statistique, théorie et gestion de portefeuille - Docs at ISFA

110 3. Distributions exponentielles étirées contre distributions régulièrement variables
(a) Independent data
Stretched-Exponential c=0.7 Stretched-Exponential c=0.3 Pareto Distribution b=3
quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1
N/k=4 N/k=4 N/k=4
mean 0.0791 0.0689 0.1172 0.1432 mean 0.3843 0.4340 0.6301 0.7768 mean 0.3460 0.3496 0.3359 0.3334
emp. Std 0.5156 0.3614 0.1652 0.1154 emp. Std 0.5643 0.3925 0.1800 0.1340 emp. Std 0.5384 0.3631 0.1760 0.1175
th. Std 0.5148 0.3635 0.1636 0.1161 th. Std 0.5393 0.3847 0.1786 0.1302 th. Std 0.5358 0.3791 0.1691 0.1195
N/k=10 N/k=10 N/k=10
mean 0.0726 0.0652 0.0880 0.1083 mean 0.3613 0.3819 0.5028 0.5950 mean 0.3134 0.3379 0.3213 0.3382
emp. Std 0.8218 0.5925 0.2602 0.1839 emp. Std 0.8961 0.6141 0.2741 0.1963 emp. Std 0.9003 0.6056 0.2551 0.1933
th. Std 0.8133 0.5745 0.2577 0.1827 th. Std 0.8493 0.6027 0.2755 0.1983 th. Std 0.8425 0.5982 0.2668 0.1892
(b) Dependent data
Stretched-Exponential c=0.7 with long memory Stretched-Exponential c=0.3 with long memory Pareto b=3 with long memory
quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1
N/k=4 N/k=4 N/k=4
mean -0.0439 -0.0736 -0.2134 -0.3395 mean 0.1297 0.1215 0.0410 0.0235 mean 0.1458 0.1261 -0.0023 -0.0790
emp. Std 0.5396 0.3599 0.1643 0.1128 emp. Std 0.5557 0.3756 0.1633 0.1130 emp. Std 0.5572 0.3765 0.1633 0.1125
th. Std 0.5073 0.3576 0.1579 0.1108 th. Std 0.5183 0.3661 0.1620 0.1143 th. Std 0.5194 0.3663 0.1612 0.1130
46
N/k=10 N/k=10 N/k=10
mean -0.0371 -0.0307 -0.0928 -0.1781 mean 0.0961 0.1275 0.0682 0.0521 mean 0.1239 0.1461 0.0584 0.0196
emp. Std 0.8325 0.5754 0.2561 0.1821 emp. Std 0.8314 0.5796 0.2607 0.1801 emp. Std 0.8352 0.5811 0.2608 0.1801
th. Std 0.8028 0.5680 0.2524 0.1771 th. Std 0.8158 0.5793 0.2570 0.1814 th. Std 0.8188 0.5808 0.2567 0.1807
Table 3: Pickands estimates (9) of the parameter ξ for the Generalized Pareto Distribution (7) for thresholds u corresponding to quantiles 90%,95%,99%
ans 99.5% and two different values of the ratio N/k respectively equal to 4 and 10. In panel (a), we have used iid samples of size 10000 drawn from a
Stretched-Exponential distribution with c = 0.7 and c = 0.3 and a Pareto distribution with tail index b = 3, while in panel (b) the samples are drawn from
a long memory process with Stretched-Exponential marginals and regularly-varying marginal.