Logarithmic series distribution maximum likelihood estimation

For the density function of the Logarithmic series distribution see Logarithmic series. For an example data set, see corbet.

Usage

mllgser(x, na.rm = FALSE, ...)

Arguments

x: a (non-empty) numeric vector of data values.
na.rm: logical. Should missing values be removed?
...: Not in use.

Value

mllgser returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for theta.

model: The name of the model.
density: The density associated with the estimates.
logLik: The loglikelihood at the maximum.
support: The support of the density.
n: The number of observations.
call: The call as captured my match.call

References

Fisher, R. A., Corbet, A. S., & Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12(1), 42. https://doi.org/10.2307/1411

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

Examples

theta_hat <- mllgser(corbet)

# The corbert data contains observations from 1 to 24.
observed <- table(corbet)

# The chi square test evaluated at the maximum likelihood is highly significant.
expected <- extraDistr::dlgser(1:24, theta_hat)
chisq.test(observed, p = expected / sum(expected))
#> Warning: Chi-squared approximation may be incorrect
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  observed
#> X-squared = 94.357, df = 23, p-value = 1.305e-10
#> 

# But chi square test evaluated at 0.997 (used in Corbet) is not.
expected <- extraDistr::dlgser(1:24, 0.997)
chisq.test(observed, p = expected / sum(expected))
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  observed
#> X-squared = 22.925, df = 23, p-value = 0.4651
#> 

# The chi square for `dzipf` is similar.
expected <- sads::dzipf(1:24, mlzipf(corbet)[1], mlzipf(corbet)[2]) * length(corbet)
chisq.test(observed, p = expected / sum(expected))
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  observed
#> X-squared = 18.99, df = 23, p-value = 0.7018
#>