Research Article | Open Access
D. J. Best, J. C. W. Rayner, O. Thas, "Tests of Fit for the Logarithmic Distribution", Advances in Decision Sciences, vol. 2008, Article ID 463781, 8 pages, 2008. https://doi.org/10.1155/2008/463781
Tests of Fit for the Logarithmic Distribution
Smooth tests for the logarithmic distribution are compared with three tests: the first is a test due to Epps and is based on a probability generating function, the second is the Anderson-Darling test, and the third is due to Klar and is based on the empirical integrated distribution function. These tests all have substantially better power than the traditional Pearson-Fisher test of fit for the logarithmic. These traditional chi-squared tests are the only logarithmic tests of fit commonly applied by ecologists and other scientists.
Species diversity data can sometimes be modeled by a zero-truncated negative binomial distribution with index parameter near zero. Fisher et al.  examined the limit as the index parameter of this distribution approached zero and so derived the logarithmic distribution. A random variable X has this distribution if and only if in which 0 < β < 1 and . The logarithmic or log-series distribution is often applied to species diversity data.
As an example of species diversity data which the logarithmic distribution may fit, consider the following data on insect catches from the Sierra Tarahuma, Mexico, reported by Aldrete . Ten species were caught precisely once, three species were caught precisely twice, and so on according to Table 1. The expected line in Table 1 shows the expected counts on fitting a logarithmic distribution. For these data, the alpha index of diversity is 9.01, where in which n is the total number of insects and is the maximum likelihood estimator of β. The AI quoted here is defined in Krebs [3, 12.13]; for a discussion of the index of diversity see Krebs [3, Section 12.4.1]. Note that larger AI implies more diversity while smaller AI implies less.
It would seem sensible to test the data for consistency with the logarithmic distribution before quoting an AI value. However, the only statistic that appears to be commonly used by ecologists as a test of fit for the logarithmic distribution is the so-called chi-squared test, which, as Krebs [3, Section 12.4.1] notes, may not always have good power.
“The goodness-of-fit of the logarithmic series … can be tested by the usual chi-squared goodness-of-fit test … this means low power …. Thus in most cases the decision to use the logarithmic series … must be made on ecological grounds, rather than statistical goodness-of-fit criteria.” [3, page 429].
In this paper, we will examine a number of statistical tests which are considerably more powerful than the traditional Pearson-Fisher test. These include tests of fit based on components of Neyman’s smooth test statistic, the Anderson-Darling test discussed by Lockhart et al. , an empirical integrated distribution function test given by Klar , and a test due to Epps  based on a probability generating function (pgf). We suggest that these could be used to help make a decision as to whether or not to use the logarithmic series based on statistical as well as ecological criteria. In particular, the dispersion statistic, , defined subsequently, should be useful for identifying the not infrequent case of data for which the abundance species are more abundant than predicted by the logarithmic series.
Our second example is included for its somewhat curious interest and is not involved with conventional species diversity. Collins and Hand  have counted the number of times, in the period 1983 to 2000, that a Hollywood film won one Oscar, two Oscars, three Oscars, and so on, giving the data in Table 2. The film with 11 Oscars was “Titanic.”
2. Tests of Fit for the Logarithmic
A discussion of smooth tests of fit and their components, particularly when testing for the logarithmic, is given in Appendix A. These tests may be derived as a routine application of Rayner and Best [8, Theorem 6.1.1]. The first-order component is identically zero when β is estimated by maximum likelihood, or, equivalently, by method of moments. The test based on the component suggests whether or not the data are consistent with the logarithmic variance while the test based on suggests whether or not the data are consistent with logarithmic moments up to the third. To find p-values for these tests, it is suggested that the parametric bootstrap is to be used as convergence to the asymptotic standard normal distribution is very slow. See Gürtler and Henze  and Appendix B for details of the parametric bootstrap in a goodness of fit context.
In Section 3, we give powers for the Anderson-Darling test based on the statistic in which , and Ox is the number of observations equal to x. Summation is halted when x is the minimum such that Ox = 0 and . We also give powers for a test given by Klar , based on the empirical integrated distribution function with test statistic in which M is the largest observation. Finally, for comparison purposes, we quote powers of the pgf and tests given by Epps .
3. Power Comparisons
Random deviates from the logarithmic (L), positive Poisson (P+), and positive geometric (G+) distributions were generated using IMSL  routines RNLGR, RNPOI, and RNGEO. Random zeta deviates (Z) and random Yule deviates (Y) were found using algorithms of Devroye [11, pages 551 and 553]. Table 3 gives powers for the same alternatives as used by Epps , but with the addition of two Yule alternatives. For convenience, we reproduce the powers given by Epps for his pgf and tests. The powers we give for , Tn, , , and + were found using parametric bootstrap with 1000 simulations both for the inner and the outer loops. Note that the calculation of can involve large numbers, and calculation of the pgf and statistics can involve small numbers. Care with rounding error may be needed. The statistics Tn and are less prone to rounding error. Klar  notes that the smooth tests, the test, and the pgf test are not consistent against all alternatives.
From Table 3 our powers for Tn are a little greater than those of Klar , and we observe that the power for the Z(1.0) alternative is 0.73, somewhat larger than the 0.40 reported by Klar . Also from Table 3, we see that the test is not generally competitive with the other tests.
The test based on + performs reasonably well. The test based on the Tn statistic has power a little less than that for the pgf- and -based tests. An advantage of the test based on Tn is that Klar  showed it is consistent.
The test based on the dispersion statistic has good power for the zeta and Yule alternatives, while the and pgf tests generally have competitive powers for all alternatives. Clearly, the test based on will not have good power for alternatives with similar dispersion to the logarithmic distribution. If the test based on is not significant but that based on is, this suggests a skewness departure from the logarithmic distribution. However, if the test based on is significant, then this suggests that the test based on may be significant due to either a dispersion or a skewness departure of the data from the logarithmic distribution. Notice that we say the test based on suggests how the data deviate from the logarithmic. We do not claim that the data actually do deviate in this manner. See the comments of Henze and Klar .
On the basis of Table 3 powers, we suggest that the tests based on and Klar’s Tn are considered as tests of fit for the logarithmic distribution. These tests have good power and are consistent. We recommend that the tests based on Tn and are augmented by the use of and in a data analytic fashion.
In the following parametric bootstrap, p-values for the tests based on , , , and Tn are given. These use 1000 random samples of the logarithmic distribution with parameter as given below. We give and values because they may suggest how the data deviate from the logarithmic. We give the and Tn values because the tests based on these statistics are consistent and have good power.
4.1. Insect Data
From the data in Table 1, we find with p-value 0.52, with p-value 0.19, with p-value 0.82, and with p-value 0.33. It appears that the logarithmic distribution is a good fit. In agreement with this, the Pearson-Fisher statistic takes the value 4.56 on 11 degrees of freedom when data greater than 12 have been combined.
4.2. Oscars Data
We find that and on 7 degrees of freedom if the classes greater than or equal to 9 are combined. The corresponding p-values are 0.58, 0.08, and 0.62. It appears that a logarithmic distribution with = 0.7044 fits the data reasonably well. However, the p-value for suggests that the data may not be quite as skewed as would be expected for the logarithmic distribution. Collins and Hand  suggest a Yule distribution fits the data well. In addition, we note that with p-value 0.12 and with p-value 0.26.
In this paper, we have examined a number of statistical tests which are considerably more powerful than the traditional Pearson-Fisher test. We suggest that these could be used to help make a decision as to whether or not to use the logarithmic series based on statistical as well as ecological criteria. A test of fit could be done before quoting the index of diversity. In particular, the dispersion statistic, , should be useful for identifying the not infrequent case of data for which the abundance species are more abundant than predicted by the logarithmic series.
A. The Smooth Tests and Their Components
For distributions from exponential families the smooth tests can be derived as score statistics for testing H0: θ = 0 against K : θ≠ 0 for observations from the model in which
(i) is a probability density function that depends on a vector of nuisance parameters β and for which we test;(ii) is a complete orthonormal set on ;(iii) is a normalizing constant. For details see Rayner and Best .
The score test statistic has a particularly appealing form where Here, is the maximum likelihood estimator of β assuming that is true.
To define , central moments of up to order 2r are required. For example, to directly define components up to to test for the logarithmic, we note that the equation to estimate β is ≡ 0, as discussed below. To define and requires and , which in turn require central logarithmic moments up to order six. These are given by To calculate further orthonormal polynomials directly, we could use the result that for the logarithmic, generates cumulants and hence central moments, but it is more efficient to use recurrence as described in Rayner et al. . Proceeding directly, the first six central moments can be used to calculate in which These formulas give the first three orthonormal polynomials for any univariate distribution.
The components can be called smooth components as they are analogous to the components of the smooth test for uniformity introduced by Neyman . His smooth components also used orthonormal polynomials. When testing for distributions from exponential families these components are asymptotically independent and asymptotically have the standard normal distribution.
For the logarithmic distribution, the maximum likelihood and method of moments estimators of β coincide, given by ≡ 0 or To solve this equation, the Newton-Raphson algorithm can be used. An initial estimate of and other details helpful in the solution are given in Birch . Note also that for the logarithmic, is proportional to where , so the test based on tests for the dispersion of the logarithmic distribution. Similarly, if then the numerator of is of the form , so the test based on assesses whether the data are consistent with moments of the logarithmic up to the third.
B. P-Values via the Parametric Bootstrap
Gürtler and Henze [9, page 223] suggest that p-values can be obtained using an analogue of the parametric bootstrap. If Wn denotes a test statistic, calculate where denote, as usual, the data. Find an estimate from the data and conditional on this estimate, generate say pseudorandom samples of size n, each having the logarithmic () distribution. For compute the value on each random sample. The parametric bootstrap p-value is then the proportion of the that are at least the observed wn, namely, .
The above requires random logarithmic (β) values. Devroye [11, page 547] outlines an algorithm for generating random logarithmic deviates. Alternatively, the routine RNLGR from IMSL  can be used. To obtain p-values for two-tailed tests proceed as above and find the p-value, say P. Then if , the two-tailed p-value is 2P, while if , the two-tailed p-value is .
Research of Olivier Thas was supported by IAP research network Grant no. P6/03 of the Belgian Government (Belgian Science Policy).
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Copyright © 2008 D. J. Best et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.