399:
3373:
31:
2802:) is the probability density function. The forms of this equation, and its classical analysis by series and asymptotic solutions, for the cases of the normal, Student, gamma and beta distributions has been elucidated by Steinbrecher and Shaw (2008). Such solutions provide accurate benchmarks, and in the case of the Student, suitable series for live Monte Carlo use.
1597:
function. Unfortunately, this function has no closed-form representation using basic algebraic functions; as a result, approximate representations are usually used. Thorough composite rational and polynomial approximations have been given by
Wichura and Acklam. Non-composite rational approximations
1857:
This has historically been one of the more intractable cases, as the presence of a parameter, ν, the degrees of freedom, makes the use of rational and other approximations awkward. Simple formulas exist when the ν = 1, 2, 4 and the problem may be reduced to the solution of a
2195:
1461:
for use in diverse types of simulation calculations. A sample from a given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The demands of simulation methods, for example in modern
1072:
1453:
entry. Before the popularization of computers, it was not uncommon for books to have appendices with statistical tables sampling the quantile function. Statistical applications of quantile functions are discussed extensively by
Gilchrist.
704:
2672:
1707:
2789:
2104:
1202:
2018:
1830:
3323:
929:
593:
2349:
2115:
1445:
of a given distribution. For example, they require the median and 25% and 75% quartiles as in the example above or 5%, 95%, 2.5%, 97.5% levels for other applications such as assessing the
165:
74:
function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the
320:
3475:
1941:
218:
1764:
1389:
393:
1330:
1271:
2249:
1124:
806:
764:
2550:
is a quantile function. Two four-parametric quantile mixtures, the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture, are presented by
Karvanen.
2492:
2427:
2254:
In the above the "sign" function is +1 for positive arguments, −1 for negative arguments and zero at zero. It should not be confused with the trigonometric sine function.
1522:
to invert the cdf. Other methods rely on an approximation of the inverse via interpolation techniques. Further algorithms to evaluate quantile functions are given in the
970:
3316:
1593:, its quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal distribution, known as the
2389:
852:
2519:
2454:
2548:
3438:
3309:
1419:
2926:
608:
1530:
packages. General methods to numerically compute the quantile functions for general classes of distributions can be found in the following libraries:
2576:
3018:
Derflinger, Gerhard; Hörmann, Wolfgang; Leydold, Josef (2010). "Random variate generation by numerical inversion when only the density is known".
3291:
3286:
1620:
2981:
2029:
2683:
2562:
is a special case of that available for any quantile function whose second derivative exists. In general the equation for a quantile,
1438:, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function.
2949:
2894:
1858:
polynomial when ν is even. In other cases the quantile functions may be developed as power series. The simple cases are as follows:
1495:
1132:
717:, which is equivalent to the previous probability statement in the special case that the distribution is continuous. Note that the
721:
can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing.
3356:
1952:
1415:
436:
112:
91:
3443:
2821:
1552:
1475:
3184:
1772:
875:
3350:
1407:
449:
2272:
2190:{\displaystyle q={\frac {\cos \left({\frac {1}{3}}\arccos \left({\sqrt {\alpha }}\,\right)\right)}{\sqrt {\alpha }}}\!}
815:
is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have:
2831:
1852:
435:
In the general case of distribution functions that are not strictly monotonic and therefore do not permit an inverse
3432:
2811:
117:
3344:
1511:
1503:
1411:
598:
It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of
258:
3333:
1839:
approach. From this solutions of arbitrarily high accuracy may be developed (see
Steinbrecher and Shaw, 2008).
1446:
941:
67:
3211:
Shaw, W.T. (2006). "Sampling
Student's T distribution – Use of the inverse cumulative distribution function".
1875:
177:
1718:
1406:
The quantile function is one way of prescribing a probability distribution, and it is an alternative to the
2853:"Of quantiles and expectiles: Consistent scoring functions, Choquet representations, and forecast rankings"
1339:
335:
1560:
1515:
1491:
1283:
1224:
2206:
1590:
1548:
1507:
1467:
1463:
1084:
769:
3270:"Applying series expansion to the inverse beta distribution to find percentiles of the F-distribution"
733:
2263:
1527:
1499:
718:
1067:{\displaystyle F(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}
1000:
2559:
2459:
2394:
1867:
1586:
1580:
1556:
1471:
1466:, are focusing increasing attention on methods based on quantile functions, as they work well with
1458:
43:
398:
3404:
3164:
3129:
3073:
2864:
1568:
1400:
2945:
2920:
2358:
1564:
1523:
1487:
821:
725:
3087:
Baumgarten, Christoph; Patel, Tirth (2022). "Automatic random variate generation in Python".
1547:
Quantile functions may also be characterized as solutions of non-linear ordinary and partial
3277:
3247:
3238:
Karvanen, J. (2006). "Estimation of quantile mixtures via L-moments and trimmed L-moments".
3220:
3156:
3121:
3092:
3027:
2993:
2874:
2816:
1519:
1490:, such as the exponential distribution above, which is one of the few distributions where a
1442:
1427:
862:
326:
3147:
Wichura, M.J. (1988). "Algorithm AS241: The
Percentage Points of the Normal Distribution".
2497:
2432:
3372:
3188:
2524:
869:
behaves as an "almost sure left inverse" for the distribution function, in the sense that
168:
63:
2901:
3394:
3389:
953:
3301:
17:
3469:
1514:). When the cdf itself has a closed-form expression, one can always use a numerical
3133:
1836:
1457:
Monte-Carlo simulations employ quantile functions to produce non-uniform random or
699:{\displaystyle Q(p)\ =\ \inf \left\{\ x\in \mathbb {R} \ :\ p\leq F(x)\ \right\}~.}
439:, the quantile is a (potentially) set valued functional of a distribution function
3181:
3097:
709:
Here we capture the fact that the quantile function returns the minimum value of
51:
3296:
3269:
3251:
3384:
3182:
An algorithm for computing the inverse normal cumulative distribution function
3125:
3045:
2667:{\displaystyle {\frac {d^{2}Q}{dp^{2}}}=H(Q)\left({\frac {dQ}{dp}}\right)^{2}}
79:
55:
3059:
27:
Statistical function that defines the quantiles of a probability distribution
3454:
3031:
1702:{\displaystyle {\frac {d^{2}w}{dp^{2}}}=w\left({\frac {dw}{dp}}\right)^{2}}
3281:
3224:
2997:
1589:
is perhaps the most important case. Because the normal distribution is a
3449:
3419:
3414:
3409:
3399:
2826:
1526:
series of books. Algorithms for common distributions are built into many
1450:
1212:
71:
3074:"Random Number Generators (Scipy.stats.sampling) — SciPy v1.13.0 Manual"
1835:
This equation may be solved by several methods, including the classical
30:
3200:
Computational
Finance: Differential Equations for Monte Carlo Recycling
3168:
2879:
2852:
70:
is less than or equal to an input probability value. Intuitively, the
2099:{\displaystyle Q(p)=\operatorname {sign} (p-1/2)\,2\,{\sqrt {q-1}}\!}
1606:
A non-linear ordinary differential equation for the normal quantile,
1594:
1275:
35:
3160:
2784:{\displaystyle H(x)=-{\frac {f'(x)}{f(x)}}=-{\frac {d}{dx}}\ln f(x)}
2869:
3199:
2982:"Continuous random variate generation by fast numerical inversion"
1541:
1441:
Consider a statistical application where a user needs to know key
397:
29:
3060:"Runuran: R Interface to the 'UNU.RAN' Random Variate Generators"
1399:
Quantile functions are used in both statistical applications and
957:
3305:
414:
values. The quantile function does the opposite: it gives the
1197:{\displaystyle Q(p;\lambda )={\frac {-\ln(1-p)}{\lambda }},\!}
3112:
Steinbrecher, G.; Shaw, W.T. (2008). "Quantile mechanics".
2013:{\displaystyle Q(p)=2(p-1/2){\sqrt {\frac {2}{\alpha }}}\!}
1060:
3046:"UNU.RAN - Universal Non-Uniform RANdom number generators"
2558:
The non-linear ordinary differential equation given for
2554:
Non-linear differential equations for quantile functions
713:
from amongst all those values whose c.d.f value exceeds
1449:
of an observation whose distribution is known; see the
2851:
Ehm, W.; Gneiting, T.; Jordan, A.; Krüger, F. (2016).
1602:
Ordinary differential equation for the normal quantile
1434:. The derivative of the quantile function, namely the
111:
With reference to a continuous and strictly monotonic
2686:
2579:
2527:
2500:
2462:
2435:
2397:
2361:
2275:
2209:
2118:
2032:
1955:
1878:
1825:{\displaystyle w'\left(1/2\right)={\sqrt {2\pi }}.\,}
1775:
1721:
1623:
1342:
1286:
1227:
1135:
1087:
973:
940:
For example, the cumulative distribution function of
878:
824:
772:
736:
611:
452:
338:
261:
180:
120:
3089:
Proceedings of the 21st Python in
Science Conference
3020:
ACM Transactions on
Modeling and Computer Simulation
2986:
ACM Transactions on
Modeling and Computer Simulation
2266:, distributions can be defined as quantile mixtures
1486:
The evaluation of quantile functions often involves
3292:
New Methods for Managing "Student's" T Distribution
924:{\displaystyle Q{\bigl (}\ F(X)\ {\bigr )}=X\quad }
724:The quantile is the unique function satisfying the
2783:
2666:
2542:
2513:
2486:
2448:
2421:
2383:
2343:
2243:
2189:
2098:
2012:
1935:
1824:
1758:
1701:
1383:
1324:
1265:
1196:
1118:
1066:
923:
857:In general, even though the distribution function
846:
800:
758:
698:
588:{\displaystyle Q(p)\ =\ {\boldsymbol {\biggl }}~.}
587:
387:
314:
212:
159:
3268:Abernathy, Roger W. and Smith, Robert P. (1993) *
2677:augmented by suitable boundary conditions, where
2344:{\displaystyle Q(p)=\sum _{i=1}^{m}a_{i}Q_{i}(p)}
2240:
2186:
2095:
2009:
1932:
1193:
1081:) is derived by finding the value of Q for which
574:
476:
633:
532:
484:
284:
402:The cumulative distribution function (shown as
3476:Functions related to probability distributions
3317:
2942:Statistical Modelling with Quantile Functions
909:
884:
8:
3240:Computational Statistics & Data Analysis
1581:Normal distribution § Quantile function
160:{\displaystyle F_{X}\colon \mathbb {R} \to }
315:{\displaystyle F_{X}(x):=\Pr(X\leq x)=p\,,}
3324:
3310:
3302:
2980:Hörmann, Wolfgang; Leydold, Josef (2003).
1571:distributions have been given and solved.
3096:
2878:
2868:
2748:
2705:
2685:
2658:
2634:
2605:
2587:
2580:
2578:
2526:
2505:
2499:
2494:are the model parameters. The parameters
2461:
2440:
2434:
2396:
2366:
2360:
2326:
2316:
2306:
2295:
2274:
2208:
2167:
2160:
2139:
2125:
2117:
2082:
2081:
2077:
2066:
2031:
1997:
1986:
1954:
1918:
1877:
1821:
1808:
1792:
1774:
1755:
1733:
1720:
1693:
1669:
1649:
1631:
1624:
1622:
1380:
1369:
1358:
1341:
1321:
1313:
1302:
1285:
1262:
1254:
1243:
1226:
1157:
1134:
1098:
1086:
1013:
995:
972:
908:
907:
883:
882:
877:
835:
823:
771:
735:
651:
650:
610:
573:
572:
524:
475:
474:
451:
381:
363:
358:
337:
308:
266:
260:
206:
205:
179:
135:
134:
125:
119:
3297:ACM Algorithm 396: Student's t-Quantiles
1936:{\displaystyle Q(p)=\tan(\pi (p-1/2))\!}
1430:of its cumulative distribution function
240:. In terms of the distribution function
213:{\displaystyle Q\colon \to \mathbb {R} }
107:Strictly monotonic distribution function
88:inverse cumulative distribution function
3114:European Journal of Applied Mathematics
2843:
1426:, of a probability distribution is the
2925:: CS1 maint: archived copy as title (
2918:
1759:{\displaystyle w\left(1/2\right)=0,\,}
1077:The quantile function for Exponential(
1712:with the centre (initial) conditions
1384:{\displaystyle -\ln(1/4)/\lambda .\,}
388:{\displaystyle Q(p)=F_{X}^{-1}(p)\,.}
7:
3155:(3). Blackwell Publishing: 477–484.
1325:{\displaystyle -\ln(1/2)/\lambda \,}
1266:{\displaystyle -\ln(3/4)/\lambda \,}
426:in red is a horizontal line segment.
2244:{\displaystyle \alpha =4p(1-p).\!}
1119:{\displaystyle 1-e^{-\lambda Q}=p}
801:{\displaystyle \quad p\leq F(x)~.}
25:
3287:Refinement of the Normal Quantile
1494:can be found (others include the
1474:or quasi-Monte-Carlo methods and
422:values. Note that the portion of
3371:
3357:cumulative distribution function
3213:Journal of Computational Finance
1862:ν = 1 (Cauchy distribution)
1416:cumulative distribution function
759:{\displaystyle Q(p)\leq x\quad }
525:
113:cumulative distribution function
92:cumulative distribution function
3444:probability-generating function
1553:ordinary differential equations
920:
773:
755:
2967:Monte Carlo methods in finance
2822:Probability integral transform
2778:
2772:
2736:
2730:
2722:
2716:
2696:
2690:
2626:
2620:
2537:
2531:
2378:
2372:
2338:
2332:
2285:
2279:
2234:
2222:
2074:
2054:
2042:
2036:
1994:
1974:
1965:
1959:
1929:
1926:
1906:
1900:
1888:
1882:
1842:
1540:Python subpackage sampling in
1476:Monte Carlo methods in finance
1366:
1352:
1310:
1296:
1251:
1237:
1181:
1169:
1151:
1139:
989:
977:
901:
895:
789:
783:
746:
740:
679:
673:
621:
615:
555:
549:
507:
501:
462:
456:
378:
372:
348:
342:
299:
287:
278:
272:
202:
199:
187:
154:
142:
139:
1:
2487:{\displaystyle i=1,\ldots ,m}
2422:{\displaystyle i=1,\ldots ,m}
1598:have been developed by Shaw.
431:General distribution function
96:inverse distribution function
3351:probability density function
3098:10.25080/majora-212e5952-007
1408:probability density function
418:values as a function of the
410:values as a function of the
2429:are quantile functions and
1470:techniques based on either
228:so that the probability of
3492:
3433:moment-generating function
3252:10.1016/j.csda.2005.09.014
2812:Inverse transform sampling
1865:
1850:
1578:
3428:
3380:
3369:
3345:probability mass function
3340:
3334:probability distributions
3126:10.1017/S0956792508007341
2521:must be selected so that
2264:the mixtures of densities
1436:quantile density function
1422:. The quantile function,
1412:probability mass function
232:being less or equal than
2944:. Taylor & Francis.
2384:{\displaystyle Q_{i}(p)}
1853:Student's t-distribution
1447:statistical significance
1334:third quartile (p = 3/4)
1219:first quartile (p = 1/4)
1211: < 1. The
865:, the quantile function
847:{\displaystyle Q=F^{-1}}
443:, given by the interval
325:which can be written as
244:, the quantile function
174:, the quantile function
3439:characteristic function
3276:, 9 (4), 478–480
3274:ACM Trans. Math. Softw.
3032:10.1145/1842722.1842723
2570:), may be given. It is
1614:), may be given. It is
1420:characteristic function
62:outputs the value of a
3007:– via WU Vienna.
2940:Gilchrist, W. (2000).
2832:Rank–size distribution
2785:
2668:
2544:
2515:
2488:
2450:
2423:
2385:
2345:
2311:
2245:
2191:
2100:
2014:
1937:
1826:
1760:
1703:
1549:differential equations
1516:root-finding algorithm
1492:closed-form expression
1385:
1326:
1267:
1198:
1120:
1068:
925:
861:may fail to possess a
848:
802:
760:
700:
589:
427:
389:
316:
214:
161:
84:percent-point function
47:
18:Percent point function
3282:10.1145/168173.168387
3225:10.21314/JCF.2006.150
2998:10.1145/945511.945517
2786:
2669:
2545:
2516:
2514:{\displaystyle a_{i}}
2489:
2451:
2449:{\displaystyle a_{i}}
2424:
2386:
2346:
2291:
2246:
2192:
2101:
2015:
1938:
1851:Further information:
1827:
1761:
1704:
1591:location-scale family
1555:for the cases of the
1464:computational finance
1386:
1327:
1268:
1199:
1121:
1069:
948:(i.e. intensity
926:
863:left or right inverse
849:
803:
761:
701:
590:
401:
390:
317:
224:to a threshold value
215:
162:
33:
3187:May 5, 2007, at the
2965:Jaeckel, P. (2002).
2684:
2577:
2543:{\displaystyle Q(p)}
2525:
2498:
2460:
2433:
2395:
2359:
2273:
2207:
2116:
2030:
1953:
1876:
1773:
1719:
1621:
1528:statistical software
1506:(which includes the
1459:pseudorandom numbers
1340:
1284:
1225:
1133:
1085:
971:
876:
822:
770:
734:
609:
450:
336:
259:
178:
118:
2560:normal distribution
1868:Cauchy distribution
1587:normal distribution
1575:Normal distribution
1401:Monte Carlo methods
726:Galois inequalities
371:
76:percentile function
44:normal distribution
3405:standard deviation
3149:Applied Statistics
3091:. pp. 46–51.
3062:. 17 January 2023.
2880:10.1111/rssb.12154
2857:J. R. Stat. Soc. B
2781:
2664:
2540:
2511:
2484:
2446:
2419:
2381:
2341:
2241:
2187:
2096:
2010:
1933:
1822:
1756:
1699:
1537:R library Runuran
1534:C library UNU.RAN
1381:
1322:
1263:
1207:for 0 ≤
1194:
1116:
1064:
1059:
921:
844:
798:
756:
696:
585:
428:
385:
354:
312:
248:returns the value
210:
157:
48:
3463:
3462:
3363:quantile function
2907:on March 24, 2012
2761:
2740:
2652:
2612:
2258:Quantile mixtures
2184:
2183:
2165:
2147:
2093:
2007:
2006:
1816:
1687:
1656:
1524:Numerical Recipes
1488:numerical methods
1443:percentage points
1188:
906:
891:
794:
692:
684:
663:
657:
643:
632:
626:
581:
571:
531:
523:
483:
473:
467:
60:quantile function
40:quantile function
16:(Redirected from
3483:
3375:
3326:
3319:
3312:
3303:
3256:
3255:
3235:
3229:
3228:
3208:
3202:
3197:
3191:
3179:
3173:
3172:
3144:
3138:
3137:
3109:
3103:
3102:
3100:
3084:
3078:
3077:
3070:
3064:
3063:
3056:
3050:
3049:
3042:
3036:
3035:
3015:
3009:
3008:
3006:
3004:
2977:
2971:
2970:
2962:
2956:
2955:
2937:
2931:
2930:
2924:
2916:
2914:
2912:
2906:
2900:. Archived from
2899:
2891:
2885:
2884:
2882:
2872:
2848:
2817:Percentage point
2790:
2788:
2787:
2782:
2762:
2760:
2749:
2741:
2739:
2725:
2715:
2706:
2673:
2671:
2670:
2665:
2663:
2662:
2657:
2653:
2651:
2643:
2635:
2613:
2611:
2610:
2609:
2596:
2592:
2591:
2581:
2549:
2547:
2546:
2541:
2520:
2518:
2517:
2512:
2510:
2509:
2493:
2491:
2490:
2485:
2455:
2453:
2452:
2447:
2445:
2444:
2428:
2426:
2425:
2420:
2390:
2388:
2387:
2382:
2371:
2370:
2350:
2348:
2347:
2342:
2331:
2330:
2321:
2320:
2310:
2305:
2250:
2248:
2247:
2242:
2196:
2194:
2193:
2188:
2185:
2179:
2178:
2177:
2173:
2172:
2168:
2166:
2161:
2148:
2140:
2126:
2105:
2103:
2102:
2097:
2094:
2083:
2070:
2019:
2017:
2016:
2011:
2008:
1999:
1998:
1990:
1942:
1940:
1939:
1934:
1922:
1831:
1829:
1828:
1823:
1817:
1809:
1804:
1800:
1796:
1783:
1765:
1763:
1762:
1757:
1745:
1741:
1737:
1708:
1706:
1705:
1700:
1698:
1697:
1692:
1688:
1686:
1678:
1670:
1657:
1655:
1654:
1653:
1640:
1636:
1635:
1625:
1520:bisection method
1390:
1388:
1387:
1382:
1373:
1362:
1331:
1329:
1328:
1323:
1317:
1306:
1272:
1270:
1269:
1264:
1258:
1247:
1203:
1201:
1200:
1195:
1189:
1184:
1158:
1125:
1123:
1122:
1117:
1109:
1108:
1073:
1071:
1070:
1065:
1063:
1062:
1024:
1023:
930:
928:
927:
922:
913:
912:
904:
889:
888:
887:
868:
860:
853:
851:
850:
845:
843:
842:
814:
811:If the function
807:
805:
804:
799:
792:
766:if and only if
765:
763:
762:
757:
719:infimum function
716:
712:
705:
703:
702:
697:
690:
689:
685:
682:
661:
655:
654:
641:
630:
624:
601:
594:
592:
591:
586:
579:
578:
577:
569:
568:
564:
529:
528:
521:
520:
516:
481:
480:
479:
471:
465:
442:
394:
392:
391:
386:
370:
362:
321:
319:
318:
313:
271:
270:
219:
217:
216:
211:
209:
166:
164:
163:
158:
138:
130:
129:
21:
3491:
3490:
3486:
3485:
3484:
3482:
3481:
3480:
3466:
3465:
3464:
3459:
3424:
3376:
3367:
3336:
3330:
3265:
3263:Further reading
3260:
3259:
3237:
3236:
3232:
3210:
3209:
3205:
3198:
3194:
3189:Wayback Machine
3180:
3176:
3161:10.2307/2347330
3146:
3145:
3141:
3111:
3110:
3106:
3086:
3085:
3081:
3072:
3071:
3067:
3058:
3057:
3053:
3044:
3043:
3039:
3017:
3016:
3012:
3002:
3000:
2979:
2978:
2974:
2964:
2963:
2959:
2952:
2939:
2938:
2934:
2917:
2910:
2908:
2904:
2897:
2895:"Archived copy"
2893:
2892:
2888:
2850:
2849:
2845:
2840:
2808:
2753:
2726:
2708:
2707:
2682:
2681:
2644:
2636:
2630:
2629:
2601:
2597:
2583:
2582:
2575:
2574:
2556:
2523:
2522:
2501:
2496:
2495:
2458:
2457:
2436:
2431:
2430:
2393:
2392:
2362:
2357:
2356:
2322:
2312:
2271:
2270:
2262:Analogously to
2260:
2205:
2204:
2159:
2155:
2138:
2134:
2127:
2114:
2113:
2028:
2027:
1951:
1950:
1874:
1873:
1870:
1855:
1849:
1788:
1784:
1776:
1771:
1770:
1729:
1725:
1717:
1716:
1679:
1671:
1665:
1664:
1645:
1641:
1627:
1626:
1619:
1618:
1604:
1583:
1577:
1484:
1397:
1338:
1337:
1282:
1281:
1223:
1222:
1215:are therefore:
1159:
1131:
1130:
1094:
1083:
1082:
1058:
1057:
1046:
1040:
1039:
1025:
1009:
996:
969:
968:
938:
874:
873:
866:
858:
831:
820:
819:
812:
768:
767:
732:
731:
714:
710:
640:
636:
607:
606:
599:
539:
535:
491:
487:
448:
447:
440:
433:
334:
333:
262:
257:
256:
220:maps its input
176:
175:
169:random variable
121:
116:
115:
109:
104:
64:random variable
28:
23:
22:
15:
12:
11:
5:
3489:
3487:
3479:
3478:
3468:
3467:
3461:
3460:
3458:
3457:
3452:
3447:
3441:
3436:
3429:
3426:
3425:
3423:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3390:central moment
3387:
3381:
3378:
3377:
3370:
3368:
3366:
3365:
3360:
3354:
3348:
3341:
3338:
3337:
3331:
3329:
3328:
3321:
3314:
3306:
3300:
3299:
3294:
3289:
3284:
3264:
3261:
3258:
3257:
3246:(2): 947–956.
3230:
3203:
3192:
3174:
3139:
3104:
3079:
3065:
3051:
3037:
3034:. Art. No. 18.
3010:
2992:(4): 347–362.
2972:
2957:
2950:
2932:
2886:
2863:(3): 505–562.
2842:
2841:
2839:
2836:
2835:
2834:
2829:
2824:
2819:
2814:
2807:
2804:
2792:
2791:
2780:
2777:
2774:
2771:
2768:
2765:
2759:
2756:
2752:
2747:
2744:
2738:
2735:
2732:
2729:
2724:
2721:
2718:
2714:
2711:
2704:
2701:
2698:
2695:
2692:
2689:
2675:
2674:
2661:
2656:
2650:
2647:
2642:
2639:
2633:
2628:
2625:
2622:
2619:
2616:
2608:
2604:
2600:
2595:
2590:
2586:
2555:
2552:
2539:
2536:
2533:
2530:
2508:
2504:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2443:
2439:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2380:
2377:
2374:
2369:
2365:
2353:
2352:
2340:
2337:
2334:
2329:
2325:
2319:
2315:
2309:
2304:
2301:
2298:
2294:
2290:
2287:
2284:
2281:
2278:
2259:
2256:
2252:
2251:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2198:
2197:
2182:
2176:
2171:
2164:
2158:
2154:
2151:
2146:
2143:
2137:
2133:
2130:
2124:
2121:
2107:
2106:
2092:
2089:
2086:
2080:
2076:
2073:
2069:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2025:
2021:
2020:
2005:
2002:
1996:
1993:
1989:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1948:
1944:
1943:
1931:
1928:
1925:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1866:Main article:
1864:
1863:
1848:
1841:
1833:
1832:
1820:
1815:
1812:
1807:
1803:
1799:
1795:
1791:
1787:
1782:
1779:
1767:
1766:
1754:
1751:
1748:
1744:
1740:
1736:
1732:
1728:
1724:
1710:
1709:
1696:
1691:
1685:
1682:
1677:
1674:
1668:
1663:
1660:
1652:
1648:
1644:
1639:
1634:
1630:
1603:
1600:
1579:Main article:
1576:
1573:
1545:
1544:
1538:
1535:
1483:
1480:
1418:(cdf) and the
1396:
1393:
1392:
1391:
1379:
1376:
1372:
1368:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1335:
1332:
1320:
1316:
1312:
1309:
1305:
1301:
1298:
1295:
1292:
1289:
1279:
1273:
1261:
1257:
1253:
1250:
1246:
1242:
1239:
1236:
1233:
1230:
1220:
1205:
1204:
1192:
1187:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1115:
1112:
1107:
1104:
1101:
1097:
1093:
1090:
1075:
1074:
1061:
1056:
1053:
1050:
1047:
1045:
1042:
1041:
1038:
1035:
1032:
1029:
1026:
1022:
1019:
1016:
1012:
1008:
1005:
1002:
1001:
999:
994:
991:
988:
985:
982:
979:
976:
954:expected value
937:
936:Simple example
934:
933:
932:
931:almost surely.
919:
916:
911:
903:
900:
897:
894:
886:
881:
855:
854:
841:
838:
834:
830:
827:
809:
808:
797:
791:
788:
785:
782:
779:
776:
754:
751:
748:
745:
742:
739:
707:
706:
695:
688:
681:
678:
675:
672:
669:
666:
660:
653:
649:
646:
639:
635:
629:
623:
620:
617:
614:
596:
595:
584:
576:
567:
563:
560:
557:
554:
551:
548:
545:
542:
538:
534:
527:
519:
515:
512:
509:
506:
503:
500:
497:
494:
490:
486:
478:
470:
464:
461:
458:
455:
432:
429:
396:
395:
384:
380:
377:
374:
369:
366:
361:
357:
353:
350:
347:
344:
341:
329:of the c.d.f.
323:
322:
311:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
269:
265:
208:
204:
201:
198:
195:
192:
189:
186:
183:
156:
153:
150:
147:
144:
141:
137:
133:
128:
124:
108:
105:
103:
100:
94:or c.d.f.) or
66:such that its
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3488:
3477:
3474:
3473:
3471:
3456:
3453:
3451:
3448:
3445:
3442:
3440:
3437:
3434:
3431:
3430:
3427:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3382:
3379:
3374:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3342:
3339:
3335:
3327:
3322:
3320:
3315:
3313:
3308:
3307:
3304:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3279:
3275:
3271:
3267:
3266:
3262:
3253:
3249:
3245:
3241:
3234:
3231:
3226:
3222:
3218:
3214:
3207:
3204:
3201:
3196:
3193:
3190:
3186:
3183:
3178:
3175:
3170:
3166:
3162:
3158:
3154:
3150:
3143:
3140:
3135:
3131:
3127:
3123:
3120:(2): 87–112.
3119:
3115:
3108:
3105:
3099:
3094:
3090:
3083:
3080:
3075:
3069:
3066:
3061:
3055:
3052:
3047:
3041:
3038:
3033:
3029:
3025:
3021:
3014:
3011:
2999:
2995:
2991:
2987:
2983:
2976:
2973:
2968:
2961:
2958:
2953:
2951:1-58488-174-7
2947:
2943:
2936:
2933:
2928:
2922:
2903:
2896:
2890:
2887:
2881:
2876:
2871:
2866:
2862:
2858:
2854:
2847:
2844:
2837:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2809:
2805:
2803:
2801:
2797:
2775:
2769:
2766:
2763:
2757:
2754:
2750:
2745:
2742:
2733:
2727:
2719:
2712:
2709:
2702:
2699:
2693:
2687:
2680:
2679:
2678:
2659:
2654:
2648:
2645:
2640:
2637:
2631:
2623:
2617:
2614:
2606:
2602:
2598:
2593:
2588:
2584:
2573:
2572:
2571:
2569:
2565:
2561:
2553:
2551:
2534:
2528:
2506:
2502:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2441:
2437:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2375:
2367:
2363:
2335:
2327:
2323:
2317:
2313:
2307:
2302:
2299:
2296:
2292:
2288:
2282:
2276:
2269:
2268:
2267:
2265:
2257:
2255:
2237:
2231:
2228:
2225:
2219:
2216:
2213:
2210:
2203:
2202:
2201:
2180:
2174:
2169:
2162:
2156:
2152:
2149:
2144:
2141:
2135:
2131:
2128:
2122:
2119:
2112:
2111:
2110:
2090:
2087:
2084:
2078:
2071:
2067:
2063:
2060:
2057:
2051:
2048:
2045:
2039:
2033:
2026:
2023:
2022:
2003:
2000:
1991:
1987:
1983:
1980:
1977:
1971:
1968:
1962:
1956:
1949:
1946:
1945:
1923:
1919:
1915:
1912:
1909:
1903:
1897:
1894:
1891:
1885:
1879:
1872:
1871:
1869:
1861:
1860:
1859:
1854:
1847:-distribution
1846:
1840:
1838:
1818:
1813:
1810:
1805:
1801:
1797:
1793:
1789:
1785:
1780:
1777:
1769:
1768:
1752:
1749:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1715:
1714:
1713:
1694:
1689:
1683:
1680:
1675:
1672:
1666:
1661:
1658:
1650:
1646:
1642:
1637:
1632:
1628:
1617:
1616:
1615:
1613:
1609:
1601:
1599:
1596:
1592:
1588:
1582:
1574:
1572:
1570:
1566:
1562:
1558:
1554:
1550:
1543:
1539:
1536:
1533:
1532:
1531:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1489:
1481:
1479:
1477:
1473:
1469:
1465:
1460:
1455:
1452:
1448:
1444:
1439:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1404:
1402:
1394:
1377:
1374:
1370:
1363:
1359:
1355:
1349:
1346:
1343:
1336:
1333:
1318:
1314:
1307:
1303:
1299:
1293:
1290:
1287:
1280:
1277:
1274:
1259:
1255:
1248:
1244:
1240:
1234:
1231:
1228:
1221:
1218:
1217:
1216:
1214:
1210:
1190:
1185:
1178:
1175:
1172:
1166:
1163:
1160:
1154:
1148:
1145:
1142:
1136:
1129:
1128:
1127:
1113:
1110:
1105:
1102:
1099:
1095:
1091:
1088:
1080:
1054:
1051:
1048:
1043:
1036:
1033:
1030:
1027:
1020:
1017:
1014:
1010:
1006:
1003:
997:
992:
986:
983:
980:
974:
967:
966:
965:
963:
959:
955:
951:
947:
945:
935:
917:
914:
898:
892:
879:
872:
871:
870:
864:
839:
836:
832:
828:
825:
818:
817:
816:
795:
786:
780:
777:
774:
752:
749:
743:
737:
730:
729:
728:
727:
722:
720:
693:
686:
676:
670:
667:
664:
658:
647:
644:
637:
627:
618:
612:
605:
604:
603:
582:
565:
561:
558:
552:
546:
543:
540:
536:
517:
513:
510:
504:
498:
495:
492:
488:
468:
459:
453:
446:
445:
444:
438:
430:
425:
421:
417:
413:
409:
405:
400:
382:
375:
367:
364:
359:
355:
351:
345:
339:
332:
331:
330:
328:
309:
305:
302:
296:
293:
290:
281:
275:
267:
263:
255:
254:
253:
251:
247:
243:
239:
235:
231:
227:
223:
196:
193:
190:
184:
181:
173:
170:
151:
148:
145:
131:
126:
122:
114:
106:
101:
99:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
45:
41:
37:
32:
19:
3362:
3273:
3243:
3239:
3233:
3219:(4): 37–73.
3216:
3212:
3206:
3195:
3177:
3152:
3148:
3142:
3117:
3113:
3107:
3088:
3082:
3068:
3054:
3040:
3023:
3019:
3013:
3001:. Retrieved
2989:
2985:
2975:
2966:
2960:
2941:
2935:
2909:. Retrieved
2902:the original
2889:
2860:
2856:
2846:
2799:
2795:
2793:
2676:
2567:
2563:
2557:
2354:
2261:
2253:
2199:
2108:
1856:
1844:
1837:power series
1834:
1711:
1611:
1607:
1605:
1584:
1546:
1518:such as the
1512:log-logistic
1504:Tukey lambda
1485:
1468:multivariate
1456:
1440:
1435:
1431:
1423:
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1398:
1395:Applications
1208:
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1076:
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943:
942:Exponential(
939:
856:
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723:
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3026:(4): 1–25.
1542:scipy.stats
1482:Calculation
90:(after the
78:(after the
68:probability
52:probability
3385:raw moment
3332:Theory of
2870:1503.08195
2838:References
2024:ν = 4
1947:ν = 2
1843:Student's
1510:) and the
252:such that
102:Definition
80:percentile
56:statistics
3455:combinant
2911:March 25,
2767:
2746:−
2703:−
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2411:…
2293:∑
2229:−
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2181:α
2163:α
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2132:
2088:−
2061:−
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2004:α
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1814:π
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1375:λ
1350:
1344:−
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1167:
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544::
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3470:Category
3450:cumulant
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3400:variance
3185:Archived
2921:cite web
2827:Quantile
2806:See also
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1781:′
1508:logistic
1451:quantile
72:quantile
3169:2347330
3134:6899308
3003:17 June
1561:Student
1500:Weibull
1496:uniform
1428:inverse
327:inverse
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38:is the
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2948:
2355:where
2150:arccos
2109:where
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1557:normal
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1472:copula
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3165:JSTOR
3130:S2CID
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1569:gamma
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2927:link
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