lr.test {RecordTest}

Likelihood-Ratio Test for the Likelihood of the Record Indicators

Description

This function performs likelihood-ratio tests for the likelihood of the record indicators I_t to study the hypothesis of the classical record model (i.e., of IID continuous RVs).

Usage

lr.test(
  X,
  record = c("upper", "lower"),
  alternative = c("two.sided", "greater", "less"),
  probabilities = c("different", "equal"),
  simulate.p.value = FALSE,
  B = 1000
)

Arguments

X

A numeric vector, matrix (or data frame).

record

A character string indicating the type of record, "upper" or "lower".

alternative

A character indicating the alternative hypothesis ("two.sided", "greater" or "less"). Different statistics are used in the one-sided and two-sided alternatives (see Details).

probabilities

A character indicating if the alternative hypothesis assume all series with "equal" or "different" probabilities of record.

simulate.p.value

Logical. Indicates whether to compute p-values by Monte Carlo simulation.

B

An integer specifying the number of replicates used in the Monte Carlo estimation.

Details

The null hypothesis of the likelihood-ratio tests is that in every vector (columns of the matrix X), the probability of record at time t is 1 / t as in the classical record model, and the alternative depends on the alternative and probabilities arguments. The probability at time t is any value, but equal in the M series if probabilities = "equal" or different in the M series if probabilities = "different". The alternative hypothesis is more specific in the first case than in the second one. Furthermore, the "two.sided" alternative is tested with the usual likelihood ratio statistic, while the one-sided alternatives use specific statistics based on likelihoods (see Cebrián, Castillo-Mateo and Asín, 2022, for the details).

If alternative = "two.sided" & probabilities = "equal", under the null, the likelihood ratio statistic has an asymptotic \chi^2 distribution with T-1 degrees of freedom. It has been seen that for the approximation to be adequate M must be between 4 and 5 times greater than T. Otherwise, a simulate.p.value is recommended.

If alternative = "two.sided" & probabilities = "different", the asymptotic behaviour is not fulfilled, but the Monte Carlo approach to simulate the p-value is applied. This statistic is the same as \ell below multiplied by a factor of 2, so the p-value is the same.

If alternative is one-sided and probabilities = "equal", the statistic of the test is

-2 \sum_{t=2}^T \left\{-S_t \log\left(\frac{tS_t}{M}\right)+(M-S_t)\left( \log\left(1-\frac{1}{t}\right) - \log\left(1-\frac{S_t}{M}\right) I_{\{S_t<M\}} \right) \right\} I_{\{S_t > M/t\}}.

The p-value of this test is estimated with Monte Carlo simulations, because the computation of its exact distribution is very expensive.

If alternative is one-sided and probabilities = "different", the statistic of the test is

\ell = \sum_{t=2}^T S_{t} \log(t-1) - M \log\left(1-\frac{1}{t}\right).

The p-value of this test is estimated with Monte Carlo simulations. However, it is equivalent to the statistic of the weighted number of records N.test with weights \omega_t = \log(t-1) (t=2,\ldots,T).

Value

A list of class "htest" with the following elements:

statistic

Value of the statistic.

parameter

Degrees of freedom of the approximate \chi^2 distribution.

p.value

(Estimated) P-value.

method

A character string indicating the type of test.

data.name

A character string giving the name of the data.

alternative

A character string indicating the alternative hypothesis.

Author(s)

Jorge Castillo-Mateo

References

Cebrián AC, Castillo-Mateo J, Asín J (2022). “Record Tests to Detect Non Stationarity in the Tails with an Application to Climate Change.” Stochastic Environmental Research and Risk Assessment, 36(2): 313-330. doi:10.1007/s00477-021-02122-w.

See Also

global.test, score.test

Examples

set.seed(23)
# two-sided and different probabilities of record, always simulated the p-value
lr.test(ZaragozaSeries, probabilities = "different")
# equal probabilities
lr.test(ZaragozaSeries, probabilities = "equal")
# equal probabilities with simulated p-value
lr.test(ZaragozaSeries, probabilities = "equal", simulate.p.value = TRUE)

# one-sided and different probabilities of record
lr.test(ZaragozaSeries, alternative = "greater", probabilities = "different")
# different probabilities with simulated p-value
lr.test(ZaragozaSeries, alternative = "greater", probabilities = "different", 
  simulate.p.value = TRUE)
# equal probabilities, always simulated the p-value
lr.test(ZaragozaSeries, alternative = "greater", probabilities = "equal")