Performs Brown's method on the p-values of N.test
as proposed by Cebri<U+00E1>n, Castillo-Mateo and As<U+00ED>n (2021). The null
hypothesis of the classical record model (i.e., of randomness) is tested
against the alternative hypothesis.
brown.method(
X,
weights = function(t) 1,
record = c(FU = 1, FL = 1, BU = 1, BL = 1),
alternative = c(FU = "greater", FL = "less", BU = "less", BL = "greater"),
correct = TRUE
)A numeric vector, matrix (or data frame).
A function indicating the weight given to the different
records according to their position in the series,
e.g., if function(t) t-1 then \(\omega_t = t-1\).
Logical vector. Vector with four elements indicating if the p-value of the test for forward upper, forward lower, backward upper and backward lower are going to be used, respectively. Logical values or 0,1 values are accepted.
Vector with four character string taking values
"greater" or "less" indicating the alternative hypothesis
in every test (for forward upper, forward lower, backward upper and
backward lower records , respectively). Under the alternative hypothesis
of linear trend the FU and BL records will be greater and the FL and BU
records will be less than under the null, but other combinations (e.g.,
for trend in variability) could be considered.
Logical. Indicates, whether a continuity correction
should be applied in N.test; defaults to TRUE.
A "htest" object with elements:
Value of the chi-square statistic (not scaled).
Degrees of freedom \(df\) and scale parameter \(c\).
P-value.
A character string indicating the type of test performed.
A character string giving the name of the data.
In this function, the test is implemented as given by Cebri<U+00E1>n,
Castillo-Mateo and As<U+00ED>n (2021), where the
p-values \(p^{(FU)}\), \(p^{(FL)}\), \(p^{(BU)}\) and \(p^{(BL)}\)
of the test N.test for the four types of record are used for
the statistic:
$$-2 \left(\log(p^{(FU)}) + \log(p^{(FL)}) + \log(p^{(BU)}) + \log(p^{(BL)})\right).$$
(Any other combination of p-values for the test is also allowed.)
According to Brown's method (Brown, 1975) for the union of dependent p-values, the statistic follows a \(c \chi^2_{df}\) distribution, with a scale parameter \(c\) and \(df\) degrees of freedom that depend on the covariance of the p-values. This covariances are approximated according to Kost and McDermott (2002): $$\textrm{COV}\left(-2 \log(p^{(i)}), -2 \log(p^{(j)})\right) \approx 3.263 \rho_{ij} + 0.710 \rho_{ij}^2 + 0.027 \rho_{ij}^3,$$ where \(\rho_{ij}\) is the correlation between their respective statistics.
Power studies indicate that this and foster.test using all
four types of records and linear weights are the two most powerful records
tests against a linear drift model. In particular, this is more powerful
than Mann-Kendall test against alternatives with linear drift in series of
generalized Pareto variables and some cases of the generalized extreme
value variables. See Cebri<U+00E1>n, Castillo-Mateo and As<U+00ED>n (2021) for more
details.
Brown M (1975). <U+201C>A Method for Combining Non-Independent, One-Sided Tests of Significance.<U+201D> Biometrics. 31(4), 987<U+2013>992.
Cebri<U+00E1>n A, Castillo-Mateo J, As<U+00ED>n J (2021). <U+201C>Record Tests to detect non stationarity in the tails with an application to climate change.<U+201D> Unpublished manuscript.
Kost JT, McDermott MP (2002). <U+201C>Combining Dependent P-Values.<U+201D> Statistics & Probability Letters, 60(2), 183-190.
# NOT RUN {
brown.method(ZaragozaSeries)
brown.method(ZaragozaSeries, weights = function(t) t-1)
brown.method(ZaragozaSeries, weights = function(t) t-1, correct = FALSE)
# Join p-values of upper records
brown.method(ZaragozaSeries, weights = function(t) t-1, record = c(1,0,1,0))
# Join p-values of lower records
brown.method(ZaragozaSeries, weights = function(t) t-1, record = c(0,1,0,1))
# }
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