Semiparametric maxima estimators
These estimators are based on the approximate relation \(G(u_b) = F(u_b)^{b\theta}\) between the marginal distribution function \(F\) of the process and the distribution function \(G\) of the block maximum \(M\) of \(b\) consecutive variables, which applies provided that \(b\) and \(u_b\) are large. If \(G = F^{b\theta}\) then \(Y = -b\log F(M)\) has an exponential(\(\theta\)) distribution and \(Z = b(1 - F(M))\) is a multiple \(b\) of a beta(\(1, b\theta\)) distribution.
Let \((M_1, \ldots, M_k)\) be a sample of block maxima of \(b\) consecutive values. The marginal distribution function \(F\) is not known so we estimate it using the empirical distribution function \(\hat{F}\) set \[ \quad Y_i = -b \log \hat{F}(M_i), \quad Z_i = b(1 - \hat{F}(M_i)), \quad i = 1, \ldots, k \] and define the estimators \[ \hat{\theta}_N = \left( \frac{1}{k} \sum_{i=1}^k Y_i \right)^{-1} \quad \hat{\theta}_{BB} = \left( \frac{1}{k} \sum_{i=1}^k Z_i \right)^{-1}. \] These two estimators are equivalent asymptotically. The former is the estimator proposed in Northrop (2015). Berghaus and Bücher (2018) derived asymptotic properties of the latter, because it is more amenable to mathematical analysis.
The function spm provides three estimators, estimated using both sliding and disjoint block maxima and four types of bias-adjustment. The estimators are \(\hat{\theta}_N, \hat{\theta}_{BB}\) and \(\hat{\theta}_{BB}-1/b\). The latter is motivated by the observation that E\((Z) = (\theta + 1/b)^{-1} < \theta^{-1}\) and therefore the subtraction of \(1 / b\) provides further bias-adjustment. In fact empirical results indicate that \(\hat{\theta}_N\) and \(\hat{\theta}_{BB} - 1/b\) produce similar estimates.
Newlyn sea surges
We provide an illustration of the spm function using the newlyn data, a time series of 2894 maximum seas surges measured at Newlyn, Cornwall, UK over the period 1971-1976. We use a block size \(b = 20\). We explain in Block size selection how this value was chosen.
library(exdex)
theta <- spm(newlyn, 20)
# Estimates: BB2018b is BB2018 - 1/b
theta
#>
#> Call:
#> spm(data = newlyn, b = 20)
#>
#> Estimates of the extremal index theta:
#> N2015 BB2018 BB2018b
#> sliding 0.2392 0.3078 0.2578
#> disjoint 0.2350 0.3042 0.2542
# Estimates, SEs and bias-adjustments
summary(theta)
#>
#> Call:
#> spm(data = newlyn, b = 20)
#>
#> Estimate Std. Error Bias adj.
#> N2015, sliding 0.2392 0.01990 0.003317
#> BB2018, sliding 0.3078 0.01642 0.003026
#> BB2018b, sliding 0.2578 0.01642 0.053030
#> N2015, disjoint 0.2350 0.02222 0.003726
#> BB2018, disjoint 0.3042 0.02101 0.003571
#> BB2018b, disjoint 0.2542 0.02101 0.053570There is a confint method for calculating confidence intervals using objects returned from spm. The likelihood-based intervals are based on an adjustment (performed by the chandwich package Northrop and Chandler (2021)) of the naive (pseudo-)loglikelihood so that the curvature of the adjusted loglikelihood agrees with the estimated standard errors.
# Sliding maxima, symmetric intervals
conf <- confint(theta)
# Sliding maxima, likelihood-based intervals
conf <- confint(theta, interval_type = "lik")
plot(conf)
For this small block size of \(b = 20\) there is an appreciable difference between the inferences from the N2015 and BB2018 estimates.
S&P 500 index
We also consider a dataset (sp500) that is similar to one analysed in Berghaus and Bücher (2018): daily log returns of the S&P 500 index, from 3rd January 1990 to 9th October 2018. We use a block size \(b = 225\), which is similar to that chosen by Berghaus and Bücher (2018).
theta <- spm(sp500, 225)
summary(theta)
#>
#> Call:
#> spm(data = sp500, b = 225)
#>
#> Estimate Std. Error Bias adj.
#> N2015, sliding 0.2454 0.04075 0.01451
#> BB2018, sliding 0.2497 0.04069 0.01452
#> BB2018b, sliding 0.2453 0.04069 0.01897
#> N2015, disjoint 0.2402 0.04150 0.01472
#> BB2018, disjoint 0.2446 0.04152 0.01475
#> BB2018b, disjoint 0.2401 0.04152 0.01920
conf <- confint(theta, interval_type = "lik")
plot(conf)
For this large block size there is very little difference between the inferences produce by the three estimators.
Block size selection
An informal way to choose the value of \(b\) is plots estimates of \(\theta\) over a range of values of \(b\) and to select the smallest \(b\) above which these estimates appear to be constant with respect to \(b\), taking into account sampling variability. The function choose_b, and its associated plot method, does this, quantifying sampling variability using pointwise confidence intervals. The plot method allows us to select one of 3 estimators and choose whether to use sliding or disjoint block maxima. The default is to plot the estimates using the Northrop (2015) estimator applied to sliding maxima.
For the newlyn data we obtain the following plot.
# Plot like the top left of Northrop (2015)
# We remove the 14 values because 2880 has lots of factors
b_vals <- c(2,3,4,5,6,8,9,10,12,15,16,18,20,24,30,32,36,40,45,48,54,60)
res <- choose_b(newlyn[1:2880], b_vals)
# Some b are too small for the sampling variance of the sliding blocks
# estimator to be estimated
plot(res, ylim = c(0, 1))
This plots suggests that \(b = 20\) is a reasonable block size to choose. Indeed the point estimates varies smoothly with \(b\) and varies little for \(b \geq 20\). The confidence intervals tend to widen as \(b\) increases, although the method for estimating these intervals is somewhat sensitive to block size owing to its use of disjoint block maxima. See the help file for spm and Berghaus and Bücher (2018) for further information. Note that for very small values of \(b\) it may not be possible to estimate the confidence intervals, hence the missing intervals in this plot.
For the sp500 data we obtain the following plot.
b_vals <- c(10, seq(from = 25, to = 350, by = 25), 357)
res500 <- choose_b(sp500, b_vals)
plot(res500, ylim = c(0, 1))
The estimates of \(\theta\) stabilise less quickly than for the newlyn data but \(b = 225\) seems to be reasonable choice.
