lr.test: 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
)

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.

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.

Author

Jorge Castillo-Mateo

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 $\omega_t = \log(t-1)$ $(t=2,\ldots,T)$.

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. tools:::Rd_expr_doi("10.1007/s00477-021-02122-w").

Examples

Run this code

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")

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