Test for Finiteness of Moments in a Distribution

Description

The purpose of this package is to tests whether a given moment of the distribution of a given sample is finite or not. For heavy-tailed distributions with tail exponent b, only moments of order smaller than b are finite. Tail exponent and heavy- tailedness are notoriously difficult to ascertain. But the finiteness of moments (including fractional moments) can be tested directly. This package does that following the test suggested by Trapani (2016) <doi:10.1016/j.jeconom.2015.08.006>.

Details

The main function of the package, which performs the test for the finiteness of moments, is

finite_moment_test().

The test assumes a sample drawn from an unknown distrubution F. Given this sample, the finiteness or infiniteness of any moment of order k of distribution F can be ascertained. For this, the test follows a randomised testing procedure with artificial randomness. The absolute sample moment \mu_k of the desired order k of the sample is transformed into a test statistic, which follows a \chi^2 distribution with one degree of freedom exactly if the moment of the same order k of F is not finite. The null hypothesis in the test is that the moment is infinite; the alternative is that it is finite.

It should be noted that while the moment of order k of F may be infinite, the sample moment \mu_k is always finite because the sample is of finite size. The sample moment will, however, diverge with growing sample size if the moment of the same order k of the original distribution F is not finite.

The test works as follows: A standard normal distribution is rescaled with \sqrt{\exp(\mu_k)}, yielding a normal distribution with mean 0 and either finite or infinite variance, depending on whether the hypothesis holds. For every observation of the resulting distribution, it is then tested, if the observation is located within an interval [-u, u]. The resulting binary quantity \zeta (0, or 1, true or false) follows a Bernoulli distribution with mean 1/2 exactly if the kth moment of F is infinite. Otherwise the mean is not 1/2. Sampling a number of different intervals characterized by different bounds u drawn from a distribution with finite support, the test aggregates over quantities \zeta such that the resulting test statistic follows a \chi^2 exactly if E(\zeta)=1/2, i.e., if the kth moment of F is infinite.

Trapani (2016) offers some insights into the performance of the test and the impact its parameters have. These parameters are optional arguments of the testing function finite_moment_test().

Author(s)

Torsten Heinrich [aut, cre], Julian Winkler [aut]

Maintainer: Torsten Heinrich <torsten.heinrich@posteo.net>

References

Trapani, Lorenzo (2016), 'Testing for (in)finite moments', Journal of Econometrics 191(1), 57 – 68.

See Also

https://github.com/x0range/finity

Examples

# Generate sample
rvs <- stabledist::rstable(100000, 1.9, 0.5, 1, 0, pm = 0)
# Perform test
result <- finite_moment_test(rvs, 2)
# Print results
message(paste("Test statistic:", result[1], "p-value:", result[2], "\n\n"))

# More examples are included in https://github.com/x0range/finity/examples

Absolute Moment of Order k

Description

Computes the absolute moment of order k of a sample of observations.

Usage

compute_absolute_moment(obs, k)

Arguments

Value

Moment value (type: double)

Examples

rvs <- stabledist::rstable(100000, 1.9, 0.5, 1, 0, pm = 0)
absolute_moment <- compute_absolute_moment(rvs, 2)

Finite Moment Test

Description

Computes Trapani's (2016) finite moment test for moment of order k of the distribution of a given the sample of observations obs. Knowledge of the identity of the distribution is not required. The null hypothesis is that the moment is infinite; the alternative is that it is finite. The function takes parameters of the test as optional arguments; some insights into the impact the choice of parameter values has are given in Trapani (2016).

Usage

finite_moment_test(
  obs,
  k,
  r = 0L,
  psi = 2,
  u = 1,
  force_random_variate_sample = 0L,
  ignore_errors = 0L,
  verbose = 0L,
  random_salting = 0L
)

Arguments

Value

Trapani's Theta test statistic (type: double).

Corresponding p-value (Chi^2(1) percentile) (type: double).

Examples

rvs <- stabledist::rstable(100000, 1.9, 0.5, 1, 0, pm = 0)
result <- finite_moment_test(rvs, 2)

Chi^2(1) Percentile

Description

Returns the Chi^2(1) percentile for the test statistic.

Usage

get_chisq1_percentile(value)

Arguments

Value

Chi^2(1) percentile (type: double).

Examples

get_chisq1_percentile(20.0)