PEkit (version 1.0.0.1000)

sample.test: Lagrange Multiplier Test for \(\psi\)

Description

Performs the Lagrange Multiplier test for the equality of the dispersion parameter \(\psi\) of a sample. The null hypothesis of the test is \(H_0: \psi = \psi_0\), where \(\psi_0\) is given as input here.

Usage

sample.test(abund, psi = "a")

Arguments

abund

An abundance vector of a sample.

psi

Target positive number \(\psi_0\) to be tested. Accepted values are "a" for absolute value 1, "r" for relative value \(n\) (sample size), or any positive number.

Value

The statistic \(S\) and a p-value of the two-sided test of the hypothesis.

Details

Calculates the Lagrange Multiplier test statistic $$S\, = \,U(\psi_0)^2 / I(\psi_0),$$ where \(U\) is the log-likelihood function of \(\psi\) and \(I\) is its Fisher information. The statistic \(S\) follows \(\chi^2\)-distribution with 1 degree of freedom when the null hypothesis \(H_0:\psi=\psi_0\) is true.

References

Radhakrishna Rao, C, (1948), Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 44(1), 50-57. <10.1017/S0305004100023987>

Examples

# NOT RUN {
## Test the psi of a sample from the Poisson-Dirichlet distribution:
set.seed(10000)
x<-rPD(1000, 10)
## Find the abundance of the data vector:
abund=abundance(x)
## Test for the psi that was used, as well as a higher and a lower one:
sample.test(abund, 10)
sample.test(abund, 15)
sample.test(abund, 5)
sample.test(abund)       #test for psi=1
sample.test(abund, "r")  #test for psi=n
# }