542 Fossil Record the diversity estimate is the number of taxa occurring in only one sample, f1, the number occurring in two samples, f2, and so on, as well as the total number of occurrences in all samples Burnham and Overton (1979) utilized a statistical approach known as the jackknife The mathematics of the derivations are too complex to review here, but with this approach they obtained a series of possible estimators The simplest of these utilizes only the number of samples and the number of taxa occurring in only one sample: ^ ẳ Dobs ỵ f1 k ; D k ½1 ^ is the first-order jackknife estimate of diversity, Dobs where D is the number of distinct taxa appearing in the sample, f1 is the number of taxa occurring in exactly one sample, and k is the number of samples They developed additional estimators by incorporating the number of taxa occurring in more than one sample (for instance, their second-order jackknife uses the number of taxa appearing in exactly two samples, as well as the number appearing in just one) Nichols and Pollock (1983) applied these models to a molluscan dataset from the Middle Miocene (16–12 Ma) of Denmark They found that, even under a relatively intensive sampling regime, sampled diversity was as much as 30% lower than estimated diversity An alternative model, proposed by Chao (1987), uses the number of taxa occurring in either exactly one or exactly two samples: f2 S2 ¼ Dobs þ 2f2 ½2 Wing and DiMichele (1995) used this estimator, usually termed ‘‘Chao-2,’’ to compare regional vegetation diversity in the late Paleozoic and early Cenozoic for North American river and delta floodplains Somewhat surprisingly, they found similar biodiversity levels during the two periods for these regions, despite the markedly higher apparent diversities for the latter interval at the global level The same formula has also been applied to estimate local diversity, but with the number of individuals representing each taxon in a single sample used rather than the frequency of occurrences in a set of multiple samples That is, f1 is the number of taxa represented by only one individual in a sample, f2 is the number represented by two individuals, and so on Wing and DiMichele (1995) used this approach to examine local vegetation biodiversities in the Paleozoic and Cenozoic floodplain data set discussed previously They found that, on average, local diversity in the late Paleozoic was similar to local diversity in the early Cenozoic This was consistent with their findings for regional diversity using the Chao-2 estimator However, they did find greater variation in diversity levels among sites in the Cenozoic; in particular, the most diverse Cenozoic sites were much more diverse than the most diverse Paleozoic sites Yet another approach, developed by Chao and Lee (1992), utilizes the entire frequency distribution of occurrences in a set of samples, rather than just the number occurring in only one or a few samples Wang and Dodson (2006) used this method to estimate that vertebrate paleontologists have discovered and named only B29% of an estimated total of B1850 nonavian dinosaur genera Anderson et al (1996) estimated South African late Triassic (228–200 Ma) plant and insect diversities with this method These approaches have some advantages over sampling standardization Perhaps foremost of these is that it is clear what is being estimated – the number of taxa in the sampling universe from which the occurrences have come Secondly, the estimators are explicitly derived from a set of statistical assumptions, so it is possible to quantitatively assess how well those assumptions are met Unfortunately, some of these assumptions are violated in fossil data, often quite severely, so they are likely to produce biased estimates in many applications For instance, the jackknife and Chao estimators assume that each taxon has an equal probability of occurring in each sample That is, taxon A may have a different probability of being present in a sample than taxon B, but that taxonspecific probability is the same for every sample This assumption may be violated if samples differ in extent or quality However, even if great care is taken to minimize this problem, the assumption may still be violated For instance, if relative abundances of particular taxa differed among sampling locations, then the associated relative probabilities of sampling may differ accordingly Parametric Estimates Nonparametric methods seek to make minimal assumptions about the true underlying probability distribution of abundance or occurrences However, parametric approaches assume that these distributions follow a particular mathematical form, they estimate the parameters of this distribution for a particular sample or set of samples (hence the term parametric), and they then use those parameter estimates to infer the fraction of species predicted not to be sampled at all Various distributions have been proposed as good fits to fossil data, and it has been suggested that the best fit distributions may even change through time reflecting ecological changes in the biosphere In one such case, such a fit has been used to estimate unsampled diversity The particular probability distribution used was the Generalized Inverse Gaussian-Poisson (GIGP) distribution Anderson et al (1996) fitted this model to plant and insect data mentioned in the previous section They used these estimates to argue that late Triassic plant and insect diversity in the sampled habitats was comparable to those of the present day, again in contrast to apparent global diversity patterns, which record increasing biodiversity levels through time The primary limitation of this approach is that it assumes that the underlying frequency distribution of occurrences follows a particular statistical distribution As a general rule, parametric methods are more precise than nonparametric analogs when the assumptions about the underlying distribution are met When they are not, however, the estimates can be very inaccurate Another limitation, shared by the nonparametric estimates, is that the uncertainty associated with the estimated diversity increases as the proportion of unsampled taxa increases That is, when the probability of a taxon appearing in a sample is low on average (or when