469:
107:
464:{\displaystyle {\begin{aligned}&(1-\tau )\int _{-\infty }^{t}(t-x)\,dF(x)=\tau \int _{t}^{\infty }(x-t)\,dF(x)\\&\int _{-\infty }^{t}|t-x|\,dF(x)=\tau \int _{-\infty }^{\infty }|x-t|\,dF(x)\\&t-\operatorname {E} ={\frac {2\tau -1}{1-\tau }}\int _{t}^{\infty }(x-t)\,dF(x)\end{aligned}}}
112:
487:
Werner Ehm, Tilmann
Gneiting, Alexander Jordan, Fabian Krüger, "Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings,"
79:
99:
535:
25:
504:
46:
29:
529:
84:
499:
Yuwen Gu and Hui Zou, "Aggregated
Expectile Regression by Exponential Weighting,"
81:
expectile of the probability distribution with cumulative distribution function
17:
33:
515:
Whitney K. Newey, "Asymmetric Least
Squares Estimation and Testing,"
37:
488:
505:
https://www3.stat.sinica.edu.tw/preprint/SS-2016-0285_Preprint.pdf
101:
is characterized by any of the following equivalent conditions:
32:
of the distribution in a way analogous to that in which the
87:
49:
110:
463:
93:
73:
8:
441:
420:
415:
382:
337:
332:
318:
312:
304:
278:
273:
259:
253:
245:
218:
197:
192:
166:
145:
137:
111:
109:
86:
48:
480:
36:of the distribution are related to the
7:
519:, volume 55, number 4, pp. 819–47.
421:
364:
313:
308:
249:
198:
141:
14:
454:
448:
438:
426:
376:
370:
350:
344:
333:
319:
291:
285:
274:
260:
231:
225:
215:
203:
179:
173:
163:
151:
130:
118:
68:
56:
16:In the mathematical theory of
1:
74:{\textstyle \tau \in (0,1)}
552:
536:Probability distributions
26:probability distribution
465:
95:
75:
466:
96:
76:
108:
85:
47:
425:
317:
258:
202:
150:
28:are related to the
461:
459:
411:
300:
241:
188:
133:
91:
71:
501:Statistica Sinica
409:
543:
520:
513:
507:
497:
491:
485:
470:
468:
467:
462:
460:
424:
419:
410:
408:
397:
383:
356:
336:
322:
316:
311:
277:
263:
257:
252:
237:
201:
196:
149:
144:
114:
100:
98:
97:
92:
80:
78:
77:
72:
551:
550:
546:
545:
544:
542:
541:
540:
526:
525:
524:
523:
514:
510:
498:
494:
486:
482:
477:
458:
457:
398:
384:
354:
353:
235:
234:
106:
105:
83:
82:
45:
44:
12:
11:
5:
549:
547:
539:
538:
528:
527:
522:
521:
508:
492:
479:
478:
476:
473:
472:
471:
456:
453:
450:
447:
444:
440:
437:
434:
431:
428:
423:
418:
414:
407:
404:
401:
396:
393:
390:
387:
381:
378:
375:
372:
369:
366:
363:
360:
357:
355:
352:
349:
346:
343:
340:
335:
331:
328:
325:
321:
315:
310:
307:
303:
299:
296:
293:
290:
287:
284:
281:
276:
272:
269:
266:
262:
256:
251:
248:
244:
240:
238:
236:
233:
230:
227:
224:
221:
217:
214:
211:
208:
205:
200:
195:
191:
187:
184:
181:
178:
175:
172:
169:
165:
162:
159:
156:
153:
148:
143:
140:
136:
132:
129:
126:
123:
120:
117:
115:
113:
94:{\textstyle F}
90:
70:
67:
64:
61:
58:
55:
52:
30:expected value
13:
10:
9:
6:
4:
3:
2:
548:
537:
534:
533:
531:
518:
512:
509:
506:
502:
496:
493:
490:
484:
481:
474:
451:
445:
442:
435:
432:
429:
416:
412:
405:
402:
399:
394:
391:
388:
385:
379:
373:
367:
361:
358:
347:
341:
338:
329:
326:
323:
305:
301:
297:
294:
288:
282:
279:
270:
267:
264:
254:
246:
242:
239:
228:
222:
219:
212:
209:
206:
193:
189:
185:
182:
176:
170:
167:
160:
157:
154:
146:
138:
134:
127:
124:
121:
116:
104:
103:
102:
88:
65:
62:
59:
53:
50:
41:
39:
35:
31:
27:
23:
19:
517:Econometrica
516:
511:
500:
495:
483:
42:
21:
15:
18:probability
475:References
22:expectiles
433:−
422:∞
413:∫
406:τ
403:−
392:−
389:τ
368:
362:−
327:−
314:∞
309:∞
306:−
302:∫
298:τ
268:−
250:∞
247:−
243:∫
210:−
199:∞
190:∫
186:τ
158:−
142:∞
139:−
135:∫
128:τ
125:−
54:∈
51:τ
34:quantiles
530:Category
38:median
20:, the
489:arxiv
24:of a
43:For
532::
503:,
40:.
455:)
452:x
449:(
446:F
443:d
439:)
436:t
430:x
427:(
417:t
400:1
395:1
386:2
380:=
377:]
374:X
371:[
365:E
359:t
351:)
348:x
345:(
342:F
339:d
334:|
330:t
324:x
320:|
295:=
292:)
289:x
286:(
283:F
280:d
275:|
271:x
265:t
261:|
255:t
232:)
229:x
226:(
223:F
220:d
216:)
213:t
207:x
204:(
194:t
183:=
180:)
177:x
174:(
171:F
168:d
164:)
161:x
155:t
152:(
147:t
131:)
122:1
119:(
89:F
69:)
66:1
63:,
60:0
57:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.