Extreme Value Statistics in Semi-Supervised Models

Figures & data

Fig. 1 RMSE using the pseudo-MLE and the SSE-MLE. From left to right: $\gamma =-0.25,0,0.25$.

Table 1 Variance reduction for different extreme value indices.

(i) $(n,m)=(1000,500)$
g	γ				g	γ
	–0.3	–0.2	–0.1	0		0	0.1	0.2	0.3
–0.25	13.8%	14.0%	13.4%	12.0%	0	15.4%	15.6%	15.4%	14.8%
–0.125	12.6%	14.2%	14.8%	14.1%	0.125	15.3%	15.9%	16.1%	15.8%
0	10.4%	13.0%	14.6%	15.4%	0.25	14.7%	15.5%	16.1%	16.4%
(ii) $(n,m)=(1000,1000)$
	–0.3	–0.2	–0.1	0		0	0.1	0.2	0.3
–0.25	20.2%	21.1%	20.5%	18.8%	0	23.6%	23.5%	23.1%	22.2%
–0.125	18.9%	21.5%	22.4%	22.0%	0.125	23.4%	24.1%	24.3%	25.0%
0	16.3%	19.9%	22.3%	23.6%	0.25	22.4%	23.7%	24.6%	24.6%
(iii) $(n,m)=(500,1000)$
	–0.3	–0.2	–0.1	0		0	0.1	0.2	0.3
–0.25	22.4%	25.8%	25.7%	24.5%	0	30.8%	30.2%	29.9%	29.4%
–0.125	21.6%	26.2%	28.0%	28.0%	0.125	31.0%	30.4%	31.0%	30.8%
0	18.6%	24.6%	28.1%	30.8%	0.25	30.0%	30.1%	31.4%	31.5%

Fig. 2 Variance reduction for various combinations of γ and g.

Table 2 Variance reduction for different numbers of unlabeled data m.

m	γ
	–0.3	–0.2	–0.1	0	0.1	0.2	0.3
500	6.0%	8.3%	9.6%	11.4%	11.7%	11.9%	11.8%
1000	8.7%	12.3%	14.4%	15.6%	12.9%	13.0%	15.5%
5000	11.2%	16.4%	19.5%	22.5%	21.8%	22.0%	21.7%
10,000	11.9%	17.5%	20.8%	24.6%	23.0%	23.2%	22.9%

Table 3 Variance reduction for the extreme quantile estimator.

g	γ
	–0.3	–0.2	–0.1	0	0.1	0.2	0.3
−0.25	24.07%	25.12%	23.6%	22.52%	20.57%	19.10%	16.56%
−0.125	27.92%	30.07%	29.26%	28.03%	26.74%	25.54%	22.98%
0	27.57%	31.34%	31.89%	31.28%	30.78%	30.02%	27.81%
0.125	23.99%	29.85%	31.95%	32.44%	32.60%	32.42%	30.75%
0.25	20.57%	27.74%	30.95%	32.34%	33.09%	33.46%	32.42%

Fig. 3 Relative variance reduction for the extreme quantile estimators.

Table 4 Variance reduction with two covariates.

	$s=0,r=0$			$s=0.8,r=0.8$			$s=0.8,r=0.3$
g	γ			γ			γ
	–0.2	0	0.2	–0.2	0	0.2	–0.2	0	0.2
−0.25	19.8%	11.9%	0.9%	37.4%	28.4%	8.0%	48.9%	42.1%	21.0%
−0.125	20.6%	19.5%	16.2%	37.8%	37.2%	29.3%	48.4%	49.3%	41.9%
0	19.2%	21.8%	21.3%	35.1%	39.1%	36.3%	44.5%	49.8%	48.2%
0.125	16.6%	22.1%	23.0%	30.6%	38.7%	38.5%	38.6%	48.2%	49.8%
0.25	13.1%	21.5%	23.5%	24.8%	36.9%	38.9%	31.0%	45.1%	49.5%

Table 5 Estimation results for three stations.

	Nîmes	Dieppe	Ciboure
(LAT, LON)	(43.86, 4.41)	(49.93, 1.09)	(43.39,–1.69)
$\widehat{\gamma }$ (MLE)	0.358	–0.017	–0.056
	(0.166,0.549)	(–0.156,0.121)	(–0.189,0.077)
${\widehat{\gamma }}_0$ (SSE)	0.517	0.005	–0.088
	(0.341,0.692)	(–0.121,0.130)	(–0.201,0.025)
$\widehat{R}(1,1)$	0.478	0.456	0.537
Reduction	16.0%	17.5%	28.4%
Quantile (MLE)	114.85	33.05	59.32
	(37.77,191.93)	(22.37,43.74)	(41.72,76.93)
Quantile (SSE)	167.82	32.48	56.36
	(110.37,225.26)	(23.35,41.61)	(43.64,69.07)
${\widehat{\gamma }}_{\mathit{\text{obs}}}$ (MLE)	0.421	0.052	0.003
	(0.326, 0.516)	(–0.019,0.122)	(–0.064,0.071)
Quantile (Obs)	175.29	42.13	81.08
	(119.39, 231.20)	(35.31,48.95)	(68.77, 93.38)

Fig. 4 Heatmap of the variance reduction factor across 100 stations.