Threshold Estimation for a Spectrally Negative Lévy Process

<div>The estimator of <svg height="14.3554pt" id="M165" style="vertical-align:-5.34108pt" version="1.1" viewbox="-0.0498162 -9.01432 23.8177 14.3554" width="23.8177pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M608 643C551 682 486 687 443 687C299 687 188 608 188 504C188 361 341 312 465 312C483 312 501 313 518 315C454 195 366 93 314 59C255 82 201 109 141 109C73 109 33 83 33 52C33 13 76 -9 159 -9C218 -9 267 -1 314 15C354 0 402 -15 478 -15C575 -15 671 36 759 141L740 155C657 63 571 16 475 16C448 16 411 26 372 39C458 74 565 199 627 333C806 381 931 499 931 587C931 634 908 669 848 669C767 669 671 582 620 506C591 462 569 411 534 347C503 343 490 342 471 342C330 342 229 399 229 500C229 603 346 657 451 657C486 657 540 657 596 624L608 643ZM890 588C890 519 822 457 747 414C712 395 678 378 643 369C683 484 760 632 841 632C873 632 890 615 890 588ZM266 35C235 24 200 21 161 21C103 21 74 32 74 54C74 75 122 79 141 79C172 79 209 69 266 36V35Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.298,3.132)"><path d="M498 658H277V626C335 621 342 616 342 579C220 574 46 510 46 330C46 148 226 88 342 86C342 41 334 37 271 32V0H498V32C435 37 427 41 427 86C546 88 723 153 723 337C723 518 550 575 427 579C427 616 434 621 498 626V658ZM342 123C255 129 141 183 141 338C141 489 257 538 342 542V123ZM427 542C512 538 629 489 629 333C629 178 512 129 427 123V542Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,19.296,5.233)"><path d="M489 489C489 601 457 708 332 708C138 708 25 411 25 221C25 101 58 -12 189 -12C378 -12 489 294 489 489ZM400 519C400 480 398 450 388 397H132C162 535 231 665 316 665C391 665 400 562 400 519ZM378 339C354 187 285 28 196 28C132 28 114 119 114 202C114 243 115 290 126 339H378Z"></path></g></svg> with sample sizes <i>n</i> = 5000, 10000, 30000, 50000, and 80000.</div>

Mathematical Problems in Engineering

Threshold Estimation for a Spectrally Negative Lévy Process

Figure 3