estimator *Q̂* of *Q* is well represented by a term of the variance called the extension term, Var_{E}(*Q̂*). It turns out that this term is proportional to *T*^{2q+2}, where *T* is the distance between the sampling planes and *q* is the ‘smoothness constant’ of *f*, namely a non negative real constant that represents the order of the first non-continuous (possibly fractional) derivative of *f*. So far, the following cases have been considered:

- (i)Cavalieri sections (namely
*t*= 0) and*q*∈ [0, 1], see García-Fiñana & Cruz-Orive (2000, 2004). Local errors (arising from the estimation of section areas by point counting) are considered in García-Fiñana*et al*. (2003). - (ii) Cavalieri slices (namely
*t*≥ 0) and*q*= 0, 1, … , see Gual-Arnau & Cruz-Orive (1998) and the application in McNulty*et al*. (2000). Local errors (arising from slice subsampling when the target is particle number) are considered in Cruz-Orive (1999, 2004), Cruz-Orive & Geiser (2004), and Cruz-Orive*et al*. (2004).

Kiêu (1997), Kiêu *et al.* (1999) and Gundersen *et al.* (1999) consider *q* = 0, 1, … , (denoted by ‘*m*’ in these papers) and, although they contemplate the case *t* > 0, their variance predictors do not involve the parameter *t*. As a consequence, their variance predictors do not tend to zero (as they should) when the slice thickness *t* approaches the sampling period *T* (namely the distance between slice midplanes). Earlier predictors (Matheron, 1965, 1971; Gundersen & Jensen, 1987; Cruz-Orive, 1989) involve neither *t* nor *q* (they implicitly assume *t* = 0 and *q* = 0).

The problem of predicting the variance of the estimator of *Q* from Cavalieri slices of thickness *t* > 0 for measurement functions with smoothness constant *q* ∈ [0, 1], has hitherto remained open. The purpose of this paper is to fill this gap.

The new prediction formulae are displayed in Section 3, the relevant coefficient α(*q*, τ) is computed in Section 4, and the performance of the formulae is checked in Section 5 for the brain data used in García-Fiñana & Cruz-Orive (2000) and in McNulty *et al.* (2000). For practical applications the reader may concentrate on the formulae (3.3)–(3.5), resorting to Table 1 to compute the α(*q*, τ). The derivation details are given in the Appendixes.

τ = q | 0.00 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.00 | 12.0 | 13.0 | 14.1 | 15.4 | 16.9 | 18.7 | 20.8 | 23.4 | 26.7 | 30.7 | 36.0 | 43.0 | 52.5 | 66.1 | 86.7 | 120.0 | 180.0 | 306.7 | 660.0 | 2520 |

0.05 | 13.7 | 14.6 | 15.7 | 17.1 | 18.8 | 20.7 | 23.1 | 26.0 | 29.6 | 34.2 | 40.1 | 47.9 | 58.7 | 74.1 | 97.5 | 135.6 | 204.3 | 350.3 | 760.0 | 2934 |

0.10 | 15.5 | 16.4 | 17.7 | 19.2 | 21.0 | 23.1 | 25.8 | 29.0 | 33.0 | 38.1 | 44.8 | 53.6 | 65.8 | 83.3 | 109.9 | 153.4 | 232.3 | 400.6 | 875.7 | 3414 |

0.15 | 17.7 | 18.6 | 19.8 | 21.5 | 23.4 | 25.8 | 28.8 | 32.4 | 36.9 | 42.6 | 50.1 | 60.1 | 73.9 | 93.8 | 124.1 | 173.9 | 264.5 | 458.7 | 1010.0 | 3972 |

0.20 | 20.2 | 21.0 | 22.4 | 24.1 | 26.3 | 28.9 | 32.2 | 36.2 | 41.3 | 47.8 | 56.2 | 67.5 | 83.2 | 105.8 | 140.5 | 197.5 | 301.8 | 526.0 | 1165.0 | 4622 |

0.25 | 23.0 | 23.8 | 25.3 | 27.2 | 29.6 | 32.5 | 36.1 | 40.7 | 46.4 | 53.7 | 63.2 | 76.0 | 93.8 | 119.7 | 159.3 | 224.8 | 344.9 | 604.2 | 1346.0 | 5379 |

0.30 | 26.2 | 27.1 | 28.6 | 30.7 | 33.4 | 36.6 | 40.7 | 45.8 | 52.2 | 60.4 | 71.3 | 85.9 | 106.2 | 135.8 | 181.2 | 256.5 | 395.1 | 695.4 | 1558.0 | 6266 |

0.35 | 30.0 | 30.9 | 32.5 | 34.8 | 37.7 | 41.4 | 46.0 | 51.7 | 59.0 | 68.3 | 80.6 | 97.3 | 120.5 | 154.4 | 206.7 | 293.4 | 453.7 | 802.0 | 1806.0 | 7308 |

0.40 | 34.4 | 35.3 | 37.1 | 39.6 | 42.8 | 46.9 | 52.1 | 58.6 | 66.8 | 77.4 | 91.5 | 110.5 | 137.2 | 176.2 | 236.4 | 336.6 | 522.4 | 927.2 | 2098.0 | 8536 |

0.45 | 39.5 | 40.4 | 42.4 | 45.1 | 48.8 | 53.4 | 59.2 | 66.6 | 75.9 | 88.1 | 104.2 | 126.0 | 156.7 | 201.6 | 271.2 | 387.4 | 603.1 | 1075.0 | 2443.0 | 9990 |

0.50 | 45.5 | 46.5 | 48.6 | 51.7 | 55.7 | 61.0 | 67.6 | 75.9 | 86.7 | 100.6 | 119.1 | 144.2 | 179.6 | 231.5 | 312.2 | 447.2 | 698.6 | 1250.0 | 2853.0 | 11720 |

0.55 | 52.6 | 53.6 | 55.9 | 59.3 | 63.9 | 69.9 | 77.4 | 87.0 | 99.3 | 115.3 | 136.6 | 165.7 | 206.6 | 266.9 | 360.8 | 518.2 | 812.0 | 1458.0 | 3341.0 | 13780 |

0.60 | 61.0 | 62.0 | 64.6 | 68.4 | 73.6 | 80.4 | 89.0 | 100.0 | 114.2 | 132.7 | 157.4 | 191.1 | 238.7 | 309.0 | 418.6 | 602.8 | 947.5 | 1706.0 | 3926.0 | 16260 |

0.65 | 70.9 | 72.0 | 74.9 | 79.3 | 85.2 | 92.9 | 102.9 | 115.6 | 132.0 | 153.5 | 182.2 | 221.5 | 277.1 | 359.3 | 487.8 | 704.3 | 1110.0 | 2006.0 | 4632.0 | 19250 |

0.70 | 82.8 | 84.1 | 87.3 | 92.3 | 99.1 | 108.0 | 119.5 | 134.2 | 153.3 | 178.4 | 211.9 | 257.9 | 323.2 | 419.9 | 571.3 | 826.8 | 1307.0 | 2369.0 | 5488.0 | 22890 |

0.75 | 97.3 | 98.6 | 102.3 | 108.0 | 115.8 | 126.2 | 139.5 | 156.8 | 179.1 | 208.5 | 247.9 | 302.1 | 379.1 | 493.4 | 672.6 | 975.9 | 1547.0 | 2812.0 | 6535.0 | 27340 |

0.80 | 114.8 | 116.4 | 120.5 | 127.1 | 136.2 | 148.3 | 163.9 | 184.2 | 210.5 | 245.2 | 291.8 | 356.0 | 447.4 | 583.2 | 796.8 | 1159.0 | 1841.0 | 3357.0 | 7825.0 | 32840 |

0.85 | 136.4 | 138.1 | 142.9 | 150.6 | 161.2 | 175.4 | 193.9 | 217.9 | 249.1 | 290.3 | 345.8 | 422.4 | 531.6 | 694.2 | 950.3 | 1385.0 | 2207.0 | 4034.0 | 9431.0 | 39690 |

0.90 | 163.2 | 165.2 | 170.8 | 179.7 | 192.4 | 209.2 | 231.2 | 259.8 | 297.1 | 346.5 | 413.0 | 505.0 | 636.5 | 832.7 | 1142.0 | 1668.0 | 2664.0 | 4884.0 | 11450.0 | 48320 |

0.95 | 197.0 | 199.3 | 205.8 | 216.4 | 231.5 | 251.6 | 278.0 | 312.4 | 357.5 | 417.1 | 497.7 | 609.3 | 769.0 | 1008.0 | 1385.0 | 2027.0 | 3245.0 | 5964.0 | 14020.0 | 59310 |

1.00 | 240.0 | 242.7 | 250.5 | 263.3 | 281.4 | 305.8 | 337.9 | 379.7 | 434.6 | 507.5 | 606.0 | 742.7 | 938.7 | 1232.0 | 1696.0 | 2488.0 | 3993.0 | 7356.0 | 17330.0 | 73530 |