2354:
3984:
2007:
1509:
2058:
3619:
2815:
318:
1764:
1189:
2349:{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {\operatorname {sgn} (y_{i}+\alpha )|y_{i}+\alpha |^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\operatorname {sgn} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}
3979:{\displaystyle y_{i}^{(\lambda )}={\begin{cases}((y_{i}+1)^{\lambda }-1)/\lambda &{\text{if }}\lambda \neq 0,y\geq 0\\\ln(y_{i}+1)&{\text{if }}\lambda =0,y\geq 0\\-((-y_{i}+1)^{(2-\lambda )}-1)/(2-\lambda )&{\text{if }}\lambda \neq 2,y<0\\-\ln(-y_{i}+1)&{\text{if }}\lambda =2,y<0\end{cases}}}
2593:
1174:
2604:
124:
3139:
Possibly, the transformation could be improved by adding a shift parameter to the log transformation. Panel (c) of the figure shows the log-likelihood. In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed. The final panel shows the transformed
2002:{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {(y_{i}+\alpha )^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}
3104:
1504:{\displaystyle {\begin{aligned}\log({\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}))&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2})+1)+n(\lambda -1)\log(\operatorname {GM} (y))\\&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2}/\operatorname {GM} (y)^{2(\lambda -1)})+1).\end{aligned}}}
460:
3143:
Note that although Box–Cox transformations can make big improvements in model fit, there are some issues that the transformation cannot help with. In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic and a
687:
2452:
3360:
2810:{\displaystyle y_{i}^{({\boldsymbol {\lambda }})}={\begin{cases}{\dfrac {(y_{i}+\lambda _{2})^{\lambda _{1}}-1}{\lambda _{1}}}&{\text{if }}\lambda _{1}\neq 0,\\\ln(y_{i}+\lambda _{2})&{\text{if }}\lambda _{1}=0,\end{cases}}}
964:
313:{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\\operatorname {GM} (y)\ln {y_{i}},&{\text{if }}\lambda =0\end{cases}}}
780:
2437:
can substantially underestimate the variance when parameter values are small relative to the noise variance. However, this problem of underestimating the variance may not be a substantive problem in many applications.
3522:
resource pages contain a number of hands-on interactive activities demonstrating the Box–Cox (power) transformation using Java applets and charts. These directly illustrate the effects of this transform on
3253:
2969:
1194:
3122:. Suppose we are interested in using log(γGT) to predict ALT. A plot of the data appears in panel (a) of the figure. There appears to be non-constant variance, and a Box–Cox transformation might help.
1566:
4148:
Peters, J. L.; Rushton, L.; Sutton, A. J.; Jones, D. R.; Abrams, K. R.; Mugglestone, M. A. (2005). "Bayesian methods for the cross-design synthesis of epidemiological and toxicological evidence".
953:
329:
3508:
2420:
3441:
1695:
2894:
1602:
534:
1644:
887:
4527:
2588:{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda }}&{\text{if }}\lambda \neq 0,\\\ln y_{i}&{\text{if }}\lambda =0,\end{cases}}}
3611:
3136:/2 from the maximum and can be used to read off an approximate 95% confidence interval for λ. It appears as though a value close to zero would be good, so we take logs.
2851:
819:
509:
3585:
2961:
2917:
839:
529:
2381:
4588:
3268:
1169:{\displaystyle J(\lambda ;y_{1},\ldots ,y_{n})=\prod _{i=1}^{n}|dy_{i}^{(\lambda )}/dy|=\prod _{i=1}^{n}y_{i}^{\lambda -1}=\operatorname {GM} (y)^{n(\lambda -1)}}
59:
3565:
4428:
4756:
696:
4413:
4343:
3367:
51:
1605:
3177:
3099:{\displaystyle \ln {\big (}L(\lambda ){\big )}\geq \ln {\big (}L({\hat {\lambda }}){\big )}-{\frac {1}{2}}{\chi ^{2}}_{1,1-\alpha }.}
4464:
4449:
4020:
3127:
47:
904:
4331:
1517:
4724:
3119:
2030:
455:{\displaystyle \operatorname {GM} (y)=\left(\prod _{i=1}^{n}y_{i}\right)^{\frac {1}{n}}={\sqrt{y_{1}y_{2}\cdots y_{n}}}\,}
4617:
Handelsman, D. J. (2002). "Optimal Power
Transformations for Analysis of Sperm Concentration and Other Semen Variables".
4003:
Gao, Peisheng; Wu, Weilin (2006). "Power
Quality Disturbances Classification using Wavelet and Support Vector Machines".
913:
4751:
4105:
Howarth, R. J.; Earle, S. A. M. (1979-02-01). "Application of a generalized power transformation to geochemical data".
3652:
4719:
4557:
4477:
Yeo, In-Kwon; Johnson, Richard A. (2000). "A New Family of Power
Transformations to Improve Normality or Symmetry".
3460:
3132:
The log-likelihood of the power parameter appears in panel (b). The horizontal reference line is at a distance of χ
2430:
4359:
Asar, O.; Ilk, O.; Dag, O. (2017). "Estimating Box-Cox power transformation parameter via goodness-of-fit tests".
4677:
Howarth, R. J.; Earle, S. A. M. (1979). "Application of a generalized power transformation to geochemical data".
2386:
3392:
2038:
4522:
4293:
1653:
900:
66:
4729:
2434:
4714:
2856:
682:{\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}
157:
4225:
2045:
1571:
1758:
Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.
2936:
1611:
854:
3156:
Economists often characterize production relationships by some variant of the Box–Cox transformation.
4649:
4063:
4044:
Gluzman, S.; Yukalov, V. I. (2006-01-01). "Self-similar power transforms in extrapolation problems".
3376:
3115:
2426:
849:
70:
4230:
2639:
2485:
2101:
1807:
2932:
1748:
1736:
1732:
1647:
55:
4694:
4665:
4639:
4536:
4496:
4386:
4368:
4305:
4243:
4165:
4130:
4087:
4053:
4026:
2940:
2920:
1180:
39:
3590:
2823:
785:
488:
4630:
Gluzman, S.; Yukalov, V. I. (2006). "Self-similar power transforms in extrapolation problems".
4584:
4445:
4409:
4339:
4122:
4079:
4016:
3570:
3145:
2946:
2902:
824:
514:
3451:
3355:{\displaystyle Q={\big (}\alpha K^{\lambda }+(1-\alpha )N^{\lambda }{\big )}^{1/\lambda },\,}
2366:
4686:
4657:
4603:
4572:
4518:
4488:
4378:
4289:
4235:
4198:
4157:
4114:
4071:
4008:
896:
4548:
4317:
89:, typically given in piece-wise form that makes it continuous at the point of singularity (
4544:
4313:
4186:
4653:
4067:
4007:. ISDA '06. Vol. 1. Washington, DC, USA: IEEE Computer Society. pp. 201–206.
3550:
466:
43:
17:
4745:
4698:
4669:
4261:
4169:
4161:
4134:
4091:
2360:
903:(1964) introduced the geometric mean into this transformation by first including the
690:
4390:
3524:
2820:
as described in the original article. Moreover, the first transformations hold for
4202:
4030:
1752:
74:
4382:
3532:
3528:
848: − 1)th power of the geometric mean in the denominator simplifies the
85:
The power transformation is defined as a continuous function of power parameter
69:, statistical data analysis, medical research, modeling of physical processes,
4661:
4576:
4492:
4479:
4075:
1709:
1604:
produces an expression that establishes that minimizing the sum of squares of
775:{\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}
31:
4126:
4083:
4005:
Sixth
International Conference on Intelligent Systems Design and Applications
4608:
4189:; Doksum, Kjell A. (June 1981). "An analysis of transformations revisited".
3536:
4736:
3547:
The Yeo–Johnson transformation allows also for zero and negative values of
4012:
4644:
4058:
4269:(slide presentation), Sao Paulo, Brazil: University of Sao Paulo, Brazil
3126:
4690:
4540:
4500:
4309:
4247:
4216:
Sakia, R. M. (1992), "The Box–Cox transformation technique: a review",
4118:
4440:
Zarembka, P. (1974). "Transformation of
Variables in Econometrics".
4239:
3613:
produces the identity transformation. The transformation law reads:
58:-like, improve the validity of measures of association (such as the
2363:. This change in definition has little practical import as long as
4373:
77:
and many other clinical, environmental and social research areas.
4679:
Journal of the
International Association for Mathematical Geology
4107:
Journal of the
International Association for Mathematical Geology
62:
between variables), and for other data stabilization procedures.
3519:
3248:{\displaystyle \tau (Q)=\alpha \tau (K)+(1-\alpha )\tau (N).\,}
2425:
Bickel and Doksum also proved that the parameter estimates are
2433:
under appropriate regularity conditions, though the standard
1208:
3972:
2803:
2581:
2342:
1995:
306:
2931:
Confidence interval for the Box–Cox transformation can be
4361:
Communications in
Statistics - Simulation and Computation
4338:(Third ed.). New York: McGraw-Hill. pp. 61–74.
2446:
The one-parameter Box–Cox transformations are defined as
65:
Power transforms are used in multiple fields, including
3114:
The BUPA liver data set contains data on liver enzymes
1561:{\displaystyle \operatorname {GM} (y)^{2(\lambda -1)}}
3622:
3593:
3573:
3553:
3463:
3395:
3271:
3180:
3163:
as dependent on services provided by a capital stock
2972:
2949:
2905:
2859:
2826:
2643:
2607:
2489:
2455:
2389:
2369:
2105:
2061:
1811:
1767:
1735:
transformation, but done in such a way as to make it
1656:
1614:
1574:
1520:
1192:
967:
916:
857:
827:
788:
701:
699:
537:
517:
491:
332:
161:
127:
2048:
by extending the range of the transformation to all
889:, because the units of measurement do not change as
4558:"On prediction and the power transformation family"
4181:
4179:
3386:= 1, this produces the linear production function:
1724:is a version of some other variable scaled to give
1697:and the log of the Jacobian of the transformation.
958:with the likelihood. This Jacobian is as follows:
948:{\displaystyle {\frac {y^{\lambda }-1}{\lambda }}.}
850:
scientific interpretation of any equation involving
4528:Journal of the Royal Statistical Society, Series B
4298:Journal of the Royal Statistical Society, Series B
4150:Journal of the Royal Statistical Society, Series C
3978:
3605:
3579:
3559:
3502:
3435:
3354:
3247:
3098:
2955:
2911:
2888:
2845:
2809:
2587:
2414:
2375:
2348:
2001:
1689:
1646:is equivalent to maximizing the sum of the normal
1638:
1596:
1560:
1503:
1168:
947:
881:
833:
813:
774:
681:
523:
503:
454:
312:
2598:and the two-parameter Box–Cox transformations as
3262:by inverting the Box–Cox transformation we find
4191:Journal of the American Statistical Association
3159:Consider a common representation of production
2044:Bickel and Doksum eliminated the need to use a
4444:. New York: Academic Press. pp. 81–104.
3503:{\displaystyle Q=K^{\alpha }N^{1-\alpha }.\,}
3329:
3280:
3044:
3016:
3000:
2981:
38:is a family of functions applied to create a
8:
2943:function to find all the possible values of
2415:{\displaystyle \operatorname {min} (y_{i})}
4525:(1964). "An analysis of transformations".
4406:Statistical Theory: A Concise Introduction
4404:Abramovich, Felix; Ritov, Ya'acov (2013).
4296:(1964). "An analysis of transformations".
3436:{\displaystyle Q=\alpha K+(1-\alpha )N.\,}
3140:data with a superimposed regression line.
2923:function and using goodness-of-fit tests.
118: > 0, the power transform is
4643:
4607:
4372:
4229:
4057:
3943:
3926:
3877:
3855:
3828:
3812:
3766:
3749:
3706:
3696:
3681:
3665:
3647:
3632:
3627:
3621:
3592:
3572:
3552:
3499:
3484:
3474:
3462:
3432:
3394:
3351:
3338:
3334:
3328:
3327:
3320:
3292:
3279:
3278:
3270:
3244:
3179:
3075:
3068:
3063:
3052:
3043:
3042:
3028:
3027:
3015:
3014:
2999:
2998:
2980:
2979:
2971:
2948:
2904:
2880:
2864:
2858:
2831:
2825:
2785:
2776:
2765:
2752:
2720:
2711:
2700:
2681:
2676:
2666:
2653:
2642:
2634:
2621:
2617:
2612:
2606:
2561:
2553:
2522:
2501:
2496:
2488:
2480:
2465:
2460:
2454:
2403:
2388:
2368:
2322:
2305:
2226:
2208:
2163:
2158:
2145:
2136:
2121:
2104:
2096:
2072:
2060:
1975:
1958:
1900:
1882:
1837:
1821:
1810:
1802:
1778:
1766:
1690:{\displaystyle (y^{\lambda }-1)/\lambda }
1679:
1664:
1655:
1624:
1619:
1613:
1588:
1577:
1576:
1573:
1537:
1519:
1461:
1440:
1434:
1423:
1422:
1392:
1307:
1296:
1295:
1265:
1232:
1231:
1217:
1216:
1207:
1206:
1193:
1191:
1145:
1114:
1109:
1099:
1088:
1076:
1065:
1053:
1048:
1036:
1030:
1019:
1003:
984:
966:
924:
917:
915:
867:
862:
856:
826:
802:
787:
748:
713:
708:
700:
698:
670:
657:
620:
582:
565:
547:
542:
536:
516:
490:
451:
444:
437:
424:
414:
407:
393:
382:
372:
361:
331:
289:
277:
272:
233:
212:
173:
168:
160:
152:
137:
132:
126:
4284:
4282:
3148:approach leads to a more precise model.
2963:that fulfill the following restriction:
3995:
2622:
2889:{\displaystyle y_{i}>-\lambda _{2}}
531:approaches 0. To see this, note that
67:multi-resolution and wavelet analysis
27:Family of functions to transform data
7:
4556:Carroll, R. J.; Ruppert, D. (1981).
4263:Box–Cox Transformations: An Overview
1597:{\displaystyle {\hat {\sigma }}^{2}}
3368:constant elasticity of substitution
2021: + α > 0 for all
1728:= 1 at some sort of average value.
93: = 0). For data vectors (
1639:{\displaystyle y_{i}^{(\lambda )}}
882:{\displaystyle y_{i}^{(\lambda )}}
25:
4737:Yeo-Johnson Power Transformations
4632:Journal of Mathematical Chemistry
4046:Journal of Mathematical Chemistry
3375:The CES production function is a
907:of rescaled power transformation
4589:"A Conversation with George Box"
4162:10.1111/j.1467-9876.2005.00476.x
3125:
4757:Statistical data transformation
4408:. CRC Press. pp. 121–122.
4260:Li, Fengfei (April 11, 2005),
4203:10.1080/01621459.1981.10477649
3938:
3916:
3872:
3860:
3852:
3841:
3829:
3825:
3802:
3799:
3761:
3742:
3693:
3678:
3658:
3655:
3639:
3633:
3587:can be any real number, where
3423:
3411:
3313:
3301:
3238:
3232:
3226:
3214:
3208:
3202:
3190:
3184:
3039:
3033:
3024:
2995:
2989:
2771:
2745:
2673:
2646:
2626:
2618:
2472:
2466:
2409:
2396:
2317:
2298:
2289:
2277:
2268:
2256:
2205:
2201:
2189:
2180:
2159:
2137:
2133:
2114:
2090:
2065:
1970:
1951:
1942:
1930:
1879:
1875:
1863:
1854:
1834:
1814:
1796:
1771:
1747:= 0. It has proved popular in
1676:
1657:
1631:
1625:
1582:
1553:
1541:
1534:
1527:
1491:
1482:
1477:
1465:
1458:
1451:
1428:
1412:
1403:
1400:
1383:
1370:
1367:
1361:
1352:
1343:
1331:
1322:
1313:
1301:
1285:
1276:
1273:
1256:
1246:
1243:
1237:
1222:
1213:
1203:
1161:
1149:
1142:
1135:
1077:
1060:
1054:
1037:
1009:
971:
874:
868:
808:
795:
769:
763:
754:
741:
676:
667:
663:
650:
638:
635:
626:
613:
592:
588:
575:
562:
345:
339:
263:
257:
209:
205:
199:
190:
144:
138:
1:
4465:Power Transform Family Graphs
3514:Activities and demonstrations
3450:→ 0 this produces the famous
2031:truncated normal distribution
1181:log likelihood at its maximum
4383:10.1080/03610918.2014.957839
4720:Encyclopedia of Mathematics
4429:BUPA liver disorder dataset
4773:
3606:{\displaystyle \lambda =1}
3543:Yeo–Johnson transformation
2933:asymptotically constructed
2846:{\displaystyle y_{i}>0}
1183:to be written as follows:
814:{\displaystyle \ln(y_{i})}
504:{\displaystyle \lambda =0}
4662:10.1007/s10910-005-9003-7
4442:Frontiers in Econometrics
4076:10.1007/s10910-005-9003-7
71:geochemical data analysis
4715:"Box–Cox transformation"
3580:{\displaystyle \lambda }
2956:{\displaystyle \lambda }
2912:{\displaystyle \lambda }
1731:The transformation is a
1568:into the expression for
834:{\displaystyle \lambda }
524:{\displaystyle \lambda }
40:monotonic transformation
4577:10.1093/biomet/68.3.609
4493:10.1093/biomet/87.4.954
3152:Econometric application
2919:is estimated using the
2422:, which it usually is.
2376:{\displaystyle \alpha }
1179:This allows the normal
821:becomes negligible for
3980:
3607:
3581:
3561:
3504:
3437:
3365:which is known as the
3356:
3249:
3100:
2957:
2913:
2890:
2847:
2811:
2589:
2442:Box–Cox transformation
2435:Cramér–Rao lower bound
2416:
2377:
2350:
2046:truncated distribution
2003:
1691:
1640:
1598:
1562:
1505:
1170:
1104:
1035:
949:
883:
844:The inclusion of the (
835:
815:
776:
683:
525:
505:
456:
377:
314:
18:Box–Cox transformation
4609:10.1214/ss/1177013223
4013:10.1109/ISDA.2006.217
3981:
3608:
3582:
3562:
3505:
3454:production function:
3438:
3372:production function.
3357:
3250:
3101:
2958:
2914:
2891:
2853:, and the second for
2848:
2812:
2590:
2431:asymptotically normal
2417:
2378:
2351:
2004:
1692:
1641:
1599:
1563:
1514:From here, absorbing
1506:
1171:
1084:
1015:
950:
884:
836:
816:
782:, and everything but
777:
684:
526:
506:
457:
357:
315:
54:, make the data more
4713:Nishii, R. (2001) ,
4619:Journal of Andrology
3620:
3591:
3571:
3551:
3461:
3393:
3377:homogeneous function
3269:
3178:
2970:
2947:
2903:
2857:
2824:
2605:
2453:
2387:
2367:
2359:where sgn(.) is the
2059:
2039:Box–Cox distribution
2037:is said to follow a
1765:
1654:
1612:
1572:
1518:
1190:
965:
914:
855:
841:sufficiently small.
825:
786:
697:
535:
515:
489:
469:of the observations
330:
125:
4752:Normal distribution
4654:2006cond.mat..6104G
4596:Statistical Science
4336:Econometric Methods
4068:2006cond.mat..6104G
3643:
3167:and by labor hours
2927:Confidence interval
2630:
2506:
2476:
1749:regression analysis
1739:with the parameter
1716:there is 1 for any
1650:of deviations from
1635:
1125:
1064:
878:
718:
552:
178:
148:
60:Pearson correlation
56:normal distribution
48:data transformation
4735:Sanford Weisberg,
4691:10.1007/BF01043245
4119:10.1007/BF01043245
3976:
3971:
3623:
3603:
3577:
3557:
3500:
3433:
3352:
3245:
3096:
2953:
2941:profile likelihood
2921:profile likelihood
2909:
2886:
2843:
2807:
2802:
2707:
2608:
2585:
2580:
2518:
2492:
2456:
2412:
2373:
2346:
2341:
2222:
2029:, λ, α) follows a
1999:
1994:
1896:
1687:
1636:
1615:
1594:
1558:
1501:
1499:
1166:
1105:
1044:
945:
879:
858:
831:
811:
772:
730:
704:
679:
538:
521:
501:
452:
310:
305:
226:
164:
128:
52:stabilize variance
50:technique used to
4519:Box, George E. P.
4415:978-1-4398-5184-5
4345:978-0-07-032685-9
4290:Box, George E. P.
3946:
3880:
3769:
3709:
3560:{\displaystyle y}
3146:robust regression
3060:
3036:
2779:
2714:
2706:
2564:
2525:
2517:
2325:
2229:
2221:
1978:
1903:
1895:
1585:
1431:
1304:
1240:
1225:
940:
729:
476:, ...,
449:
401:
292:
236:
225:
16:(Redirected from
4764:
4727:
4702:
4673:
4647:
4645:cond-mat/0606104
4626:
4613:
4611:
4593:
4580:
4562:
4552:
4505:
4504:
4474:
4468:
4462:
4456:
4455:
4437:
4431:
4426:
4420:
4419:
4401:
4395:
4394:
4376:
4356:
4350:
4349:
4328:
4322:
4321:
4286:
4277:
4276:
4275:
4274:
4268:
4257:
4251:
4250:
4233:
4218:The Statistician
4213:
4207:
4206:
4197:(374): 296–311.
4187:Bickel, Peter J.
4183:
4174:
4173:
4145:
4139:
4138:
4102:
4096:
4095:
4061:
4059:cond-mat/0606104
4041:
4035:
4034:
4000:
3985:
3983:
3982:
3977:
3975:
3974:
3947:
3944:
3931:
3930:
3881:
3878:
3859:
3845:
3844:
3817:
3816:
3770:
3767:
3754:
3753:
3710:
3707:
3700:
3686:
3685:
3670:
3669:
3642:
3631:
3612:
3610:
3609:
3604:
3586:
3584:
3583:
3578:
3566:
3564:
3563:
3558:
3509:
3507:
3506:
3501:
3495:
3494:
3479:
3478:
3442:
3440:
3439:
3434:
3361:
3359:
3358:
3353:
3347:
3346:
3342:
3333:
3332:
3325:
3324:
3297:
3296:
3284:
3283:
3254:
3252:
3251:
3246:
3129:
3105:
3103:
3102:
3097:
3092:
3091:
3074:
3073:
3072:
3061:
3053:
3048:
3047:
3038:
3037:
3029:
3020:
3019:
3004:
3003:
2985:
2984:
2962:
2960:
2959:
2954:
2918:
2916:
2915:
2910:
2895:
2893:
2892:
2887:
2885:
2884:
2869:
2868:
2852:
2850:
2849:
2844:
2836:
2835:
2816:
2814:
2813:
2808:
2806:
2805:
2790:
2789:
2780:
2777:
2770:
2769:
2757:
2756:
2725:
2724:
2715:
2712:
2708:
2705:
2704:
2695:
2688:
2687:
2686:
2685:
2671:
2670:
2658:
2657:
2644:
2629:
2625:
2616:
2594:
2592:
2591:
2586:
2584:
2583:
2565:
2562:
2558:
2557:
2526:
2523:
2519:
2513:
2505:
2500:
2490:
2475:
2464:
2421:
2419:
2418:
2413:
2408:
2407:
2382:
2380:
2379:
2374:
2355:
2353:
2352:
2347:
2345:
2344:
2326:
2323:
2310:
2309:
2246:
2230:
2227:
2223:
2220:
2219:
2218:
2175:
2168:
2167:
2162:
2150:
2149:
2140:
2126:
2125:
2106:
2077:
2076:
2008:
2006:
2005:
2000:
1998:
1997:
1979:
1976:
1963:
1962:
1920:
1904:
1901:
1897:
1894:
1893:
1892:
1849:
1842:
1841:
1826:
1825:
1812:
1783:
1782:
1712:with respect to
1696:
1694:
1693:
1688:
1683:
1669:
1668:
1645:
1643:
1642:
1637:
1634:
1623:
1603:
1601:
1600:
1595:
1593:
1592:
1587:
1586:
1578:
1567:
1565:
1564:
1559:
1557:
1556:
1510:
1508:
1507:
1502:
1500:
1481:
1480:
1444:
1439:
1438:
1433:
1432:
1424:
1396:
1376:
1312:
1311:
1306:
1305:
1297:
1269:
1242:
1241:
1233:
1227:
1226:
1218:
1212:
1211:
1175:
1173:
1172:
1167:
1165:
1164:
1124:
1113:
1103:
1098:
1080:
1069:
1063:
1052:
1040:
1034:
1029:
1008:
1007:
989:
988:
954:
952:
951:
946:
941:
936:
929:
928:
918:
888:
886:
885:
880:
877:
866:
840:
838:
837:
832:
820:
818:
817:
812:
807:
806:
781:
779:
778:
773:
753:
752:
731:
725:
717:
712:
702:
688:
686:
685:
680:
675:
674:
662:
661:
625:
624:
591:
587:
586:
551:
546:
530:
528:
527:
522:
511:is the limit as
510:
508:
507:
502:
485:. The case for
461:
459:
458:
453:
450:
448:
443:
442:
441:
429:
428:
419:
418:
408:
403:
402:
394:
392:
388:
387:
386:
376:
371:
319:
317:
316:
311:
309:
308:
293:
290:
283:
282:
281:
237:
234:
227:
224:
223:
222:
185:
177:
172:
162:
147:
136:
109:) in which each
21:
4772:
4771:
4767:
4766:
4765:
4763:
4762:
4761:
4742:
4741:
4712:
4709:
4676:
4629:
4616:
4591:
4583:
4560:
4555:
4517:
4514:
4509:
4508:
4476:
4475:
4471:
4467:, SOCR webpages
4463:
4459:
4452:
4439:
4438:
4434:
4427:
4423:
4416:
4403:
4402:
4398:
4358:
4357:
4353:
4346:
4330:
4329:
4325:
4288:
4287:
4280:
4272:
4270:
4266:
4259:
4258:
4254:
4240:10.2307/2348250
4231:10.1.1.469.7176
4215:
4214:
4210:
4185:
4184:
4177:
4147:
4146:
4142:
4104:
4103:
4099:
4043:
4042:
4038:
4023:
4002:
4001:
3997:
3992:
3970:
3969:
3941:
3922:
3904:
3903:
3875:
3824:
3808:
3793:
3792:
3764:
3745:
3733:
3732:
3704:
3677:
3661:
3648:
3618:
3617:
3589:
3588:
3569:
3568:
3549:
3548:
3545:
3516:
3480:
3470:
3459:
3458:
3391:
3390:
3379:of degree one.
3326:
3316:
3288:
3267:
3266:
3176:
3175:
3154:
3135:
3112:
3064:
3062:
2968:
2967:
2945:
2944:
2937:Wilks's theorem
2929:
2901:
2900:
2876:
2860:
2855:
2854:
2827:
2822:
2821:
2801:
2800:
2781:
2774:
2761:
2748:
2736:
2735:
2716:
2709:
2696:
2677:
2672:
2662:
2649:
2645:
2635:
2603:
2602:
2579:
2578:
2559:
2549:
2540:
2539:
2520:
2491:
2481:
2451:
2450:
2444:
2399:
2385:
2384:
2365:
2364:
2340:
2339:
2320:
2301:
2247:
2244:
2243:
2224:
2204:
2176:
2157:
2141:
2117:
2107:
2097:
2068:
2057:
2056:
2020:
2012:which holds if
1993:
1992:
1973:
1954:
1921:
1918:
1917:
1898:
1878:
1850:
1833:
1817:
1813:
1803:
1774:
1763:
1762:
1660:
1652:
1651:
1610:
1609:
1575:
1570:
1569:
1533:
1516:
1515:
1498:
1497:
1457:
1421:
1374:
1373:
1294:
1249:
1188:
1187:
1141:
999:
980:
963:
962:
920:
919:
912:
911:
853:
852:
823:
822:
798:
784:
783:
744:
703:
695:
694:
666:
653:
616:
578:
533:
532:
513:
512:
487:
486:
484:
475:
433:
420:
410:
409:
378:
356:
352:
351:
328:
327:
304:
303:
287:
273:
248:
247:
231:
208:
186:
163:
153:
123:
122:
117:
108:
99:
83:
44:power functions
36:power transform
28:
23:
22:
15:
12:
11:
5:
4770:
4768:
4760:
4759:
4754:
4744:
4743:
4740:
4739:
4733:
4708:
4707:External links
4705:
4704:
4703:
4674:
4627:
4614:
4602:(3): 239–258.
4585:DeGroot, M. H.
4581:
4571:(3): 609–615.
4553:
4535:(2): 211–252.
4513:
4510:
4507:
4506:
4487:(4): 954–959.
4469:
4457:
4450:
4432:
4421:
4414:
4396:
4351:
4344:
4323:
4304:(2): 211–252.
4278:
4252:
4224:(2): 169–178,
4208:
4175:
4140:
4097:
4036:
4021:
3994:
3993:
3991:
3988:
3987:
3986:
3973:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3942:
3940:
3937:
3934:
3929:
3925:
3921:
3918:
3915:
3912:
3909:
3906:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3876:
3874:
3871:
3868:
3865:
3862:
3858:
3854:
3851:
3848:
3843:
3840:
3837:
3834:
3831:
3827:
3823:
3820:
3815:
3811:
3807:
3804:
3801:
3798:
3795:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3765:
3763:
3760:
3757:
3752:
3748:
3744:
3741:
3738:
3735:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3705:
3703:
3699:
3695:
3692:
3689:
3684:
3680:
3676:
3673:
3668:
3664:
3660:
3657:
3654:
3653:
3651:
3646:
3641:
3638:
3635:
3630:
3626:
3602:
3599:
3596:
3576:
3556:
3544:
3541:
3515:
3512:
3511:
3510:
3498:
3493:
3490:
3487:
3483:
3477:
3473:
3469:
3466:
3444:
3443:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3363:
3362:
3350:
3345:
3341:
3337:
3331:
3323:
3319:
3315:
3312:
3309:
3306:
3303:
3300:
3295:
3291:
3287:
3282:
3277:
3274:
3256:
3255:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3153:
3150:
3133:
3111:
3108:
3107:
3106:
3095:
3090:
3087:
3084:
3081:
3078:
3071:
3067:
3059:
3056:
3051:
3046:
3041:
3035:
3032:
3026:
3023:
3018:
3013:
3010:
3007:
3002:
2997:
2994:
2991:
2988:
2983:
2978:
2975:
2952:
2928:
2925:
2908:
2899:The parameter
2883:
2879:
2875:
2872:
2867:
2863:
2842:
2839:
2834:
2830:
2818:
2817:
2804:
2799:
2796:
2793:
2788:
2784:
2775:
2773:
2768:
2764:
2760:
2755:
2751:
2747:
2744:
2741:
2738:
2737:
2734:
2731:
2728:
2723:
2719:
2710:
2703:
2699:
2694:
2691:
2684:
2680:
2675:
2669:
2665:
2661:
2656:
2652:
2648:
2641:
2640:
2638:
2633:
2628:
2624:
2620:
2615:
2611:
2596:
2595:
2582:
2577:
2574:
2571:
2568:
2560:
2556:
2552:
2548:
2545:
2542:
2541:
2538:
2535:
2532:
2529:
2521:
2516:
2512:
2509:
2504:
2499:
2495:
2487:
2486:
2484:
2479:
2474:
2471:
2468:
2463:
2459:
2443:
2440:
2411:
2406:
2402:
2398:
2395:
2392:
2372:
2357:
2356:
2343:
2338:
2335:
2332:
2329:
2321:
2319:
2316:
2313:
2308:
2304:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2248:
2245:
2242:
2239:
2236:
2233:
2225:
2217:
2214:
2211:
2207:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2174:
2171:
2166:
2161:
2156:
2153:
2148:
2144:
2139:
2135:
2132:
2129:
2124:
2120:
2116:
2113:
2110:
2103:
2102:
2100:
2095:
2092:
2089:
2086:
2083:
2080:
2075:
2071:
2067:
2064:
2052:, as follows:
2016:
2010:
2009:
1996:
1991:
1988:
1985:
1982:
1974:
1972:
1969:
1966:
1961:
1957:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1922:
1919:
1916:
1913:
1910:
1907:
1899:
1891:
1888:
1885:
1881:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1848:
1845:
1840:
1836:
1832:
1829:
1824:
1820:
1816:
1809:
1808:
1806:
1801:
1798:
1795:
1792:
1789:
1786:
1781:
1777:
1773:
1770:
1708:is 0, and the
1686:
1682:
1678:
1675:
1672:
1667:
1663:
1659:
1648:log likelihood
1633:
1630:
1627:
1622:
1618:
1591:
1584:
1581:
1555:
1552:
1549:
1546:
1543:
1540:
1536:
1532:
1529:
1526:
1523:
1512:
1511:
1496:
1493:
1490:
1487:
1484:
1479:
1476:
1473:
1470:
1467:
1464:
1460:
1456:
1453:
1450:
1447:
1443:
1437:
1430:
1427:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1395:
1391:
1388:
1385:
1382:
1379:
1377:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1310:
1303:
1300:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1268:
1264:
1261:
1258:
1255:
1252:
1250:
1248:
1245:
1239:
1236:
1230:
1224:
1221:
1215:
1210:
1205:
1202:
1199:
1196:
1195:
1177:
1176:
1163:
1160:
1157:
1154:
1151:
1148:
1144:
1140:
1137:
1134:
1131:
1128:
1123:
1120:
1117:
1112:
1108:
1102:
1097:
1094:
1091:
1087:
1083:
1079:
1075:
1072:
1068:
1062:
1059:
1056:
1051:
1047:
1043:
1039:
1033:
1028:
1025:
1022:
1018:
1014:
1011:
1006:
1002:
998:
995:
992:
987:
983:
979:
976:
973:
970:
956:
955:
944:
939:
935:
932:
927:
923:
876:
873:
870:
865:
861:
830:
810:
805:
801:
797:
794:
791:
771:
768:
765:
762:
759:
756:
751:
747:
743:
740:
737:
734:
728:
724:
721:
716:
711:
707:
678:
673:
669:
665:
660:
656:
652:
649:
646:
643:
640:
637:
634:
631:
628:
623:
619:
615:
612:
609:
606:
603:
600:
597:
594:
590:
585:
581:
577:
574:
571:
568:
564:
561:
558:
555:
550:
545:
541:
520:
500:
497:
494:
480:
473:
467:geometric mean
463:
462:
447:
440:
436:
432:
427:
423:
417:
413:
406:
400:
397:
391:
385:
381:
375:
370:
367:
364:
360:
355:
350:
347:
344:
341:
338:
335:
321:
320:
307:
302:
299:
296:
288:
286:
280:
276:
271:
268:
265:
262:
259:
256:
253:
250:
249:
246:
243:
240:
232:
230:
221:
218:
215:
211:
207:
204:
201:
198:
195:
192:
189:
184:
181:
176:
171:
167:
159:
158:
156:
151:
146:
143:
140:
135:
131:
113:
104:
97:
82:
79:
42:of data using
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4769:
4758:
4755:
4753:
4750:
4749:
4747:
4738:
4734:
4731:
4726:
4722:
4721:
4716:
4711:
4710:
4706:
4700:
4696:
4692:
4688:
4684:
4680:
4675:
4671:
4667:
4663:
4659:
4655:
4651:
4646:
4641:
4637:
4633:
4628:
4624:
4620:
4615:
4610:
4605:
4601:
4597:
4590:
4586:
4582:
4578:
4574:
4570:
4566:
4559:
4554:
4550:
4546:
4542:
4538:
4534:
4530:
4529:
4524:
4520:
4516:
4515:
4511:
4502:
4498:
4494:
4490:
4486:
4482:
4481:
4473:
4470:
4466:
4461:
4458:
4453:
4451:0-12-776150-0
4447:
4443:
4436:
4433:
4430:
4425:
4422:
4417:
4411:
4407:
4400:
4397:
4392:
4388:
4384:
4380:
4375:
4370:
4367:(1): 91–105.
4366:
4362:
4355:
4352:
4347:
4341:
4337:
4333:
4327:
4324:
4319:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4285:
4283:
4279:
4265:
4264:
4256:
4253:
4249:
4245:
4241:
4237:
4232:
4227:
4223:
4219:
4212:
4209:
4204:
4200:
4196:
4192:
4188:
4182:
4180:
4176:
4171:
4167:
4163:
4159:
4155:
4151:
4144:
4141:
4136:
4132:
4128:
4124:
4120:
4116:
4112:
4108:
4101:
4098:
4093:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4060:
4055:
4051:
4047:
4040:
4037:
4032:
4028:
4024:
4022:9780769525280
4018:
4014:
4010:
4006:
3999:
3996:
3989:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3935:
3932:
3927:
3923:
3919:
3913:
3910:
3907:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3869:
3866:
3863:
3856:
3849:
3846:
3838:
3835:
3832:
3821:
3818:
3813:
3809:
3805:
3796:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3758:
3755:
3750:
3746:
3739:
3736:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3701:
3697:
3690:
3687:
3682:
3674:
3671:
3666:
3662:
3649:
3644:
3636:
3628:
3624:
3616:
3615:
3614:
3600:
3597:
3594:
3574:
3554:
3542:
3540:
3538:
3534:
3530:
3526:
3521:
3513:
3496:
3491:
3488:
3485:
3481:
3475:
3471:
3467:
3464:
3457:
3456:
3455:
3453:
3449:
3429:
3426:
3420:
3417:
3414:
3408:
3405:
3402:
3399:
3396:
3389:
3388:
3387:
3385:
3380:
3378:
3373:
3371:
3369:
3348:
3343:
3339:
3335:
3321:
3317:
3310:
3307:
3304:
3298:
3293:
3289:
3285:
3275:
3272:
3265:
3264:
3263:
3261:
3241:
3235:
3229:
3223:
3220:
3217:
3211:
3205:
3199:
3196:
3193:
3187:
3181:
3174:
3173:
3172:
3170:
3166:
3162:
3157:
3151:
3149:
3147:
3141:
3137:
3130:
3128:
3123:
3121:
3117:
3109:
3093:
3088:
3085:
3082:
3079:
3076:
3069:
3065:
3057:
3054:
3049:
3030:
3021:
3011:
3008:
3005:
2992:
2986:
2976:
2973:
2966:
2965:
2964:
2950:
2942:
2938:
2934:
2926:
2924:
2922:
2906:
2897:
2881:
2877:
2873:
2870:
2865:
2861:
2840:
2837:
2832:
2828:
2797:
2794:
2791:
2786:
2782:
2766:
2762:
2758:
2753:
2749:
2742:
2739:
2732:
2729:
2726:
2721:
2717:
2701:
2697:
2692:
2689:
2682:
2678:
2667:
2663:
2659:
2654:
2650:
2636:
2631:
2613:
2609:
2601:
2600:
2599:
2575:
2572:
2569:
2566:
2554:
2550:
2546:
2543:
2536:
2533:
2530:
2527:
2514:
2510:
2507:
2502:
2497:
2493:
2482:
2477:
2469:
2461:
2457:
2449:
2448:
2447:
2441:
2439:
2436:
2432:
2428:
2423:
2404:
2400:
2393:
2390:
2383:is less than
2370:
2362:
2361:sign function
2336:
2333:
2330:
2327:
2314:
2311:
2306:
2302:
2295:
2292:
2286:
2283:
2280:
2274:
2271:
2265:
2262:
2259:
2253:
2250:
2240:
2237:
2234:
2231:
2215:
2212:
2209:
2198:
2195:
2192:
2186:
2183:
2177:
2172:
2169:
2164:
2154:
2151:
2146:
2142:
2130:
2127:
2122:
2118:
2111:
2108:
2098:
2093:
2087:
2084:
2081:
2078:
2073:
2069:
2062:
2055:
2054:
2053:
2051:
2047:
2042:
2040:
2036:
2032:
2028:
2024:
2019:
2015:
1989:
1986:
1983:
1980:
1967:
1964:
1959:
1955:
1948:
1945:
1939:
1936:
1933:
1927:
1924:
1914:
1911:
1908:
1905:
1889:
1886:
1883:
1872:
1869:
1866:
1860:
1857:
1851:
1846:
1843:
1838:
1830:
1827:
1822:
1818:
1804:
1799:
1793:
1790:
1787:
1784:
1779:
1775:
1768:
1761:
1760:
1759:
1756:
1754:
1750:
1746:
1742:
1738:
1734:
1729:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1700:The value at
1698:
1684:
1680:
1673:
1670:
1665:
1661:
1649:
1628:
1620:
1616:
1607:
1589:
1579:
1550:
1547:
1544:
1538:
1530:
1524:
1521:
1494:
1488:
1485:
1474:
1471:
1468:
1462:
1454:
1448:
1445:
1441:
1435:
1425:
1418:
1415:
1409:
1406:
1397:
1393:
1389:
1386:
1380:
1378:
1364:
1358:
1355:
1349:
1346:
1340:
1337:
1334:
1328:
1325:
1319:
1316:
1308:
1298:
1291:
1288:
1282:
1279:
1270:
1266:
1262:
1259:
1253:
1251:
1234:
1228:
1219:
1200:
1197:
1186:
1185:
1184:
1182:
1158:
1155:
1152:
1146:
1138:
1132:
1129:
1126:
1121:
1118:
1115:
1110:
1106:
1100:
1095:
1092:
1089:
1085:
1081:
1073:
1070:
1066:
1057:
1049:
1045:
1041:
1031:
1026:
1023:
1020:
1016:
1012:
1004:
1000:
996:
993:
990:
985:
981:
977:
974:
968:
961:
960:
959:
942:
937:
933:
930:
925:
921:
910:
909:
908:
906:
902:
898:
894:
892:
871:
863:
859:
851:
847:
842:
828:
803:
799:
792:
789:
766:
760:
757:
749:
745:
738:
735:
732:
726:
722:
719:
714:
709:
705:
692:
691:Taylor series
671:
658:
654:
647:
644:
641:
632:
629:
621:
617:
610:
607:
604:
601:
598:
595:
583:
579:
572:
569:
566:
559:
556:
553:
548:
543:
539:
518:
498:
495:
492:
483:
479:
472:
468:
445:
438:
434:
430:
425:
421:
415:
411:
404:
398:
395:
389:
383:
379:
373:
368:
365:
362:
358:
353:
348:
342:
336:
333:
326:
325:
324:
300:
297:
294:
284:
278:
274:
269:
266:
260:
254:
251:
244:
241:
238:
228:
219:
216:
213:
202:
196:
193:
187:
182:
179:
174:
169:
165:
154:
149:
141:
133:
129:
121:
120:
119:
116:
112:
107:
103:
96:
92:
88:
80:
78:
76:
72:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4718:
4685:(1): 45–62.
4682:
4678:
4638:(1): 47–56.
4635:
4631:
4622:
4618:
4599:
4595:
4568:
4564:
4532:
4526:
4484:
4478:
4472:
4460:
4441:
4435:
4424:
4405:
4399:
4364:
4360:
4354:
4335:
4332:Johnston, J.
4326:
4301:
4297:
4271:, retrieved
4262:
4255:
4221:
4217:
4211:
4194:
4190:
4153:
4149:
4143:
4113:(1): 45–62.
4110:
4106:
4100:
4052:(1): 47–56.
4049:
4045:
4039:
4004:
3998:
3546:
3529:scatterplots
3517:
3452:Cobb–Douglas
3447:
3445:
3383:
3381:
3374:
3366:
3364:
3259:
3258:Solving for
3257:
3168:
3164:
3160:
3158:
3155:
3142:
3138:
3131:
3124:
3113:
2930:
2898:
2819:
2597:
2445:
2424:
2358:
2049:
2043:
2034:
2026:
2022:
2017:
2013:
2011:
1757:
1753:econometrics
1751:, including
1744:
1740:
1730:
1725:
1721:
1720:. Sometimes
1717:
1713:
1705:
1704:= 1 for any
1701:
1699:
1513:
1178:
957:
895:
890:
845:
843:
481:
477:
470:
464:
322:
114:
110:
105:
101:
94:
90:
86:
84:
75:epidemiology
64:
35:
29:
4156:: 159–172.
3533:time-series
100:,...,
4746:Categories
4730:fixed link
4565:Biometrika
4523:Cox, D. R.
4512:References
4480:Biometrika
4294:Cox, D. R.
4273:2014-11-02
3537:histograms
3535:plots and
2427:consistent
1737:continuous
1710:derivative
81:Definition
46:. It is a
32:statistics
4725:EMS Press
4699:121582755
4670:118965098
4374:1401.3812
4226:CiteSeerX
4170:121909404
4135:121582755
4127:1573-8868
4092:118965098
4084:1572-8897
3949:λ
3920:−
3914:
3908:−
3886:≠
3883:λ
3870:λ
3867:−
3847:−
3839:λ
3836:−
3806:−
3797:−
3787:≥
3772:λ
3740:
3727:≥
3715:≠
3712:λ
3702:λ
3688:−
3683:λ
3637:λ
3595:λ
3575:λ
3525:Q–Q plots
3492:α
3489:−
3476:α
3421:α
3418:−
3403:α
3344:λ
3322:λ
3311:α
3308:−
3294:λ
3286:α
3230:τ
3224:α
3221:−
3200:τ
3197:α
3182:τ
3089:α
3086:−
3066:χ
3050:−
3034:^
3031:λ
3012:
3006:≥
2993:λ
2977:
2951:λ
2907:λ
2878:λ
2874:−
2783:λ
2763:λ
2743:
2727:≠
2718:λ
2698:λ
2690:−
2679:λ
2664:λ
2623:λ
2567:λ
2547:
2531:≠
2528:λ
2515:λ
2508:−
2503:λ
2470:λ
2394:
2371:α
2328:λ
2315:α
2296:
2287:α
2275:
2266:α
2254:
2235:≠
2232:λ
2213:−
2210:λ
2199:α
2187:
2178:λ
2170:−
2165:λ
2155:α
2131:α
2112:
2088:α
2082:λ
2063:τ
1981:λ
1968:α
1949:
1940:α
1928:
1909:≠
1906:λ
1887:−
1884:λ
1873:α
1861:
1852:λ
1844:−
1839:λ
1831:α
1794:α
1788:λ
1769:τ
1685:λ
1671:−
1666:λ
1629:λ
1606:residuals
1583:^
1580:σ
1548:−
1545:λ
1525:
1472:−
1469:λ
1449:
1429:^
1426:σ
1419:π
1410:
1387:−
1359:
1350:
1338:−
1335:λ
1302:^
1299:σ
1292:π
1283:
1260:−
1238:^
1235:σ
1223:^
1220:μ
1201:
1156:−
1153:λ
1133:
1119:−
1116:λ
1086:∏
1058:λ
1017:∏
994:…
975:λ
938:λ
931:−
926:λ
893:changes.
872:λ
829:λ
793:
767:λ
739:
727:λ
720:−
715:λ
648:
642:λ
611:
605:λ
573:
567:λ
560:
549:λ
519:λ
493:λ
431:⋯
359:∏
337:
295:λ
270:
255:
242:≠
239:λ
217:−
214:λ
197:
188:λ
180:−
175:λ
142:λ
4587:(1987).
4391:41501327
4334:(1984).
3945:if
3879:if
3768:if
3708:if
2778:if
2713:if
2563:if
2524:if
2324:if
2228:if
2025:. If τ(
1977:if
1902:if
905:Jacobian
693:. Then
689:- using
291:if
235:if
4650:Bibcode
4549:0192611
4541:2984418
4501:2673623
4318:0192611
4310:2984418
4248:2348250
4064:Bibcode
4031:2444503
3110:Example
2939:on the
2033:, then
465:is the
4697:
4668:
4547:
4539:
4499:
4448:
4412:
4389:
4342:
4316:
4308:
4246:
4228:
4168:
4133:
4125:
4090:
4082:
4029:
4019:
3527:, X–Y
2935:using
323:where
4695:S2CID
4666:S2CID
4640:arXiv
4592:(PDF)
4561:(PDF)
4537:JSTOR
4497:JSTOR
4387:S2CID
4369:arXiv
4306:JSTOR
4267:(PDF)
4244:JSTOR
4166:S2CID
4131:S2CID
4088:S2CID
4054:arXiv
4027:S2CID
3990:Notes
3446:When
3382:When
3370:(CES)
1733:power
1608:from
4625:(5).
4446:ISBN
4410:ISBN
4340:ISBN
4123:ISSN
4080:ISSN
4017:ISBN
3964:<
3898:<
3520:SOCR
3518:The
3118:and
2871:>
2838:>
2429:and
899:and
34:, a
4687:doi
4658:doi
4604:doi
4573:doi
4489:doi
4379:doi
4236:doi
4199:doi
4158:doi
4115:doi
4072:doi
4009:doi
3120:γGT
3116:ALT
2391:min
2272:sgn
2109:sgn
1743:at
1407:log
1347:log
1280:log
1198:log
901:Cox
897:Box
557:exp
30:In
4748::
4723:,
4717:,
4693:.
4683:11
4681:.
4664:.
4656:.
4648:.
4636:39
4634:.
4623:23
4621:.
4598:.
4594:.
4569:68
4567:.
4563:.
4545:MR
4543:.
4533:26
4531:.
4521:;
4495:.
4485:87
4483:.
4385:.
4377:.
4365:46
4363:.
4314:MR
4312:.
4302:26
4300:.
4292:;
4281:^
4242:,
4234:,
4222:41
4220:,
4195:76
4193:.
4178:^
4164:.
4154:54
4152:.
4129:.
4121:.
4111:11
4109:.
4086:.
4078:.
4070:.
4062:.
4050:39
4048:.
4025:.
4015:.
3911:ln
3737:ln
3567:.
3539:.
3531:,
3171::
3009:ln
2974:ln
2896:.
2740:ln
2544:ln
2293:ln
2251:GM
2184:GM
2041:.
1946:ln
1925:GM
1858:GM
1755:.
1522:GM
1446:GM
1356:GM
1130:GM
790:ln
736:ln
645:ln
608:ln
570:ln
334:GM
267:ln
252:GM
194:GM
73:,
4732:)
4728:(
4701:.
4689::
4672:.
4660::
4652::
4642::
4612:.
4606::
4600:2
4579:.
4575::
4551:.
4503:.
4491::
4454:.
4418:.
4393:.
4381::
4371::
4348:.
4320:.
4238::
4205:.
4201::
4172:.
4160::
4137:.
4117::
4094:.
4074::
4066::
4056::
4033:.
4011::
3967:0
3961:y
3958:,
3955:2
3952:=
3939:)
3936:1
3933:+
3928:i
3924:y
3917:(
3901:0
3895:y
3892:,
3889:2
3873:)
3864:2
3861:(
3857:/
3853:)
3850:1
3842:)
3833:2
3830:(
3826:)
3822:1
3819:+
3814:i
3810:y
3803:(
3800:(
3790:0
3784:y
3781:,
3778:0
3775:=
3762:)
3759:1
3756:+
3751:i
3747:y
3743:(
3730:0
3724:y
3721:,
3718:0
3698:/
3694:)
3691:1
3679:)
3675:1
3672:+
3667:i
3663:y
3659:(
3656:(
3650:{
3645:=
3640:)
3634:(
3629:i
3625:y
3601:1
3598:=
3555:y
3497:.
3486:1
3482:N
3472:K
3468:=
3465:Q
3448:λ
3430:.
3427:N
3424:)
3415:1
3412:(
3409:+
3406:K
3400:=
3397:Q
3384:λ
3349:,
3340:/
3336:1
3330:)
3318:N
3314:)
3305:1
3302:(
3299:+
3290:K
3281:(
3276:=
3273:Q
3260:Q
3242:.
3239:)
3236:N
3233:(
3227:)
3218:1
3215:(
3212:+
3209:)
3206:K
3203:(
3194:=
3191:)
3188:Q
3185:(
3169:N
3165:K
3161:Q
3134:1
3094:.
3083:1
3080:,
3077:1
3070:2
3058:2
3055:1
3045:)
3040:)
3025:(
3022:L
3017:(
3001:)
2996:)
2990:(
2987:L
2982:(
2882:2
2866:i
2862:y
2841:0
2833:i
2829:y
2798:,
2795:0
2792:=
2787:1
2772:)
2767:2
2759:+
2754:i
2750:y
2746:(
2733:,
2730:0
2722:1
2702:1
2693:1
2683:1
2674:)
2668:2
2660:+
2655:i
2651:y
2647:(
2637:{
2632:=
2627:)
2619:(
2614:i
2610:y
2576:,
2573:0
2570:=
2555:i
2551:y
2537:,
2534:0
2511:1
2498:i
2494:y
2483:{
2478:=
2473:)
2467:(
2462:i
2458:y
2410:)
2405:i
2401:y
2397:(
2337:,
2334:0
2331:=
2318:)
2312:+
2307:i
2303:y
2299:(
2290:)
2284:+
2281:y
2278:(
2269:)
2263:+
2260:y
2257:(
2241:,
2238:0
2216:1
2206:)
2202:)
2196:+
2193:y
2190:(
2181:(
2173:1
2160:|
2152:+
2147:i
2143:y
2138:|
2134:)
2128:+
2123:i
2119:y
2115:(
2099:{
2094:=
2091:)
2085:,
2079:;
2074:i
2070:y
2066:(
2050:y
2035:Y
2027:Y
2023:i
2018:i
2014:y
1990:,
1987:0
1984:=
1971:)
1965:+
1960:i
1956:y
1952:(
1943:)
1937:+
1934:y
1931:(
1915:,
1912:0
1890:1
1880:)
1876:)
1870:+
1867:y
1864:(
1855:(
1847:1
1835:)
1828:+
1823:i
1819:y
1815:(
1805:{
1800:=
1797:)
1791:,
1785:;
1780:i
1776:y
1772:(
1745:λ
1741:λ
1726:Y
1722:Y
1718:λ
1714:Y
1706:λ
1702:Y
1681:/
1677:)
1674:1
1662:y
1658:(
1632:)
1626:(
1621:i
1617:y
1590:2
1554:)
1551:1
1542:(
1539:2
1535:)
1531:y
1528:(
1495:.
1492:)
1489:1
1486:+
1483:)
1478:)
1475:1
1466:(
1463:2
1459:)
1455:y
1452:(
1442:/
1436:2
1416:2
1413:(
1404:(
1401:)
1398:2
1394:/
1390:n
1384:(
1381:=
1371:)
1368:)
1365:y
1362:(
1353:(
1344:)
1341:1
1332:(
1329:n
1326:+
1323:)
1320:1
1317:+
1314:)
1309:2
1289:2
1286:(
1277:(
1274:)
1271:2
1267:/
1263:n
1257:(
1254:=
1247:)
1244:)
1229:,
1214:(
1209:L
1204:(
1162:)
1159:1
1150:(
1147:n
1143:)
1139:y
1136:(
1127:=
1122:1
1111:i
1107:y
1101:n
1096:1
1093:=
1090:i
1082:=
1078:|
1074:y
1071:d
1067:/
1061:)
1055:(
1050:i
1046:y
1042:d
1038:|
1032:n
1027:1
1024:=
1021:i
1013:=
1010:)
1005:n
1001:y
997:,
991:,
986:1
982:y
978:;
972:(
969:J
943:.
934:1
922:y
891:λ
875:)
869:(
864:i
860:y
846:λ
809:)
804:i
800:y
796:(
770:)
764:(
761:O
758:+
755:)
750:i
746:y
742:(
733:=
723:1
710:i
706:y
677:)
672:2
668:)
664:)
659:i
655:y
651:(
639:(
636:(
633:O
630:+
627:)
622:i
618:y
614:(
602:+
599:1
596:=
593:)
589:)
584:i
580:y
576:(
563:(
554:=
544:i
540:y
499:0
496:=
482:n
478:y
474:1
471:y
446:n
439:n
435:y
426:2
422:y
416:1
412:y
405:=
399:n
396:1
390:)
384:i
380:y
374:n
369:1
366:=
363:i
354:(
349:=
346:)
343:y
340:(
301:0
298:=
285:,
279:i
275:y
264:)
261:y
258:(
245:0
229:,
220:1
210:)
206:)
203:y
200:(
191:(
183:1
170:i
166:y
155:{
150:=
145:)
139:(
134:i
130:y
115:i
111:y
106:n
102:y
98:1
95:y
91:λ
87:λ
20:)
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