619:
492:
254:
202:
487:{\displaystyle d_{0}|t|^{\beta _{0}}\exp {\big (}-|t|^{\beta }/\gamma {\big )}\leq |\varphi _{X}(t)|\leq d_{1}|t|^{\beta _{1}}\exp {\big (}-|t|^{\beta }/\gamma {\big )}\quad {\text{as }}t\to \infty }
74:
679:
65:
44:
660:
43:
is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s
234:
653:
684:
646:
230:
197:{\displaystyle d_{0}|t|^{-\beta }\leq |\varphi _{X}(t)|\leq d_{1}|t|^{-\beta }\quad {\text{as }}t\to \infty }
538:
534:
584:
226:
28:
630:
574:
40:
51:
673:
626:
618:
563:"On the optimal rates of convergence for nonparametric deconvolution problems"
32:
579:
562:
533:. Such supersmooth distributions have derivatives of all orders. Examples:
17:
588:
603:
Introduction to
Fourier analysis and generalized functions
634:
257:
77:
486:
196:
654:
464:
426:
341:
303:
8:
661:
647:
578:
470:
463:
462:
454:
448:
443:
434:
425:
424:
410:
405:
400:
391:
385:
373:
358:
349:
340:
339:
331:
325:
320:
311:
302:
301:
287:
282:
277:
268:
262:
256:
248:if its characteristic function satisfies
225:. The examples of such distributions are
180:
170:
165:
156:
150:
138:
123:
114:
102:
97:
88:
82:
76:
50:Formally, we call the distribution of a
556:
554:
550:
605:. London: Cambridge University Press.
7:
615:
613:
680:Theory of probability distributions
633:. You can help Knowledge (XXG) by
481:
191:
25:
617:
469:
179:
478:
444:
435:
401:
392:
374:
370:
364:
350:
321:
312:
278:
269:
188:
166:
157:
139:
135:
129:
115:
98:
89:
1:
497:for some positive constants
207:for some positive constants
240:The distribution is called
701:
612:
601:Lighthill, M. J. (1962).
567:The Annals of Statistics
66:characteristic function
45:characteristic function
629:-related article is a
580:10.1214/aos/1176348248
561:Fan, Jianqing (1991).
488:
198:
489:
199:
255:
75:
541:, mixture normal.
484:
194:
29:probability theory
685:Probability stubs
642:
641:
473:
183:
16:(Redirected from
692:
663:
656:
649:
621:
614:
606:
593:
592:
582:
573:(3): 1257–1272.
558:
493:
491:
490:
485:
474:
471:
468:
467:
458:
453:
452:
447:
438:
430:
429:
417:
416:
415:
414:
404:
395:
390:
389:
377:
363:
362:
353:
345:
344:
335:
330:
329:
324:
315:
307:
306:
294:
293:
292:
291:
281:
272:
267:
266:
203:
201:
200:
195:
184:
181:
178:
177:
169:
160:
155:
154:
142:
128:
127:
118:
110:
109:
101:
92:
87:
86:
41:density function
21:
700:
699:
695:
694:
693:
691:
690:
689:
670:
669:
668:
667:
610:
600:
597:
596:
560:
559:
552:
547:
532:
525:
510:
503:
442:
406:
399:
381:
354:
319:
283:
276:
258:
253:
252:
220:
213:
164:
146:
119:
96:
78:
73:
72:
58:ordinary smooth
52:random variable
23:
22:
15:
12:
11:
5:
698:
696:
688:
687:
682:
672:
671:
666:
665:
658:
651:
643:
640:
639:
622:
608:
607:
595:
594:
549:
548:
546:
543:
530:
523:
519:and constants
508:
501:
495:
494:
483:
480:
477:
466:
461:
457:
451:
446:
441:
437:
433:
428:
423:
420:
413:
409:
403:
398:
394:
388:
384:
380:
376:
372:
369:
366:
361:
357:
352:
348:
343:
338:
334:
328:
323:
318:
314:
310:
305:
300:
297:
290:
286:
280:
275:
271:
265:
261:
218:
211:
205:
204:
193:
190:
187:
176:
173:
168:
163:
159:
153:
149:
145:
141:
137:
134:
131:
126:
122:
117:
113:
108:
105:
100:
95:
91:
85:
81:
24:
14:
13:
10:
9:
6:
4:
3:
2:
697:
686:
683:
681:
678:
677:
675:
664:
659:
657:
652:
650:
645:
644:
638:
636:
632:
628:
623:
620:
616:
611:
604:
599:
598:
590:
586:
581:
576:
572:
568:
564:
557:
555:
551:
544:
542:
540:
536:
529:
522:
518:
514:
507:
500:
475:
459:
455:
449:
439:
431:
421:
418:
411:
407:
396:
386:
382:
378:
367:
359:
355:
346:
336:
332:
326:
316:
308:
298:
295:
288:
284:
273:
263:
259:
251:
250:
249:
247:
243:
238:
236:
232:
228:
224:
217:
210:
185:
174:
171:
161:
151:
147:
143:
132:
124:
120:
111:
106:
103:
93:
83:
79:
71:
70:
69:
67:
63:
59:
56:
53:
48:
46:
42:
38:
34:
30:
19:
635:expanding it
624:
609:
602:
570:
566:
527:
520:
516:
512:
505:
498:
496:
245:
241:
239:
222:
215:
208:
206:
61:
57:
54:
49:
36:
26:
627:probability
242:supersmooth
231:exponential
18:Supersmooth
674:Categories
545:References
68:satisfies
37:smoothness
33:statistics
482:∞
479:→
460:γ
450:β
432:−
422:
408:β
379:≤
356:φ
347:≤
337:γ
327:β
309:−
299:
285:β
244:of order
192:∞
189:→
175:β
172:−
144:≤
121:φ
112:≤
107:β
104:−
60:of order
472:as
182:as
589:2241949
237:, etc.
235:uniform
64:if its
587:
539:Cauchy
535:normal
625:This
585:JSTOR
227:gamma
39:of a
631:stub
31:and
575:doi
419:exp
296:exp
27:In
676::
583:.
571:19
569:.
565:.
553:^
537:,
526:,
515:,
511:,
504:,
233:,
229:,
221:,
214:,
47:.
35:,
662:e
655:t
648:v
637:.
591:.
577::
531:1
528:β
524:0
521:β
517:γ
513:β
509:1
506:d
502:0
499:d
476:t
465:)
456:/
445:|
440:t
436:|
427:(
412:1
402:|
397:t
393:|
387:1
383:d
375:|
371:)
368:t
365:(
360:X
351:|
342:)
333:/
322:|
317:t
313:|
304:(
289:0
279:|
274:t
270:|
264:0
260:d
246:β
223:β
219:1
216:d
212:0
209:d
186:t
167:|
162:t
158:|
152:1
148:d
140:|
136:)
133:t
130:(
125:X
116:|
99:|
94:t
90:|
84:0
80:d
62:β
55:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
