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without a fully defined statistical model or the classical theory of statistical inference cannot be readily applied because the family of models being considered are not amenable to such treatment. In addition to these cases where general theory does not prescribe an estimator, the concept of invariance of an estimator can be applied when seeking estimators of alternative forms, either for the sake of simplicity of application of the estimator or so that the estimator is
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22:
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One use of the concept of invariance is where a class or family of estimators is proposed and a particular formulation must be selected amongst these. One procedure is to impose relevant invariance properties and then to find the formulation within this class that has the best properties, leading to
251:
for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves are unchanged when both the measurements and the parameters are transformed in a compatible way,
298:
might be undertaken, leading to a posterior distribution for relevant parameters, but the use of a specific utility or loss function may be unclear. Ideas of invariance can then be applied to the task of summarising the posterior distribution. In other cases, statistical analyses are undertaken
350:
Parameter-transformation invariance: Here, the transformation applies to the parameters alone. The concept here is that essentially the same inference should be made from data and a model involving a parameter θ as would be made from the same data if the model used a parameter φ, where φ is a
377:, it is reasonable to impose the requirement that any estimator of any property of the common distribution should be permutation-invariant: specifically that the estimator, considered as a function of the set of data-values, should not change if items of data are swapped within the dataset.
340:, this invariance requirement immediately implies that the weights should sum to one. While the same result is often derived from a requirement for unbiasedness, the use of "invariance" does not require that a mean value exists and makes no use of any probability distribution at all.
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The concept of invariance is sometimes used on its own as a way of choosing between estimators, but this is not necessarily definitive. For example, a requirement of invariance may be incompatible with the requirement that the
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For a given problem, the invariant estimator with the lowest risk is termed the "best invariant estimator". Best invariant estimator cannot always be achieved. A special case for which it can be achieved is the case when
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There are several types of transformations that are usefully considered when dealing with invariant estimators. Each gives rise to a class of estimators which are invariant to those particular types of transformation.
4304:
290:. Similarly, the theory of classical statistical inference can sometimes lead to strong conclusions about what estimator should be used. However, the usefulness of these theories depends on having a fully prescribed
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is used in formal mathematical contexts that include a precise description of the relation of the way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of
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4537:{\displaystyle {\frac {\int _{-\infty }^{\infty }L(\delta (x)-\theta )f(x_{1}-\theta ,\dots ,x_{n}-\theta )d\theta }{\int _{-\infty }^{\infty }f(x_{1}-\theta ,\dots ,x_{n}-\theta )d\theta }},}
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should be invariant to simple shifts of the data values. If all data values are increased by a given amount, the estimate should change by the same amount. When considering estimation using a
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4740:{\displaystyle \delta (x)={\frac {\int _{-\infty }^{\infty }\theta f(x_{1}-\theta ,\dots ,x_{n}-\theta )d\theta }{\int _{-\infty }^{\infty }f(x_{1}-\theta ,\dots ,x_{n}-\theta )d\theta }}.}
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858:, the rule which assigns a class to a new data-item can be considered to be a special type of estimator. A number of invariance-type considerations can be brought to bear in formulating
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367:. Though the asymptotic properties of the estimator might be invariant, the small sample properties can be different, and a specific distribution needs to be derived.
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dataset using a weighted average implies that the weights should be identical and sum to one. Of course, estimators other than a weighted average may be preferable.
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Scale invariance: Note that this topic about the invariance of the estimator scale parameter not to be confused with the more general
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To define an invariant or equivariant estimator formally, some definitions related to groups of transformations are needed first. Let
5365:
see section 5.2.1 in
Gourieroux, C. and Monfort, A. (1995). Statistics and econometric models, volume 1. Cambridge University Press.
879:
Principle of
Rational Invariance: The action taken in a decision problem should not depend on transformation on the measurement used
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that can be used to decide immediately what estimators should be used according to those approaches. For example, ideas from
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Pitman, E.J.G. (1939). "The estimation of the location and scale parameters of a continuous population of any given form".
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but the meaning has been extended to allow the estimates to change in appropriate ways with such transformations. The term
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Permutation invariance: Where a set of data values can be represented by a statistical model that they are outcomes from
355:(θ). According to this type of invariance, results from transformation-invariant estimators should also be related by φ=
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The combination of permutation invariance and location invariance for estimating a location parameter from an
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is defined in terms of the estimator's sampling distribution and so is invariant under many transformations.
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Invariance
Principle: If two decision problems have the same formal structure (in terms of
308:
5413:
Freue, Gabriela V. Cohen (2007). "The Pitman estimator of the Cauchy location parameter".
312:
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and may also depend on having a relevant loss function to determine the estimator. Thus a
8:
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Pitman, E.J.G. (1939). "Tests of
Hypotheses Concerning Location and Scale Parameters".
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4192:{\displaystyle {\bar {G}}=\{g_{c}:g_{c}(\theta )=\theta +c,c\in \mathbb {R} ^{1}\},}
4085:{\displaystyle G=\{g_{c}:g_{c}(x)=(x_{1}+c,\dots ,x_{n}+c),c\in \mathbb {R} ^{1}\},}
5476:
5445:
5422:
374:
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5156:{\displaystyle \delta _{\text{Pitman}}=\sum _{k=1}^{n}{x_{k}\left},\qquad n>1,}
3329:{\displaystyle g={\bar {g}}={\tilde {g}}=\{g_{c}:g_{c}(x)=x+c,c\in \mathbb {R} \}}
5401:
3954:. This problem is invariant with the following (additive) transformation groups:
258:
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5449:
5426:
4876:{\displaystyle \delta _{\text{Pitman}}=\delta _{ML}={\frac {\sum {x_{i}}}{n}}.}
3410:{\displaystyle \delta (x+c)=\delta (x)+c,{\text{ for all }}c\in \mathbb {R} ,}
5499:
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645:
601:
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An invariant estimator is an estimator which obeys the following two rules:
4299:{\displaystyle {\tilde {G}}=\{g_{c}:g_{c}(a)=a+c,c\in \mathbb {R} ^{1}\}.}
5488:
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5457:
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is a criterion that can be used to compare the properties of different
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about the behavior of systems under aggregate properties (in physics).
248:
124:
3654:{\displaystyle R(\theta ,\delta )=R(0,\delta )=\operatorname {E} }
2911:{\displaystyle R(\theta ,\delta )=R({\bar {g}}(\theta ),\delta )}
3661:. The best invariant estimator is the one that brings the risk
1303:(That is, each transformation has an inverse within the group.)
979:), then the same decision rule should be used in each problem.
2991:
The risk function of an invariant estimator with transitive
1047:, is a set of (measurable) 1:1 and onto transformations of
2256:{\displaystyle L(\theta ,a)=L({\bar {g}}(\theta ),a^{*})}
580:, is a function of the measurements and belongs to a set
4993:{\displaystyle \delta _{\text{Pitman}}\neq \delta _{ML}}
2788:{\displaystyle \delta (g(x))={\tilde {g}}(\delta (x)).}
1067:
into itself, which satisfies the following conditions:
393:
Under this setting, we are given a set of measurements
322:
4942:{\displaystyle x\sim C(\theta 1_{n},I\sigma ^{2})\,\!}
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An estimation problem is invariant(equivariant) under
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which contains information about an unknown parameter
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3894:{\displaystyle f(x_{1}-\theta ,\dots ,x_{n}-\theta )}
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3336:. The invariant estimator in this case must satisfy
2494:{\displaystyle {\tilde {G}}=\{{\tilde {g}}:g\in G\}}
149:. Unsourced material may be challenged and removed.
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5388:Statistical decision theory and Bayesian Analysis
5342:{\displaystyle w_{k}=\prod _{j\neq k}\left\left.}
4938:
4807:with independent, unit-variance components) then
4792:
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3762:{\displaystyle \delta (x)=x-\operatorname {E} .}
2412:{\displaystyle {\bar {G}}=\{{\bar {g}}:g\in G\}}
319:what is called the optimal invariant estimator.
4550:For the squared error loss case, the result is
388:
3905:is a parameter to be estimated, and where the
3057:
363:have this property when the transformation is
5415:Journal of Statistical Planning and Inference
2804:The risk function of an invariant estimator,
332:Shift invariance: Notionally, estimates of a
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1680:{\displaystyle X(x_{0})=\{g(x_{0}):g\in G\}}
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1640:
600:. The quality of the result is defined by a
5390:(2nd ed.). New York: Springer-Verlag.
4796:{\displaystyle x\sim N(\theta 1_{n},I)\,\!}
1003:denote the set of possible data-samples. A
50:Learn how and when to remove these messages
3171:{\displaystyle \Theta =A=\mathbb {R} ^{1}}
3082:is a location parameter if the density of
1331:(i.e. there is an identity transformation
4937:
4791:
4280:
4173:
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2587:{\displaystyle G,{\bar {G}},{\tilde {G}}}
1524:. Such an equivalence class is called an
1356:
1289:
1170:
227:Learn how and when to remove this message
209:Learn how and when to remove this message
107:Learn how and when to remove this message
3699:In the case that L is the squared error
2981:{\displaystyle {\bar {g}}\in {\bar {G}}}
1750:is said to be invariant under the group
70:This article includes a list of general
4547:and this is Pitman's estimator (1939).
865:
860:prior knowledge for pattern recognition
383:independent and identically distributed
372:independent and identically distributed
5498:
5464:
5433:
5382:
3827:{\displaystyle X=(X_{1},\dots ,X_{n})}
1848:{\displaystyle \theta ^{*}\in \Theta }
311:; on the other hand, the criterion of
5412:
497:which depends on a parameter vector
1520:. All the equivalent points form an
849:
323:Some classes of invariant estimators
147:adding citations to reliable sources
118:
56:
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2501:is a group of transformations from
2419:is a group of transformations from
1992:{\displaystyle {\bar {g}}(\theta )}
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3689:{\displaystyle R(\theta ,\delta )}
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3144:
2937:{\displaystyle \theta \in \Theta }
2931:
2831:
2426:
2282:{\displaystyle \theta \in \Theta }
2276:
1842:
1815:{\displaystyle \theta \in \Theta }
1809:
909:
813:
351:one-to-one transformation of θ, φ=
278:, there are several approaches to
264:
76:it lacks sufficient corresponding
14:
5522:
4949:(independent components having a
3482:{\displaystyle K\in \mathbb {R} }
3219:, the problem is invariant under
1296:{\displaystyle g^{-1}(g(x))=x\,.}
1174:{\displaystyle g_{1}g_{2}\in G\,}
726:. The sets of possible values of
31:This article has multiple issues.
4805:multivariate normal distribution
2646:is an invariant estimator under
1927:{\displaystyle f(y|\theta ^{*})}
1707:consists of a single orbit then
719:{\displaystyle R=R(a,\theta )=E}
123:
61:
20:
5140:
3776:The estimation problem is that
3538:so the risk does not vary with
2347:{\displaystyle {\tilde {g}}(a)}
261:" in more general mathematics.
134:needs additional citations for
39:or discuss these issues on the
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3947:{\displaystyle L(|a-\theta |)}
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3212:{\displaystyle L=L(a-\theta )}
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637:{\displaystyle L=L(a,\theta )}
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1:
5374:Gouriéroux and Monfort (1995)
5352:
4309:The best invariant estimator
2798:
2022:is invariant under the group
870:
361:Maximum likelihood estimators
269:
3130:{\displaystyle f(x-\theta )}
2544:if there exist three groups
2070:{\displaystyle L(\theta ,a)}
952:{\displaystyle f(x|\theta )}
490:{\displaystyle f(x|\theta )}
459:probability density function
389:Optimal invariant estimators
7:
3058:Example: Location parameter
2824:, is constant on orbits of
1954:{\displaystyle \theta ^{*}}
1238:{\displaystyle g^{-1}\in G}
560:. The estimate, denoted by
520:The problem is to estimate
10:
5527:
5427:10.1016/j.jspi.2006.05.002
4338:is the one that minimizes
4331:{\displaystyle \delta (x)}
3511:{\displaystyle {\bar {g}}}
3047:{\displaystyle {\bar {g}}}
3013:{\displaystyle {\bar {g}}}
2639:{\displaystyle \delta (x)}
2175:{\displaystyle a^{*}\in A}
1727:is said to be transitive.
1130:{\displaystyle g_{2}\in G}
1097:{\displaystyle g_{1}\in G}
856:statistical classification
309:estimator be mean-unbiased
243:, the concept of being an
5481:10.1093/biomet/31.1-2.200
5450:10.1093/biomet/30.3-4.391
5000:,. However the result is
2289:. The transformed value
1604:{\displaystyle X(x_{0})}
1360:{\displaystyle e(x)=x\,}
1005:group of transformations
3551:{\displaystyle \theta }
3531:{\displaystyle \Theta }
3420:thus it is of the form
3075:{\displaystyle \theta }
2837:{\displaystyle \Theta }
2817:{\displaystyle \delta }
2432:{\displaystyle \Theta }
2042:then the loss function
915:{\displaystyle \Theta }
819:{\displaystyle \Theta }
759:{\displaystyle \theta }
533:{\displaystyle \theta }
510:{\displaystyle \theta }
426:{\displaystyle \theta }
286:would lead directly to
91:more precise citations.
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2711:{\displaystyle g\in G}
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2685:{\displaystyle x\in X}
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2142:{\displaystyle a\in A}
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1883:{\displaystyle Y=g(x)}
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1822:there exists a unique
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1789:{\displaystyle g\in G}
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1730:A family of densities
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1513:{\displaystyle g\in G}
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1324:{\displaystyle e\in G}
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1202:{\displaystyle g\in G}
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455:vector random variable
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3049:
3015:
2983:
2939:
2913:
2839:
2819:
2790:
2713:
2687:
2661:
2641:
2612:
2589:
2539:
2516:
2496:
2434:
2414:
2349:
2311:
2309:{\displaystyle a^{*}}
2284:
2258:
2177:
2144:
2118:
2092:
2072:
2037:
2017:
1994:
1956:
1929:
1885:
1850:
1817:
1791:
1765:
1745:
1722:
1702:
1682:
1606:
1570:
1568:{\displaystyle x_{0}}
1543:
1515:
1489:
1440:
1420:
1418:{\displaystyle x_{2}}
1393:
1391:{\displaystyle x_{1}}
1362:
1326:
1298:
1240:
1204:
1176:
1132:
1099:
1062:
1042:
1022:
998:
974:
954:
917:
897:
841:
821:
801:
781:
761:
741:
721:
639:
595:
575:
555:
535:
512:
492:
448:
428:
408:
276:statistical inference
254:equivariant estimator
158:"Invariant estimator"
5173:
5007:
4961:
4890:
4814:
4754:
4557:
4345:
4313:
4204:
4097:
3961:
3913:
3838:
3780:
3703:
3665:
3562:
3542:
3522:
3493:
3465:
3424:
3343:
3223:
3182:
3141:
3106:
3086:
3066:
3029:
2995:
2948:
2922:
2848:
2828:
2808:
2725:
2696:
2670:
2650:
2621:
2601:
2548:
2528:
2505:
2443:
2423:
2361:
2320:
2293:
2267:
2186:
2153:
2127:
2101:
2081:
2046:
2026:
2006:
1965:
1938:
1894:
1859:
1826:
1800:
1774:
1754:
1734:
1711:
1691:
1615:
1579:
1552:
1532:
1498:
1449:
1429:
1402:
1375:
1335:
1309:
1249:
1213:
1187:
1141:
1108:
1075:
1051:
1031:
1011:
987:
963:
926:
906:
886:
866:Mathematical setting
830:
810:
790:
770:
750:
730:
651:
607:
584:
564:
544:
524:
501:
464:
437:
417:
397:
143:improve this article
4951:Cauchy distribution
4674:
4595:
4471:
4368:
3390: for all
2316:will be denoted by
1027:, to be denoted by
644:which determines a
433:. The measurements
313:median-unbiasedness
288:Bayesian estimators
245:invariant estimator
5339:
5204:
5153:
4990:
4939:
4873:
4793:
4737:
4657:
4578:
4534:
4454:
4351:
4328:
4296:
4189:
4082:
3944:
3891:
3824:
3759:
3686:
3651:
3548:
3528:
3508:
3479:
3451:
3407:
3326:
3209:
3168:
3127:
3092:
3072:
3044:
3010:
2978:
2934:
2908:
2834:
2814:
2785:
2708:
2682:
2656:
2636:
2607:
2594:as defined above.
2584:
2534:
2511:
2491:
2429:
2409:
2344:
2306:
2279:
2253:
2172:
2139:
2113:
2087:
2067:
2032:
2012:
1989:
1951:
1924:
1880:
1845:
1812:
1786:
1760:
1740:
1717:
1697:
1677:
1601:
1565:
1538:
1510:
1484:
1445:are equivalent if
1435:
1415:
1388:
1357:
1321:
1293:
1235:
1199:
1171:
1127:
1094:
1057:
1037:
1017:
993:
969:
949:
912:
892:
836:
816:
796:
776:
756:
736:
716:
634:
590:
570:
550:
530:
507:
487:
453:are modelled as a
443:
423:
403:
334:location parameter
284:Bayesian inference
5326:
5268:
5189:
5130:
5110:
5065:
5017:
4971:
4868:
4824:
4732:
4529:
4216:
4109:
3518:is transitive on
3505:
3391:
3256:
3241:
3095:{\displaystyle X}
3041:
3007:
2975:
2960:
2887:
2761:
2659:{\displaystyle G}
2610:{\displaystyle G}
2581:
2566:
2537:{\displaystyle G}
2514:{\displaystyle A}
2473:
2455:
2391:
2373:
2332:
2225:
2090:{\displaystyle G}
2035:{\displaystyle G}
2015:{\displaystyle F}
1977:
1763:{\displaystyle G}
1743:{\displaystyle F}
1720:{\displaystyle g}
1700:{\displaystyle X}
1541:{\displaystyle X}
1522:equivalence class
1438:{\displaystyle X}
1060:{\displaystyle X}
1040:{\displaystyle G}
1020:{\displaystyle X}
996:{\displaystyle X}
972:{\displaystyle L}
895:{\displaystyle X}
850:In classification
839:{\displaystyle A}
799:{\displaystyle X}
779:{\displaystyle a}
739:{\displaystyle x}
593:{\displaystyle A}
573:{\displaystyle a}
553:{\displaystyle x}
446:{\displaystyle x}
406:{\displaystyle x}
296:Bayesian analysis
292:statistical model
280:estimation theory
237:
236:
229:
219:
218:
211:
193:
117:
116:
109:
54:
5518:
5511:Invariant theory
5492:
5475:(1/2): 200–215.
5461:
5444:(3/4): 391–421.
5430:
5421:(6): 1900–1913.
5409:
5384:Berger, James O.
5375:
5372:
5366:
5363:
5348:
5346:
5345:
5340:
5335:
5331:
5327:
5325:
5321:
5320:
5308:
5307:
5294:
5286:
5273:
5269:
5267:
5266:
5265:
5250:
5249:
5240:
5239:
5227:
5226:
5210:
5203:
5185:
5184:
5162:
5160:
5159:
5154:
5136:
5135:
5131:
5129:
5128:
5124:
5123:
5111:
5108:
5104:
5099:
5083:
5079:
5078:
5066:
5063:
5060:
5054:
5053:
5042:
5037:
5019:
5018:
5015:
4999:
4997:
4996:
4991:
4989:
4988:
4973:
4972:
4969:
4948:
4946:
4945:
4940:
4933:
4932:
4917:
4916:
4882:
4880:
4879:
4874:
4869:
4864:
4863:
4862:
4861:
4847:
4842:
4841:
4826:
4825:
4822:
4802:
4800:
4799:
4794:
4781:
4780:
4746:
4744:
4743:
4738:
4733:
4731:
4715:
4714:
4690:
4689:
4673:
4668:
4655:
4639:
4638:
4614:
4613:
4594:
4589:
4576:
4543:
4541:
4540:
4535:
4530:
4528:
4512:
4511:
4487:
4486:
4470:
4465:
4452:
4436:
4435:
4411:
4410:
4367:
4362:
4349:
4337:
4335:
4334:
4329:
4305:
4303:
4302:
4297:
4289:
4288:
4283:
4247:
4246:
4234:
4233:
4218:
4217:
4209:
4198:
4196:
4195:
4190:
4182:
4181:
4176:
4140:
4139:
4127:
4126:
4111:
4110:
4102:
4091:
4089:
4088:
4083:
4075:
4074:
4069:
4045:
4044:
4020:
4019:
3995:
3994:
3982:
3981:
3953:
3951:
3950:
3945:
3940:
3926:
3900:
3898:
3897:
3892:
3881:
3880:
3856:
3855:
3833:
3831:
3830:
3825:
3820:
3819:
3801:
3800:
3772:Pitman estimator
3768:
3766:
3765:
3760:
3743:
3695:
3693:
3692:
3687:
3660:
3658:
3657:
3652:
3638:
3557:
3555:
3554:
3549:
3537:
3535:
3534:
3529:
3517:
3515:
3514:
3509:
3507:
3506:
3498:
3488:
3486:
3485:
3480:
3478:
3460:
3458:
3457:
3452:
3416:
3414:
3413:
3408:
3403:
3392:
3389:
3335:
3333:
3332:
3327:
3322:
3287:
3286:
3274:
3273:
3258:
3257:
3249:
3243:
3242:
3234:
3218:
3216:
3215:
3210:
3177:
3175:
3174:
3169:
3167:
3166:
3161:
3136:
3134:
3133:
3128:
3101:
3099:
3098:
3093:
3081:
3079:
3078:
3073:
3053:
3051:
3050:
3045:
3043:
3042:
3034:
3019:
3017:
3016:
3011:
3009:
3008:
3000:
2987:
2985:
2984:
2979:
2977:
2976:
2968:
2962:
2961:
2953:
2943:
2941:
2940:
2935:
2917:
2915:
2914:
2909:
2889:
2888:
2880:
2843:
2841:
2840:
2835:
2823:
2821:
2820:
2815:
2794:
2792:
2791:
2786:
2763:
2762:
2754:
2717:
2715:
2714:
2709:
2691:
2689:
2688:
2683:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2616:
2614:
2613:
2608:
2593:
2591:
2590:
2585:
2583:
2582:
2574:
2568:
2567:
2559:
2543:
2541:
2540:
2535:
2520:
2518:
2517:
2512:
2500:
2498:
2497:
2492:
2475:
2474:
2466:
2457:
2456:
2448:
2438:
2436:
2435:
2430:
2418:
2416:
2415:
2410:
2393:
2392:
2384:
2375:
2374:
2366:
2353:
2351:
2350:
2345:
2334:
2333:
2325:
2315:
2313:
2312:
2307:
2305:
2304:
2288:
2286:
2285:
2280:
2262:
2260:
2259:
2254:
2249:
2248:
2227:
2226:
2218:
2181:
2179:
2178:
2173:
2165:
2164:
2149:there exists an
2148:
2146:
2145:
2140:
2122:
2120:
2119:
2114:
2096:
2094:
2093:
2088:
2076:
2074:
2073:
2068:
2041:
2039:
2038:
2033:
2021:
2019:
2018:
2013:
1998:
1996:
1995:
1990:
1979:
1978:
1970:
1961:will be denoted
1960:
1958:
1957:
1952:
1950:
1949:
1933:
1931:
1930:
1925:
1920:
1919:
1910:
1889:
1887:
1886:
1881:
1854:
1852:
1851:
1846:
1838:
1837:
1821:
1819:
1818:
1813:
1795:
1793:
1792:
1787:
1769:
1767:
1766:
1761:
1749:
1747:
1746:
1741:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1686:
1684:
1683:
1678:
1658:
1657:
1633:
1632:
1610:
1608:
1607:
1602:
1597:
1596:
1574:
1572:
1571:
1566:
1564:
1563:
1547:
1545:
1544:
1539:
1519:
1517:
1516:
1511:
1493:
1491:
1490:
1485:
1480:
1479:
1461:
1460:
1444:
1442:
1441:
1436:
1424:
1422:
1421:
1416:
1414:
1413:
1397:
1395:
1394:
1389:
1387:
1386:
1366:
1364:
1363:
1358:
1330:
1328:
1327:
1322:
1302:
1300:
1299:
1294:
1264:
1263:
1244:
1242:
1241:
1236:
1228:
1227:
1208:
1206:
1205:
1200:
1180:
1178:
1177:
1172:
1163:
1162:
1153:
1152:
1136:
1134:
1133:
1128:
1120:
1119:
1103:
1101:
1100:
1095:
1087:
1086:
1066:
1064:
1063:
1058:
1046:
1044:
1043:
1038:
1026:
1024:
1023:
1018:
1002:
1000:
999:
994:
978:
976:
975:
970:
958:
956:
955:
950:
942:
921:
919:
918:
913:
901:
899:
898:
893:
846:, respectively.
845:
843:
842:
837:
825:
823:
822:
817:
805:
803:
802:
797:
785:
783:
782:
777:
765:
763:
762:
757:
745:
743:
742:
737:
725:
723:
722:
717:
709:
643:
641:
640:
635:
599:
597:
596:
591:
579:
577:
576:
571:
559:
557:
556:
551:
539:
537:
536:
531:
516:
514:
513:
508:
496:
494:
493:
488:
480:
452:
450:
449:
444:
432:
430:
429:
424:
412:
410:
409:
404:
375:random variables
345:scale invariance
338:weighted average
232:
225:
214:
207:
203:
200:
194:
192:
151:
127:
119:
112:
105:
101:
98:
92:
87:this article by
78:inline citations
65:
64:
57:
46:
24:
23:
16:
5526:
5525:
5521:
5520:
5519:
5517:
5516:
5515:
5496:
5495:
5398:
5379:
5378:
5373:
5369:
5364:
5360:
5355:
5316:
5312:
5303:
5299:
5295:
5287:
5285:
5278:
5274:
5261:
5257:
5245:
5241:
5235:
5231:
5222:
5218:
5214:
5209:
5205:
5193:
5180:
5176:
5174:
5171:
5170:
5119:
5115:
5107:
5106:
5100:
5089:
5084:
5074:
5070:
5062:
5061:
5059:
5055:
5049:
5045:
5044:
5038:
5027:
5014:
5010:
5008:
5005:
5004:
4981:
4977:
4968:
4964:
4962:
4959:
4958:
4928:
4924:
4912:
4908:
4891:
4888:
4887:
4857:
4853:
4852:
4848:
4846:
4834:
4830:
4821:
4817:
4815:
4812:
4811:
4776:
4772:
4755:
4752:
4751:
4710:
4706:
4685:
4681:
4669:
4661:
4656:
4634:
4630:
4609:
4605:
4590:
4582:
4577:
4575:
4558:
4555:
4554:
4507:
4503:
4482:
4478:
4466:
4458:
4453:
4431:
4427:
4406:
4402:
4363:
4355:
4350:
4348:
4346:
4343:
4342:
4314:
4311:
4310:
4284:
4279:
4278:
4242:
4238:
4229:
4225:
4208:
4207:
4205:
4202:
4201:
4177:
4172:
4171:
4135:
4131:
4122:
4118:
4101:
4100:
4098:
4095:
4094:
4070:
4065:
4064:
4040:
4036:
4015:
4011:
3990:
3986:
3977:
3973:
3962:
3959:
3958:
3936:
3922:
3914:
3911:
3910:
3876:
3872:
3851:
3847:
3839:
3836:
3835:
3815:
3811:
3796:
3792:
3781:
3778:
3777:
3774:
3739:
3704:
3701:
3700:
3666:
3663:
3662:
3634:
3563:
3560:
3559:
3543:
3540:
3539:
3523:
3520:
3519:
3497:
3496:
3494:
3491:
3490:
3474:
3466:
3463:
3462:
3425:
3422:
3421:
3399:
3388:
3344:
3341:
3340:
3318:
3282:
3278:
3269:
3265:
3248:
3247:
3233:
3232:
3224:
3221:
3220:
3183:
3180:
3179:
3162:
3157:
3156:
3142:
3139:
3138:
3107:
3104:
3103:
3102:is of the form
3087:
3084:
3083:
3067:
3064:
3063:
3060:
3054:is transitive.
3033:
3032:
3030:
3027:
3026:
2999:
2998:
2996:
2993:
2992:
2967:
2966:
2952:
2951:
2949:
2946:
2945:
2923:
2920:
2919:
2879:
2878:
2849:
2846:
2845:
2844:. Equivalently
2829:
2826:
2825:
2809:
2806:
2805:
2801:
2753:
2752:
2726:
2723:
2722:
2697:
2694:
2693:
2671:
2668:
2667:
2651:
2648:
2647:
2622:
2619:
2618:
2602:
2599:
2598:
2573:
2572:
2558:
2557:
2549:
2546:
2545:
2529:
2526:
2525:
2506:
2503:
2502:
2465:
2464:
2447:
2446:
2444:
2441:
2440:
2424:
2421:
2420:
2383:
2382:
2365:
2364:
2362:
2359:
2358:
2324:
2323:
2321:
2318:
2317:
2300:
2296:
2294:
2291:
2290:
2268:
2265:
2264:
2244:
2240:
2217:
2216:
2187:
2184:
2183:
2160:
2156:
2154:
2151:
2150:
2128:
2125:
2124:
2102:
2099:
2098:
2082:
2079:
2078:
2047:
2044:
2043:
2027:
2024:
2023:
2007:
2004:
2003:
1969:
1968:
1966:
1963:
1962:
1945:
1941:
1939:
1936:
1935:
1915:
1911:
1906:
1895:
1892:
1891:
1860:
1857:
1856:
1833:
1829:
1827:
1824:
1823:
1801:
1798:
1797:
1775:
1772:
1771:
1755:
1752:
1751:
1735:
1732:
1731:
1712:
1709:
1708:
1692:
1689:
1688:
1653:
1649:
1628:
1624:
1616:
1613:
1612:
1592:
1588:
1580:
1577:
1576:
1559:
1555:
1553:
1550:
1549:
1533:
1530:
1529:
1499:
1496:
1495:
1475:
1471:
1456:
1452:
1450:
1447:
1446:
1430:
1427:
1426:
1409:
1405:
1403:
1400:
1399:
1382:
1378:
1376:
1373:
1372:
1336:
1333:
1332:
1310:
1307:
1306:
1256:
1252:
1250:
1247:
1246:
1220:
1216:
1214:
1211:
1210:
1188:
1185:
1184:
1158:
1154:
1148:
1144:
1142:
1139:
1138:
1115:
1111:
1109:
1106:
1105:
1082:
1078:
1076:
1073:
1072:
1052:
1049:
1048:
1032:
1029:
1028:
1012:
1009:
1008:
988:
985:
984:
964:
961:
960:
938:
927:
924:
923:
907:
904:
903:
887:
884:
883:
873:
868:
852:
831:
828:
827:
811:
808:
807:
791:
788:
787:
786:are denoted by
771:
768:
767:
751:
748:
747:
731:
728:
727:
705:
652:
649:
648:
608:
605:
604:
585:
582:
581:
565:
562:
561:
545:
542:
541:
525:
522:
521:
502:
499:
498:
476:
465:
462:
461:
438:
435:
434:
418:
415:
414:
398:
395:
394:
391:
325:
272:
267:
265:General setting
233:
222:
221:
220:
215:
204:
198:
195:
152:
150:
140:
128:
113:
102:
96:
93:
83:Please help to
82:
66:
62:
25:
21:
12:
11:
5:
5524:
5514:
5513:
5508:
5494:
5493:
5462:
5431:
5410:
5396:
5377:
5376:
5367:
5357:
5356:
5354:
5351:
5350:
5349:
5338:
5334:
5330:
5324:
5319:
5315:
5311:
5306:
5302:
5298:
5293:
5290:
5284:
5281:
5277:
5272:
5264:
5260:
5256:
5253:
5248:
5244:
5238:
5234:
5230:
5225:
5221:
5217:
5213:
5208:
5202:
5199:
5196:
5192:
5188:
5183:
5179:
5164:
5163:
5152:
5149:
5146:
5143:
5139:
5134:
5127:
5122:
5118:
5114:
5103:
5098:
5095:
5092:
5088:
5082:
5077:
5073:
5069:
5058:
5052:
5048:
5041:
5036:
5033:
5030:
5026:
5022:
5013:
4987:
4984:
4980:
4976:
4967:
4936:
4931:
4927:
4923:
4920:
4915:
4911:
4907:
4904:
4901:
4898:
4895:
4884:
4883:
4872:
4867:
4860:
4856:
4851:
4845:
4840:
4837:
4833:
4829:
4820:
4790:
4787:
4784:
4779:
4775:
4771:
4768:
4765:
4762:
4759:
4748:
4747:
4736:
4730:
4727:
4724:
4721:
4718:
4713:
4709:
4705:
4702:
4699:
4696:
4693:
4688:
4684:
4680:
4677:
4672:
4667:
4664:
4660:
4654:
4651:
4648:
4645:
4642:
4637:
4633:
4629:
4626:
4623:
4620:
4617:
4612:
4608:
4604:
4601:
4598:
4593:
4588:
4585:
4581:
4574:
4571:
4568:
4565:
4562:
4545:
4544:
4533:
4527:
4524:
4521:
4518:
4515:
4510:
4506:
4502:
4499:
4496:
4493:
4490:
4485:
4481:
4477:
4474:
4469:
4464:
4461:
4457:
4451:
4448:
4445:
4442:
4439:
4434:
4430:
4426:
4423:
4420:
4417:
4414:
4409:
4405:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4366:
4361:
4358:
4354:
4327:
4324:
4321:
4318:
4307:
4306:
4295:
4292:
4287:
4282:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4245:
4241:
4237:
4232:
4228:
4224:
4221:
4215:
4212:
4199:
4188:
4185:
4180:
4175:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4138:
4134:
4130:
4125:
4121:
4117:
4114:
4108:
4105:
4092:
4081:
4078:
4073:
4068:
4063:
4060:
4057:
4054:
4051:
4048:
4043:
4039:
4035:
4032:
4029:
4026:
4023:
4018:
4014:
4010:
4007:
4004:
4001:
3998:
3993:
3989:
3985:
3980:
3976:
3972:
3969:
3966:
3943:
3939:
3935:
3932:
3929:
3925:
3921:
3918:
3890:
3887:
3884:
3879:
3875:
3871:
3868:
3865:
3862:
3859:
3854:
3850:
3846:
3843:
3823:
3818:
3814:
3810:
3807:
3804:
3799:
3795:
3791:
3788:
3785:
3773:
3770:
3758:
3755:
3752:
3749:
3746:
3742:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3685:
3682:
3679:
3676:
3673:
3670:
3650:
3647:
3644:
3641:
3637:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3547:
3527:
3504:
3501:
3477:
3473:
3470:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3418:
3417:
3406:
3402:
3398:
3395:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3325:
3321:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3285:
3281:
3277:
3272:
3268:
3264:
3261:
3255:
3252:
3246:
3240:
3237:
3231:
3228:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3165:
3160:
3155:
3152:
3149:
3146:
3126:
3123:
3120:
3117:
3114:
3111:
3091:
3071:
3059:
3056:
3040:
3037:
3022:
3021:
3006:
3003:
2989:
2974:
2971:
2965:
2959:
2956:
2933:
2930:
2927:
2907:
2904:
2901:
2898:
2895:
2892:
2886:
2883:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2833:
2813:
2800:
2797:
2796:
2795:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2760:
2757:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2707:
2704:
2701:
2681:
2678:
2675:
2655:
2635:
2632:
2629:
2626:
2606:
2580:
2577:
2571:
2565:
2562:
2556:
2553:
2533:
2510:
2490:
2487:
2484:
2481:
2478:
2472:
2469:
2463:
2460:
2454:
2451:
2439:to itself and
2428:
2408:
2405:
2402:
2399:
2396:
2390:
2387:
2381:
2378:
2372:
2369:
2357:In the above,
2343:
2340:
2337:
2331:
2328:
2303:
2299:
2278:
2275:
2272:
2252:
2247:
2243:
2239:
2236:
2233:
2230:
2224:
2221:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2171:
2168:
2163:
2159:
2138:
2135:
2132:
2112:
2109:
2106:
2086:
2066:
2063:
2060:
2057:
2054:
2051:
2031:
2011:
1988:
1985:
1982:
1976:
1973:
1948:
1944:
1923:
1918:
1914:
1909:
1905:
1902:
1899:
1879:
1876:
1873:
1870:
1867:
1864:
1844:
1841:
1836:
1832:
1811:
1808:
1805:
1785:
1782:
1779:
1770:if, for every
1759:
1739:
1716:
1696:
1676:
1673:
1670:
1667:
1664:
1661:
1656:
1652:
1648:
1645:
1642:
1639:
1636:
1631:
1627:
1623:
1620:
1600:
1595:
1591:
1587:
1584:
1562:
1558:
1537:
1509:
1506:
1503:
1483:
1478:
1474:
1470:
1467:
1464:
1459:
1455:
1434:
1412:
1408:
1385:
1381:
1369:
1368:
1355:
1352:
1349:
1346:
1343:
1340:
1320:
1317:
1314:
1304:
1292:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1262:
1259:
1255:
1234:
1231:
1226:
1223:
1219:
1198:
1195:
1192:
1181:
1169:
1166:
1161:
1157:
1151:
1147:
1126:
1123:
1118:
1114:
1093:
1090:
1085:
1081:
1056:
1036:
1016:
992:
981:
980:
968:
948:
945:
941:
937:
934:
931:
911:
891:
880:
872:
869:
867:
864:
851:
848:
835:
815:
795:
775:
755:
735:
715:
712:
708:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
633:
630:
627:
624:
621:
618:
615:
612:
589:
569:
549:
529:
506:
486:
483:
479:
475:
472:
469:
442:
422:
402:
390:
387:
379:
378:
368:
348:
341:
324:
321:
271:
268:
266:
263:
235:
234:
217:
216:
131:
129:
122:
115:
114:
69:
67:
60:
55:
29:
28:
26:
19:
9:
6:
4:
3:
2:
5523:
5512:
5509:
5507:
5504:
5503:
5501:
5490:
5486:
5482:
5478:
5474:
5470:
5469:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5438:
5432:
5428:
5424:
5420:
5416:
5411:
5407:
5403:
5399:
5397:0-387-96098-8
5393:
5389:
5385:
5381:
5380:
5371:
5362:
5358:
5336:
5332:
5328:
5317:
5313:
5309:
5304:
5300:
5291:
5288:
5282:
5279:
5275:
5270:
5262:
5258:
5254:
5251:
5246:
5236:
5232:
5228:
5223:
5219:
5211:
5206:
5200:
5197:
5194:
5190:
5186:
5181:
5177:
5169:
5168:
5167:
5150:
5147:
5144:
5141:
5137:
5132:
5120:
5116:
5101:
5096:
5093:
5090:
5086:
5075:
5071:
5056:
5050:
5046:
5039:
5034:
5031:
5028:
5024:
5020:
5011:
5003:
5002:
5001:
4985:
4982:
4978:
4974:
4965:
4956:
4952:
4929:
4925:
4921:
4918:
4913:
4909:
4905:
4899:
4896:
4893:
4870:
4865:
4858:
4854:
4849:
4843:
4838:
4835:
4831:
4827:
4818:
4810:
4809:
4808:
4806:
4785:
4782:
4777:
4773:
4769:
4763:
4760:
4757:
4734:
4728:
4725:
4719:
4716:
4711:
4707:
4703:
4700:
4697:
4694:
4691:
4686:
4682:
4675:
4662:
4658:
4652:
4649:
4643:
4640:
4635:
4631:
4627:
4624:
4621:
4618:
4615:
4610:
4606:
4599:
4596:
4583:
4579:
4572:
4566:
4560:
4553:
4552:
4551:
4548:
4531:
4525:
4522:
4516:
4513:
4508:
4504:
4500:
4497:
4494:
4491:
4488:
4483:
4479:
4472:
4459:
4455:
4449:
4446:
4440:
4437:
4432:
4428:
4424:
4421:
4418:
4415:
4412:
4407:
4403:
4396:
4390:
4387:
4381:
4375:
4369:
4356:
4352:
4341:
4340:
4339:
4322:
4316:
4293:
4285:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4251:
4243:
4239:
4235:
4230:
4226:
4219:
4210:
4200:
4186:
4178:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4144:
4136:
4132:
4128:
4123:
4119:
4112:
4103:
4093:
4079:
4071:
4061:
4058:
4055:
4049:
4046:
4041:
4037:
4033:
4030:
4027:
4024:
4021:
4016:
4012:
4005:
3999:
3991:
3987:
3983:
3978:
3974:
3967:
3964:
3957:
3956:
3955:
3933:
3930:
3927:
3916:
3908:
3907:loss function
3904:
3885:
3882:
3877:
3873:
3869:
3866:
3863:
3860:
3857:
3852:
3848:
3841:
3816:
3812:
3808:
3805:
3802:
3797:
3793:
3786:
3783:
3769:
3756:
3750:
3747:
3744:
3736:
3730:
3724:
3721:
3718:
3712:
3706:
3697:
3680:
3677:
3674:
3668:
3645:
3642:
3639:
3628:
3625:
3622:
3616:
3610:
3604:
3598:
3595:
3592:
3586:
3583:
3577:
3574:
3571:
3565:
3545:
3499:
3471:
3468:
3448:
3445:
3442:
3439:
3433:
3427:
3404:
3396:
3393:
3385:
3382:
3379:
3373:
3367:
3364:
3358:
3355:
3352:
3346:
3339:
3338:
3337:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3291:
3283:
3279:
3275:
3270:
3266:
3259:
3250:
3244:
3235:
3229:
3226:
3203:
3200:
3197:
3191:
3188:
3185:
3163:
3153:
3150:
3147:
3121:
3118:
3115:
3109:
3089:
3069:
3055:
3035:
3001:
2990:
2969:
2963:
2954:
2928:
2925:
2902:
2899:
2893:
2881:
2872:
2869:
2863:
2860:
2857:
2851:
2811:
2803:
2802:
2782:
2773:
2767:
2755:
2749:
2740:
2734:
2728:
2721:
2720:
2719:
2705:
2702:
2699:
2679:
2676:
2673:
2653:
2630:
2624:
2604:
2595:
2575:
2569:
2560:
2554:
2551:
2531:
2522:
2508:
2485:
2482:
2479:
2476:
2467:
2458:
2449:
2403:
2400:
2397:
2394:
2385:
2376:
2367:
2355:
2338:
2326:
2301:
2297:
2273:
2270:
2245:
2241:
2237:
2231:
2219:
2210:
2207:
2201:
2198:
2195:
2189:
2169:
2166:
2161:
2157:
2136:
2133:
2130:
2110:
2107:
2104:
2097:if for every
2084:
2061:
2058:
2055:
2049:
2029:
2009:
2000:
1983:
1971:
1946:
1942:
1916:
1912:
1903:
1897:
1874:
1868:
1865:
1862:
1839:
1834:
1830:
1806:
1803:
1783:
1780:
1777:
1757:
1737:
1728:
1714:
1694:
1671:
1668:
1665:
1662:
1654:
1650:
1643:
1637:
1629:
1625:
1618:
1611:, is the set
1593:
1589:
1582:
1560:
1556:
1535:
1527:
1523:
1507:
1504:
1501:
1476:
1472:
1465:
1462:
1457:
1453:
1432:
1410:
1406:
1383:
1379:
1353:
1350:
1344:
1338:
1318:
1315:
1312:
1305:
1290:
1286:
1283:
1274:
1268:
1260:
1257:
1253:
1232:
1229:
1224:
1221:
1217:
1196:
1193:
1190:
1182:
1167:
1164:
1159:
1155:
1149:
1145:
1124:
1121:
1116:
1112:
1091:
1088:
1083:
1079:
1070:
1069:
1068:
1054:
1034:
1014:
1006:
990:
966:
943:
935:
929:
889:
881:
878:
877:
876:
863:
861:
857:
847:
833:
793:
773:
753:
733:
710:
699:
696:
693:
687:
681:
678:
672:
669:
666:
660:
657:
654:
647:
646:risk function
628:
625:
622:
616:
613:
610:
603:
602:loss function
587:
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527:
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181:
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174:
170:
167:
163:
160: –
159:
155:
154:Find sources:
148:
144:
138:
137:
132:This article
130:
126:
121:
120:
111:
108:
100:
90:
86:
80:
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5472:
5466:
5441:
5435:
5418:
5414:
5387:
5370:
5361:
5165:
4954:
4885:
4749:
4549:
4546:
4308:
3902:
3834:has density
3775:
3698:
3696:to minimum.
3419:
3061:
3023:
3020:is constant.
2666:if, for all
2617:, estimator
2596:
2523:
2356:
2001:
1890:has density
1729:
1370:
982:
874:
853:
519:
392:
380:
356:
352:
326:
317:
305:
273:
259:equivariance
253:
244:
238:
223:
205:
196:
186:
179:
172:
165:
153:
141:Please help
136:verification
133:
103:
94:
75:
47:
40:
34:
33:Please help
30:
3558:: that is,
2521:to itself.
89:introducing
5500:Categories
5468:Biometrika
5437:Biometrika
5353:References
2799:Properties
2182:such that
1855:such that
871:Definition
270:Background
249:estimators
241:statistics
169:newspapers
72:references
36:improve it
5506:Estimator
5310:−
5292:σ
5283:−
5259:σ
5229:−
5198:≠
5191:∏
5087:∑
5025:∑
5012:δ
4979:δ
4975:≠
4966:δ
4926:σ
4906:θ
4897:∼
4850:∑
4832:δ
4819:δ
4770:θ
4761:∼
4729:θ
4720:θ
4717:−
4701:…
4695:θ
4692:−
4671:∞
4666:∞
4663:−
4659:∫
4653:θ
4644:θ
4641:−
4625:…
4619:θ
4616:−
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4592:∞
4587:∞
4584:−
4580:∫
4561:δ
4526:θ
4517:θ
4514:−
4498:…
4492:θ
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4468:∞
4463:∞
4460:−
4456:∫
4450:θ
4441:θ
4438:−
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4413:−
4391:θ
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4276:∈
4214:~
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1781:∈
1669:∈
1505:∈
1494:for some
1371:Datasets
1316:∈
1258:−
1230:∈
1222:−
1194:∈
1165:∈
1122:∈
1089:∈
944:θ
910:Θ
814:Θ
754:θ
711:θ
700:θ
673:θ
629:θ
528:θ
505:θ
482:θ
457:having a
421:θ
365:monotonic
199:July 2010
97:July 2010
42:talk page
5386:(1985).
4803:(i.e. a
3901:, where
3062:Suppose
2918:for all
2263:for all
1245:, where
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5458:2332656
5406:0804611
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1575:orbit,
1548:). The
183:scholar
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5016:Pitman
4970:Pitman
4955:σ
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3137:. For
826:, and
766:, and
540:given
301:robust
185:
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74:, but
5485:JSTOR
5454:JSTOR
5166:with
1687:. If
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1209:then
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359:(θ).
190:JSTOR
176:books
5392:ISBN
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3178:and
2944:and
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