Given a sample of Pareto-tailed samples (positive tail index), compute the lower-trimmed Hill estimator. If \(k0=k\), the estimator reduces to Hill's estimator for the shape index
Arguments
- xdat
[numeric] vector of positive observations
- k
[integer] number of order statistics for the threshold
- k0
[integer] vector of number of largest order statistics, no greater than
k- sorted
[logical] if
TRUE, data are assumed to be sorted in decreasing order.- ...
additional arguments for other routines (notably
vectorize)
Value
a scalar with the shape parameter estimate if k0 is a scalar, otherwise a data frame with columns k0 for the number of exceedances and shape for the tail index.
References
Bladt, M., Albrecher, H. & Beirlant, J. (2020) Threshold selection and trimming in extremes. Extremes, 23, 629-665 . doi:10.1007/s10687-020-00385-0
Examples
# Pareto sample
n <- 200
xdat <- 10/(1 - runif(n)) - 10
shape.lthill(xdat = xdat, k = 100, k0 = 5:100)
#> k0 shape
#> 1 5 1.086956
#> 2 6 1.083652
#> 3 7 1.087468
#> 4 8 1.092676
#> 5 9 1.099353
#> 6 10 1.105379
#> 7 11 1.113381
#> 8 12 1.122984
#> 9 13 1.133318
#> 10 14 1.142943
#> 11 15 1.150032
#> 12 16 1.157003
#> 13 17 1.163310
#> 14 18 1.170349
#> 15 19 1.176650
#> 16 20 1.183040
#> 17 21 1.189657
#> 18 22 1.194514
#> 19 23 1.200038
#> 20 24 1.205283
#> 21 25 1.211072
#> 22 26 1.217385
#> 23 27 1.223413
#> 24 28 1.229756
#> 25 29 1.235242
#> 26 30 1.239296
#> 27 31 1.243772
#> 28 32 1.248267
#> 29 33 1.251850
#> 30 34 1.254364
#> 31 35 1.257297
#> 32 36 1.260367
#> 33 37 1.262907
#> 34 38 1.264004
#> 35 39 1.265197
#> 36 40 1.266630
#> 37 41 1.268136
#> 38 42 1.269165
#> 39 43 1.270147
#> 40 44 1.271415
#> 41 45 1.272850
#> 42 46 1.274460
#> 43 47 1.275954
#> 44 48 1.277686
#> 45 49 1.279546
#> 46 50 1.281539
#> 47 51 1.283109
#> 48 52 1.284426
#> 49 53 1.285805
#> 50 54 1.287310
#> 51 55 1.288306
#> 52 56 1.289059
#> 53 57 1.289868
#> 54 58 1.290790
#> 55 59 1.291711
#> 56 60 1.292848
#> 57 61 1.294180
#> 58 62 1.295439
#> 59 63 1.296699
#> 60 64 1.297596
#> 61 65 1.298685
#> 62 66 1.299812
#> 63 67 1.301013
#> 64 68 1.302388
#> 65 69 1.303594
#> 66 70 1.304761
#> 67 71 1.305754
#> 68 72 1.306575
#> 69 73 1.307328
#> 70 74 1.307876
#> 71 75 1.308602
#> 72 76 1.309434
#> 73 77 1.310394
#> 74 78 1.311330
#> 75 79 1.312144
#> 76 80 1.313093
#> 77 81 1.313861
#> 78 82 1.314596
#> 79 83 1.314501
#> 80 84 1.314390
#> 81 85 1.314165
#> 82 86 1.313951
#> 83 87 1.313853
#> 84 88 1.313885
#> 85 89 1.313907
#> 86 90 1.314044
#> 87 91 1.314284
#> 88 92 1.314350
#> 89 93 1.314237
#> 90 94 1.314090
#> 91 95 1.314081
#> 92 96 1.314160
#> 93 97 1.314330
#> 94 98 1.314285
#> 95 99 1.314183
#> 96 100 1.314213