1934:. Bias is a property of the estimator, not of the estimate. Often, people refer to a "biased estimate" or an "unbiased estimate", but they really are talking about an "estimate from a biased estimator", or an "estimate from an unbiased estimator". Also, people often confuse the "error" of a single estimate with the "bias" of an estimator. That the error for one estimate is large, does not mean the estimator is biased. In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased. Also, an estimator's being biased does not preclude the error of an estimate from being zero in a particular instance. The ideal situation is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, have few outliers). Yet unbiasedness is not essential. Often, if just a little bias is permitted, then an estimator can be found with lower mean squared error and/or fewer outlier sample estimates.
5016:
822:
estimates (samples). Then high MSE means the average distance of the arrows from the bull's eye is high, and low MSE means the average distance from the bull's eye is low. The arrows may or may not be clustered. For example, even if all arrows hit the same point, yet grossly miss the target, the MSE is still relatively large. However, if the MSE is relatively low then the arrows are likely more highly clustered (than highly dispersed) around the target.
2675:
1828:. If the parameter is the bull's eye of a target and the arrows are estimates, then a relatively high absolute value for the bias means the average position of the arrows is off-target, and a relatively low absolute bias means the average position of the arrows is on target. They may be dispersed, or may be clustered. The relationship between bias and variance is analogous to the relationship between
4992:
5002:
4697:
MSE. The variance of the good estimator (good efficiency) would be smaller than the variance of the bad estimator (bad efficiency). The square of an estimator bias with a good estimator would be smaller than the estimator bias with a bad estimator. The MSE of a good estimator would be smaller than the MSE of the bad estimator. Suppose there are two estimator,
4696:
The first term represents the mean squared error; the second term represents the square of the estimator bias; and the third term represents the variance of the sample. The quality of the estimator can be identified from the comparison between the variance, the square of the estimator bias, or the
1243:
of the estimates. (Note the difference between MSE and variance.) If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high variance means the arrows are dispersed, and a relatively low variance means the arrows are clustered. Even if the variance is low,
2756:
A desired property for estimators is the unbiased trait where an estimator is shown to have no systematic tendency to produce estimates larger or smaller than the provided probability. Additionally, unbiased estimators with smaller variances are preferred over larger variances because it will be
1027:
821:
It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated. Consider the following analogy. Suppose the parameter is the bull's-eye of a target, the estimator is the process of shooting arrows at the target, and the individual arrows are
85:
is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given
4437:
The efficiency of an estimator is used to estimate the quantity of interest in a "minimum error" manner. In reality, there is not an explicit best estimator; there can only be a better estimator. The good or not of the efficiency of an estimator is based on the choice of a particular
4344:. Note that convergence will not necessarily have occurred for any finite "n", therefore this value is only an approximation to the true variance of the estimator, while in the limit the asymptotic variance (V/n) is simply zero. To be more specific, the distribution of the estimator
4987:
Besides using formula to identify the efficiency of the estimator, it can also be identified through the graph. If an estimator is efficient, in the frequency vs. value graph, there will be a curve with high frequency at the center and low frequency on the two sides. For example:
3550:
4691:
1244:
the cluster of arrows may still be far off-target, and even if the variance is high, the diffuse collection of arrows may still be unbiased. Finally, even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.
4915:
1826:
3845:
1941:
of the distribution of estimates agrees with the true value; thus, in the long run half the estimates will be too low and half too high. While this applies immediately only to scalar-valued estimators, it can be extended to any measure of
3272:
1354:
4982:
4816:
1237:
3081:
889:
816:
4422:
of the maximum likelihood article. However, not all estimators are asymptotically normal; the simplest examples are found when the true value of a parameter lies on the boundary of the allowable parameter region.
1628:
3966:
4316:
4493:
4029:
and theoretical distribution functions respectively. An easy example to see if something is Fisher consistent is to check the mean consistency and the variance. For example, to check consistency for the mean
2815:
1713:
196:
The definition places virtually no restrictions on which functions of the data can be called the "estimators". The attractiveness of different estimators can be judged by looking at their properties, such as
1571:
3699:
The bias-variance tradeoff will be used in model complexity, over-fitting and under-fitting. It is mainly used in the field of supervised learning and predictive modeling to diagnose the performance of
3447:
3336:
1080:
2268:
646:
3140:
3886:
An estimator can be considered Fisher
Consistent as long as the estimator is the same functional of the empirical distribution function as the true distribution function. Following the formula:
4569:
4128:
2608:
4072:
2906:
2059:
1932:
90:, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having worse properties that hold under wider conditions.
4824:
1751:
479:
395:
259:. In these problems the estimates are functions that can be thought of as point estimates in an infinite dimensional space, and there are corresponding interval estimation problems.
3694:
3639:
3586:
2844:
2297:
1980:
1862:
1742:
1657:
1501:
1432:
1383:
1283:
1129:
881:
714:
581:
508:
345:
187:
4557:
4219:
2507:
2091:
2548:
114:. A common way of phrasing it is "the estimator is the method selected to obtain an estimate of an unknown parameter". The parameter being estimated is sometimes called the
3148:
Similarly, when looking at quantities in the interest of variance as the model distribution there is also an unbiased estimator that should satisfy the two equations below.
77:, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators.
4186:
2455:
2642:
2186:
4749:
4722:
4023:
3433:
3374:
2980:
2933:
2750:
2703:
4412:
3766:
3406:
2664:
4373:
3659:
3606:
2953:
2868:
2723:
2317:
2206:
2000:
1886:
1472:
1403:
669:
316:
288:
154:
3996:
2138:
2988:
When looking at quantities in the interest of expectation for the model distribution there is an unbiased estimator which should satisfy the two equations below.
2397:
1526:
There are two kinds of estimators: biased estimators and unbiased estimators. Whether an estimator is biased or not can be identified by the relationship between
441:
3154:
2111:
1521:
1452:
848:
548:
415:
1288:
5429:
4923:
4757:
1022:{\displaystyle d(x)={\widehat {\theta }}(x)-\operatorname {E} ({\widehat {\theta }}(X))={\widehat {\theta }}(x)-\operatorname {E} ({\widehat {\theta }}),}
5008:
To put it simply, the good estimator has a narrow curve, while the bad estimator has a large curve. Plotting these two curves on one graph with a shared
1134:
2994:
3728:) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter.
193:; a particular realization of this random variable is called the "estimate". Sometimes the words "estimator" and "estimate" are used interchangeably.
4559:. These cannot in general both be satisfied simultaneously: an unbiased estimator may have a lower mean squared error than any biased estimator (see
722:
232:
When the word "estimator" is used without a qualifier, it usually refers to point estimation. The estimate in this case is a single point in the
5365:
5331:
5250:
359:
corresponding to the observed data, the estimator (itself treated as a random variable) is symbolised as a function of that random variable,
1580:
3892:
3552:
i.e. mean squared error = variance + square of bias. In particular, for an unbiased estimator, the variance equals the mean squared error.
4242:
4445:
2767:
2758:
1665:
1529:
551:
5414:
5391:
5198:
3545:{\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {Var} ({\widehat {\theta }})+(B({\widehat {\theta }}))^{2},}
5406:
5383:
5349:
3278:
1035:
5297:
4222:
4026:
1356:. It is the distance between the average of the collection of estimates, and the single parameter being estimated. The bias of
5047:
5109:
5081:
5023:
Among unbiased estimators, there often exists one with the lowest variance, called the minimum variance unbiased estimator (
5015:
2211:
589:
5307:
5113:
3087:
4686:{\displaystyle \operatorname {E} =(\operatorname {E} ({\widehat {\theta }})-\theta )^{2}+\operatorname {Var} (\theta )\ }
248:
5302:
5134:
3663:
348:
4077:
2552:
5323:
5129:
5118:
3721:
5051:
4033:
716:
is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is,
4910:{\displaystyle |\operatorname {E} (\theta _{1})-\theta |<\left|\operatorname {E} (\theta _{2})-\theta \right|}
1821:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =\operatorname {E} ({\widehat {\theta }}-\theta )}
5043:
2873:
2005:
5097:
4998:
If an estimator is not efficient, the frequency vs. value graph, there will be a relatively more gentle curve.
2757:
closer to the "true" value of the parameter. The unbiased estimator with the smallest variance is known as the
2002:. For example, if a genetic theory states there is a type of leaf (starchy green) that occurs with probability
1893:
5039:
5032:
446:
362:
5028:
4432:
4145:
3856:
3349:
we would obtain an estimator with a negative bias which would thus produce estimates that are too small for
210:
5343:
5164:
3875:
3670:
3615:
3562:
2820:
2273:
1956:
1838:
1829:
1718:
1633:
1477:
1408:
1359:
1259:
1105:
857:
690:
557:
484:
321:
163:
5315:
4502:
4379:
4195:
2459:
129:
125:
107:
2064:
5103:
4352:
4233:
4139:
3715:
2511:
1947:
1253:
206:
133:
5239:
Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005).
3870:, namely the true value divided by the asymptotic value of the estimator. This occurs frequently in
5139:
5087:
4159:
4153:
3840:{\displaystyle \lim _{n\to \infty }\Pr \left\{\left|t_{n}-\theta \right|<\varepsilon \right\}=1}
2401:
675:, depends not only on the estimator (the estimation formula or procedure), but also on the sample.
43:
5267:
5031:
exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the
2612:
2143:
5444:
4727:
4700:
4496:
4419:
4415:
4001:
3556:
3411:
3352:
2958:
2935:
being the only unbiased estimator. If the distributions overlapped and were both centered around
2911:
2728:
2681:
684:
244:
237:
74:
66:
4388:
3379:
5410:
5387:
5375:
5361:
5327:
5246:
5194:
5186:
5144:
5069:
2646:
214:
202:
111:
87:
81:
4358:
3644:
3591:
2938:
2853:
2708:
2302:
2191:
1985:
1871:
1457:
1388:
654:
301:
273:
139:
5353:
5240:
5063:
1943:
252:
121:
3974:
3267:{\displaystyle 1.\quad S_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X_{n}}})^{2}}
2116:
5191:
The
Collected Works of John W. Tukey: Philosophy and Principles of Data Analysis 1965–1986
5124:
4442:, and it is reflected by two naturally desirable properties of estimators: to be unbiased
3871:
511:
356:
233:
218:
190:
70:
62:
55:
4751:
is the bad estimator. The above relationship can be expressed by the following formulas.
3866:
of a parameter can be made into a consistent estimator by multiplying the estimator by a
2322:
420:
1349:{\displaystyle B({\widehat {\theta }})=\operatorname {E} ({\widehat {\theta }})-\theta }
5149:
4560:
4418:
estimators are asymptotically normal under fairly weak regularity conditions — see the
2096:
1888:
1865:
1506:
1437:
1239:. It is used to indicate how far, on average, the collection of estimates are from the
1083:
833:
533:
400:
99:
17:
4977:{\displaystyle \operatorname {MSE} (\theta _{1})<\operatorname {MSE} (\theta _{2})}
4811:{\displaystyle \operatorname {Var} (\theta _{1})<\operatorname {Var} (\theta _{2})}
5438:
5159:
5092:
4439:
226:
222:
2674:
5204:
3867:
1937:
An alternative to the version of "unbiased" above, is "median-unbiased", where the
1232:{\displaystyle \operatorname {Var} ({\widehat {\theta }})=\operatorname {E} )^{2}]}
291:
198:
3076:{\displaystyle 1.\quad {\overline {X}}_{n}={\frac {X_{1}+X_{2}+\cdots +X_{n}}{n}}}
4148:
estimator is a consistent estimator whose distribution around the true parameter
4383:
3725:
3609:
256:
106:(that is, a function of the data) that is used to infer the value of an unknown
51:
5357:
5154:
3851:
The consistency defined above may be called weak consistency. The sequence is
2764:
To find if your estimator is unbiased it is easy to follow along the equation
811:{\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {E} .}
157:
31:
5035:, which is an absolute lower bound on variance for statistics of a variable.
213:, etc. The construction and comparison of estimators are the subjects of the
4991:
3747:
2188:
distribution. The number can be used to express the following estimator for
103:
5001:
3612:
of the variance), or an estimate of the standard deviation of an estimator
27:
Rule for calculating an estimate of a given quantity based on observed data
1099:
481:, which is a fixed value. Often an abbreviated notation is used in which
116:
47:
39:
290:
needs to be estimated. Then an "estimator" is a function that maps the
4563:). A function relates the mean squared error with the estimator bias.
2908:
the estimator is unbiased. Looking at the figure to the right despite
1623:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta \neq 0}
3720:
A consistent sequence of estimators is a sequence of estimators that
1938:
4280:
4204:
3961:{\displaystyle {\widehat {\theta }}=h(T_{n}),\theta =h(T_{\theta })}
50:) and its result (the estimate) are distinguished. For example, the
4311:{\displaystyle {\sqrt {n}}(t_{n}-\theta ){\xrightarrow {D}}N(0,V),}
1131:
is the expected value of the squared sampling deviations; that is,
5014:
5000:
4990:
4488:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =0}
2810:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =0}
2673:
1708:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =0}
5345:
Introduction to
Empirical Processes and Semiparametric Inference
5024:
1566:{\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta }
2140:, or the number of starchy green leaves, can be modeled with a
46:: thus the rule (the estimator), the quantity of interest (the
1090:, depends not only on the estimator, but also on the sample.
3331:{\displaystyle 2.\quad \operatorname {E} \left=\sigma ^{2}}
1075:{\displaystyle \operatorname {E} ({\widehat {\theta }}(X))}
3724:
to the quantity being estimated as the index (usually the
240:, where the estimates are subsets of the parameter space.
225:, and its performance may be evaluated through the use of
189:. Being a function of the data, the estimator is itself a
3444:
The mean squared error, variance, and bias, are related:
347:. It is often convenient to express the theory using the
1748:
The bias is also the expected value of the error, since
156:
then the estimator is traditionally written by adding a
522:
The following definitions and attributes are relevant.
247:
arises in two applications. Firstly, in estimating the
73:
yield single-valued results. This is in contrast to an
5038:
Concerning such "best unbiased estimators", see also
4926:
4827:
4760:
4730:
4703:
4572:
4505:
4448:
4391:
4361:
4245:
4198:
4162:
4080:
4036:
4004:
3977:
3895:
3769:
3673:
3647:
3618:
3594:
3565:
3450:
3414:
3382:
3355:
3281:
3157:
3090:
2997:
2961:
2941:
2914:
2876:
2856:
2823:
2770:
2731:
2711:
2684:
2678:
Difference between estimators: an unbiased estimator
2649:
2615:
2555:
2514:
2462:
2404:
2325:
2305:
2276:
2263:{\displaystyle {\widehat {\theta }}=4/n\cdot N_{1}-2}
2214:
2194:
2146:
2119:
2099:
2067:
2008:
1988:
1959:
1896:
1874:
1841:
1754:
1721:
1668:
1636:
1583:
1532:
1509:
1480:
1460:
1440:
1411:
1391:
1362:
1291:
1262:
1137:
1108:
1038:
892:
860:
836:
725:
693:
657:
641:{\displaystyle e(x)={\widehat {\theta }}(x)-\theta ,}
592:
560:
536:
487:
449:
423:
403:
365:
324:
304:
276:
166:
142:
3135:{\displaystyle 2.\quad \operatorname {E} \left=\mu }
2982:
would actually be the preferred unbiased estimator.
397:. The estimate for a particular observed data value
5242:
4156:with standard deviation shrinking in proportion to
251:of random variables and secondly in estimating the
4976:
4909:
4810:
4743:
4716:
4685:
4551:
4487:
4414:as an estimator of the true mean. More generally,
4406:
4367:
4336:of the estimator. However, some authors also call
4310:
4213:
4180:
4122:
4066:
4017:
3990:
3960:
3839:
3688:
3653:
3633:
3600:
3580:
3544:
3427:
3400:
3368:
3330:
3266:
3134:
3075:
2974:
2947:
2927:
2900:
2862:
2838:
2809:
2744:
2717:
2697:
2658:
2636:
2602:
2542:
2501:
2449:
2391:
2311:
2291:
2262:
2200:
2180:
2132:
2105:
2085:
2053:
1994:
1974:
1926:
1880:
1856:
1820:
1736:
1707:
1651:
1622:
1565:
1515:
1495:
1466:
1446:
1426:
1397:
1377:
1348:
1277:
1231:
1123:
1074:
1021:
875:
842:
810:
708:
663:
640:
575:
542:
502:
473:
435:
409:
389:
339:
310:
282:
181:
148:
3786:
3771:
2870:solving the previous equation so it is shown as
3376:. It should also be mentioned that even though
236:. There also exists another type of estimator:
4123:{\displaystyle {\widehat {\sigma }}^{2}=SSD/n}
2603:{\displaystyle =4\cdot 1/4\cdot (\theta +2)-2}
5193:. Vol. 4. CRC Press. pp. 601–720 .
1982:can always have functional relationship with
671:is the parameter being estimated. The error,
8:
5234:
5232:
5012:-axis, the difference becomes more obvious.
4067:{\displaystyle {\widehat {\mu }}={\bar {X}}}
5019:Comparison between good and bad estimator.
3345: − 1 because if we divided with
2901:{\displaystyle \operatorname {E} =\theta }
2054:{\displaystyle p_{1}=1/4\cdot (\theta +2)}
1086:of the estimator. The sampling deviation,
120:. It can be either finite-dimensional (in
4965:
4940:
4925:
4887:
4861:
4846:
4828:
4826:
4799:
4774:
4759:
4735:
4729:
4708:
4702:
4656:
4632:
4631:
4607:
4586:
4585:
4571:
4540:
4519:
4518:
4504:
4459:
4458:
4447:
4393:
4392:
4390:
4360:
4275:
4260:
4246:
4244:
4199:
4197:
4171:
4166:
4161:
4112:
4094:
4083:
4082:
4079:
4053:
4052:
4038:
4037:
4035:
4009:
4003:
3982:
3976:
3949:
3921:
3897:
3896:
3894:
3803:
3774:
3768:
3731:Mathematically, a sequence of estimators
3675:
3674:
3672:
3646:
3620:
3619:
3617:
3593:
3567:
3566:
3564:
3533:
3515:
3514:
3488:
3487:
3461:
3460:
3449:
3419:
3413:
3392:
3387:
3381:
3360:
3354:
3322:
3305:
3300:
3280:
3258:
3242:
3236:
3235:
3226:
3213:
3202:
3180:
3171:
3166:
3156:
3116:
3106:
3089:
3061:
3042:
3029:
3022:
3013:
3003:
2996:
2966:
2960:
2940:
2919:
2913:
2875:
2855:
2825:
2824:
2822:
2781:
2780:
2769:
2736:
2730:
2710:
2689:
2683:
2648:
2614:
2568:
2554:
2528:
2513:
2487:
2469:
2461:
2432:
2411:
2403:
2374:
2359:
2333:
2332:
2324:
2304:
2278:
2277:
2275:
2248:
2233:
2216:
2215:
2213:
2193:
2169:
2145:
2124:
2118:
2098:
2066:
2025:
2013:
2007:
1987:
1961:
1960:
1958:
1927:{\displaystyle B({\widehat {\theta }})=0}
1904:
1903:
1895:
1873:
1843:
1842:
1840:
1798:
1797:
1765:
1764:
1753:
1723:
1722:
1720:
1679:
1678:
1667:
1638:
1637:
1635:
1594:
1593:
1582:
1543:
1542:
1531:
1508:
1482:
1481:
1479:
1459:
1439:
1413:
1412:
1410:
1390:
1364:
1363:
1361:
1326:
1325:
1299:
1298:
1290:
1264:
1263:
1261:
1220:
1202:
1201:
1178:
1177:
1148:
1147:
1136:
1110:
1109:
1107:
1049:
1048:
1037:
1002:
1001:
969:
968:
942:
941:
909:
908:
891:
862:
861:
859:
835:
796:
766:
765:
736:
735:
724:
695:
694:
692:
656:
609:
608:
591:
562:
561:
559:
535:
489:
488:
486:
451:
450:
448:
422:
402:
367:
366:
364:
326:
325:
323:
303:
275:
168:
167:
165:
141:
5320:Probability Theory: The logic of science
5177:
4074:and to check for variance confirm that
474:{\displaystyle {\widehat {\theta }}(x)}
390:{\displaystyle {\widehat {\theta }}(X)}
5217:Kosorok (2008), Section 3.1, pp 35–39.
5296:Bol'shev, Login Nikolaevich (2001) ,
5187:"Data Analysis, including Statistics"
5185:Mosteller, F.; Tukey, J. W. (1987) .
7:
4382:implies asymptotic normality of the
3689:{\displaystyle {\widehat {\theta }}}
3634:{\displaystyle {\widehat {\theta }}}
3581:{\displaystyle {\widehat {\theta }}}
2839:{\displaystyle {\widehat {\theta }}}
2292:{\displaystyle {\widehat {\theta }}}
1975:{\displaystyle {\widehat {\theta }}}
1857:{\displaystyle {\widehat {\theta }}}
1737:{\displaystyle {\widehat {\theta }}}
1652:{\displaystyle {\widehat {\theta }}}
1496:{\displaystyle {\widehat {\theta }}}
1427:{\displaystyle {\widehat {\theta }}}
1378:{\displaystyle {\widehat {\theta }}}
1278:{\displaystyle {\widehat {\theta }}}
1124:{\displaystyle {\widehat {\theta }}}
876:{\displaystyle {\widehat {\theta }}}
709:{\displaystyle {\widehat {\theta }}}
576:{\displaystyle {\widehat {\theta }}}
503:{\displaystyle {\widehat {\theta }}}
340:{\displaystyle {\widehat {\theta }}}
182:{\displaystyle {\widehat {\theta }}}
54:is a commonly used estimator of the
2759:minimum-variance unbiased estimator
1385:is a function of the true value of
4874:
4833:
4622:
4573:
4552:{\displaystyle \operatorname {E} }
4506:
4449:
4214:{\displaystyle {\xrightarrow {D}}}
3876:measures of statistical dispersion
3781:
3439:Relationships among the quantities
3286:
3095:
2877:
2771:
2502:{\displaystyle =4/n\cdot np_{1}-2}
1788:
1755:
1669:
1584:
1533:
1316:
1192:
1165:
1039:
992:
932:
753:
25:
5430:Fundamentals on Estimation Theory
5348:. Springer Series in Statistics.
3862:An estimator that converges to a
2086:{\displaystyle 0<\theta <1}
318:is usually denoted by the symbol
5245:. Springer Texts in Statistics.
3746:} is a consistent estimator for
514:, but this can cause confusion.
4027:empirical distribution function
3760:, no matter how small, we have
3285:
3161:
3094:
3001:
2850:with and parameter of interest
2543:{\displaystyle =4\cdot p_{1}-2}
136:). If the parameter is denoted
5082:Best linear unbiased estimator
4971:
4958:
4946:
4933:
4893:
4880:
4862:
4852:
4839:
4829:
4805:
4792:
4780:
4767:
4677:
4671:
4653:
4643:
4628:
4619:
4613:
4604:
4582:
4579:
4546:
4537:
4515:
4512:
4470:
4455:
4398:
4302:
4290:
4272:
4253:
4058:
3955:
3942:
3927:
3914:
3872:estimation of scale parameters
3778:
3530:
3526:
3511:
3505:
3499:
3484:
3472:
3457:
3255:
3248:
3219:
2889:
2883:
2792:
2777:
2591:
2579:
2438:
2425:
2386:
2353:
2344:
2329:
2175:
2156:
2048:
2036:
1915:
1900:
1815:
1794:
1776:
1761:
1690:
1675:
1605:
1590:
1554:
1539:
1337:
1322:
1310:
1295:
1226:
1217:
1213:
1198:
1174:
1171:
1159:
1144:
1069:
1066:
1060:
1045:
1013:
998:
986:
980:
962:
959:
953:
938:
926:
920:
902:
896:
802:
793:
783:
777:
762:
759:
747:
732:
626:
620:
602:
596:
468:
462:
384:
378:
1:
5114:generalized method of moments
5027:). In some cases an unbiased
4181:{\displaystyle 1/{\sqrt {n}}}
2450:{\displaystyle =4/n\cdot E-2}
2299:is an unbiased estimator for
510:is interpreted directly as a
249:probability density functions
42:of a given quantity based on
38:is a rule for calculating an
3111:
3008:
2637:{\displaystyle =\theta +2-2}
2181:{\displaystyle Bin(n,p_{1})}
2113:leaves, the random variable
221:, an estimator is a type of
128:), or infinite-dimensional (
5303:Encyclopedia of Mathematics
5135:Sensitivity and specificity
4744:{\displaystyle \theta _{2}}
4717:{\displaystyle \theta _{1}}
4223:convergence in distribution
4018:{\displaystyle T_{\theta }}
3428:{\displaystyle \sigma ^{2}}
3369:{\displaystyle \sigma ^{2}}
2975:{\displaystyle \theta _{1}}
2928:{\displaystyle \theta _{2}}
2745:{\displaystyle \theta _{1}}
2698:{\displaystyle \theta _{2}}
1405:so saying that the bias of
349:algebra of random variables
86:circumstances. However, in
5461:
5380:Theory of Point Estimation
5324:Cambridge University Press
5268:"Properties of Estimators"
5130:Pitman closeness criterion
5119:Minimum mean squared error
5067:
5061:
4724:is the good estimator and
4430:
4407:{\displaystyle {\bar {X}}}
4137:
3713:
1948:median-unbiased estimators
5358:10.1007/978-0-387-74978-5
5342:Kosorok, Michael (2008).
3435:the reverse is not true.
3401:{\displaystyle S_{n}^{2}}
253:spectral density function
5098:Markov chain Monte Carlo
3753:if and only if, for all
3341:Note we are dividing by
2659:{\displaystyle =\theta }
1953:In a practical problem,
5403:Mathematical Statistics
5298:"Statistical estimator"
5048:Lehmann–Scheffé theorem
4433:Efficiency (statistics)
4368:{\displaystyle \theta }
3857:converges almost surely
3722:converge in probability
3654:{\displaystyle \theta }
3601:{\displaystyle \theta }
2948:{\displaystyle \theta }
2863:{\displaystyle \theta }
2725:vs. a biased estimator
2718:{\displaystyle \theta }
2312:{\displaystyle \theta }
2201:{\displaystyle \theta }
1995:{\displaystyle \theta }
1946:of a distribution: see
1881:{\displaystyle \theta }
1467:{\displaystyle \theta }
1398:{\displaystyle \theta }
664:{\displaystyle \theta }
311:{\displaystyle \theta }
283:{\displaystyle \theta }
211:asymptotic distribution
149:{\displaystyle \theta }
18:Asymptotically unbiased
5378:; Casella, G. (1998).
5273:. University of Oxford
5165:Well-behaved statistic
5020:
5005:
4995:
4978:
4911:
4812:
4745:
4718:
4687:
4553:
4489:
4408:
4369:
4351:converges weakly to a
4312:
4215:
4182:
4124:
4068:
4019:
3992:
3962:
3841:
3690:
3655:
3635:
3602:
3582:
3546:
3429:
3402:
3370:
3332:
3268:
3218:
3136:
3077:
2976:
2949:
2929:
2902:
2864:
2840:
2811:
2753:
2746:
2719:
2699:
2660:
2638:
2604:
2544:
2503:
2451:
2393:
2313:
2293:
2264:
2202:
2182:
2134:
2107:
2087:
2055:
1996:
1976:
1928:
1882:
1858:
1830:accuracy and precision
1822:
1738:
1709:
1653:
1624:
1567:
1517:
1497:
1468:
1448:
1428:
1399:
1379:
1350:
1279:
1233:
1125:
1076:
1023:
877:
844:
812:
710:
665:
642:
577:
544:
504:
475:
437:
411:
391:
341:
312:
284:
183:
150:
126:semi-parametric models
5226:Jaynes (2007), p.172.
5068:Further information:
5052:Rao–Blackwell theorem
5018:
5004:
4994:
4979:
4912:
4813:
4746:
4719:
4688:
4554:
4490:
4409:
4380:central limit theorem
4370:
4313:
4234:asymptotically normal
4216:
4183:
4146:asymptotically normal
4125:
4069:
4020:
3993:
3991:{\displaystyle T_{n}}
3963:
3842:
3705:Behavioral properties
3691:
3656:
3636:
3603:
3583:
3547:
3430:
3403:
3371:
3333:
3269:
3198:
3137:
3078:
2977:
2950:
2930:
2903:
2865:
2841:
2812:
2747:
2720:
2700:
2677:
2661:
2639:
2605:
2545:
2504:
2452:
2394:
2314:
2294:
2265:
2203:
2183:
2135:
2133:{\displaystyle N_{1}}
2108:
2088:
2056:
1997:
1977:
1929:
1883:
1859:
1823:
1739:
1710:
1654:
1625:
1568:
1518:
1498:
1469:
1454:means that for every
1449:
1429:
1400:
1380:
1351:
1280:
1234:
1126:
1077:
1024:
878:
845:
813:
711:
666:
643:
578:
545:
518:Quantified properties
505:
476:
438:
412:
392:
342:
313:
285:
184:
151:
134:non-parametric models
5266:Lauritzen, Steffen.
5104:Maximum a posteriori
5044:Gauss–Markov theorem
4924:
4825:
4758:
4728:
4701:
4570:
4503:
4446:
4389:
4359:
4353:dirac delta function
4328:In this formulation
4243:
4196:
4160:
4140:Asymptotic normality
4134:Asymptotic normality
4078:
4034:
4002:
3975:
3893:
3767:
3716:Consistent estimator
3671:
3645:
3616:
3592:
3563:
3448:
3412:
3380:
3353:
3279:
3155:
3088:
2995:
2959:
2939:
2912:
2874:
2854:
2821:
2768:
2729:
2709:
2682:
2647:
2613:
2553:
2512:
2460:
2402:
2323:
2303:
2274:
2270:. One can show that
2212:
2192:
2144:
2117:
2097:
2065:
2006:
1986:
1957:
1894:
1872:
1839:
1752:
1719:
1666:
1634:
1581:
1530:
1507:
1478:
1458:
1438:
1409:
1389:
1360:
1289:
1260:
1135:
1106:
1036:
890:
858:
834:
723:
691:
655:
590:
558:
534:
485:
447:
421:
401:
363:
355:is used to denote a
322:
302:
274:
217:. In the context of
164:
140:
5140:Shrinkage estimator
5088:Invariant estimator
5029:efficient estimator
4420:asymptotics section
4342:asymptotic variance
4334:asymptotic variance
4284:
4208:
4188:as the sample size
4154:normal distribution
3859:to the true value.
3853:strongly consistent
3397:
3310:
3176:
2705:is centered around
2392:{\displaystyle E=E}
830:For a given sample
554:" of the estimator
530:For a given sample
436:{\displaystyle X=x}
238:interval estimators
98:An "estimator" or "
67:interval estimators
5401:Shao, Jun (1998),
5021:
5006:
4996:
4974:
4907:
4808:
4741:
4714:
4683:
4549:
4497:mean squared error
4485:
4416:maximum likelihood
4404:
4365:
4332:can be called the
4308:
4211:
4178:
4120:
4064:
4015:
3988:
3958:
3882:Fisher consistency
3837:
3785:
3686:
3651:
3631:
3598:
3578:
3557:standard deviation
3542:
3425:
3398:
3383:
3366:
3328:
3296:
3264:
3162:
3132:
3073:
2972:
2955:then distribution
2945:
2925:
2898:
2860:
2836:
2807:
2754:
2742:
2715:
2695:
2656:
2634:
2600:
2540:
2499:
2447:
2389:
2309:
2289:
2260:
2198:
2178:
2130:
2103:
2083:
2051:
1992:
1972:
1924:
1878:
1866:unbiased estimator
1854:
1818:
1734:
1705:
1649:
1620:
1563:
1513:
1493:
1464:
1444:
1424:
1395:
1375:
1346:
1275:
1229:
1121:
1072:
1019:
873:
852:sampling deviation
840:
826:Sampling deviation
808:
706:
685:mean squared error
679:Mean squared error
661:
638:
573:
540:
500:
471:
433:
407:
387:
337:
308:
298:. An estimator of
280:
245:density estimation
179:
146:
75:interval estimator
5367:978-0-387-74978-5
5333:978-0-521-59271-0
5252:978-1-85233-896-1
5145:Signal processing
5110:Method of moments
5070:Robust regression
4682:
4640:
4594:
4527:
4495:and have minimal
4467:
4401:
4285:
4251:
4209:
4176:
4091:
4061:
4046:
3905:
3770:
3683:
3628:
3575:
3523:
3496:
3469:
3251:
3196:
3114:
3071:
3011:
2846:. With estimator
2833:
2789:
2341:
2286:
2224:
2106:{\displaystyle n}
1969:
1912:
1851:
1806:
1773:
1731:
1687:
1646:
1602:
1551:
1516:{\displaystyle b}
1490:
1447:{\displaystyle b}
1421:
1372:
1334:
1307:
1272:
1210:
1186:
1156:
1118:
1057:
1010:
977:
950:
917:
870:
854:of the estimator
843:{\displaystyle x}
774:
744:
703:
617:
570:
543:{\displaystyle x}
497:
459:
410:{\displaystyle x}
375:
334:
215:estimation theory
203:mean square error
176:
160:over the symbol:
112:statistical model
88:robust statistics
82:Estimation theory
16:(Redirected from
5452:
5419:
5397:
5382:(2nd ed.).
5371:
5337:
5310:
5283:
5282:
5280:
5278:
5272:
5263:
5257:
5256:
5236:
5227:
5224:
5218:
5215:
5209:
5208:
5182:
5064:Robust estimator
5040:Cramér–Rao bound
5033:Cramér–Rao bound
4983:
4981:
4980:
4975:
4970:
4969:
4945:
4944:
4916:
4914:
4913:
4908:
4906:
4902:
4892:
4891:
4865:
4851:
4850:
4832:
4817:
4815:
4814:
4809:
4804:
4803:
4779:
4778:
4750:
4748:
4747:
4742:
4740:
4739:
4723:
4721:
4720:
4715:
4713:
4712:
4692:
4690:
4689:
4684:
4680:
4661:
4660:
4642:
4641:
4633:
4612:
4611:
4596:
4595:
4587:
4558:
4556:
4555:
4550:
4545:
4544:
4529:
4528:
4520:
4494:
4492:
4491:
4486:
4469:
4468:
4460:
4413:
4411:
4410:
4405:
4403:
4402:
4394:
4374:
4372:
4371:
4366:
4317:
4315:
4314:
4309:
4286:
4276:
4265:
4264:
4252:
4247:
4220:
4218:
4217:
4212:
4210:
4200:
4187:
4185:
4184:
4179:
4177:
4172:
4170:
4129:
4127:
4126:
4121:
4116:
4099:
4098:
4093:
4092:
4084:
4073:
4071:
4070:
4065:
4063:
4062:
4054:
4048:
4047:
4039:
4024:
4022:
4021:
4016:
4014:
4013:
3997:
3995:
3994:
3989:
3987:
3986:
3967:
3965:
3964:
3959:
3954:
3953:
3926:
3925:
3907:
3906:
3898:
3846:
3844:
3843:
3838:
3830:
3826:
3819:
3815:
3808:
3807:
3784:
3759:
3745:
3695:
3693:
3692:
3687:
3685:
3684:
3676:
3661:, is called the
3660:
3658:
3657:
3652:
3640:
3638:
3637:
3632:
3630:
3629:
3621:
3607:
3605:
3604:
3599:
3587:
3585:
3584:
3579:
3577:
3576:
3568:
3559:of an estimator
3551:
3549:
3548:
3543:
3538:
3537:
3525:
3524:
3516:
3498:
3497:
3489:
3471:
3470:
3462:
3434:
3432:
3431:
3426:
3424:
3423:
3408:is unbiased for
3407:
3405:
3404:
3399:
3396:
3391:
3375:
3373:
3372:
3367:
3365:
3364:
3337:
3335:
3334:
3329:
3327:
3326:
3314:
3309:
3304:
3273:
3271:
3270:
3265:
3263:
3262:
3253:
3252:
3247:
3246:
3237:
3231:
3230:
3217:
3212:
3197:
3195:
3181:
3175:
3170:
3141:
3139:
3138:
3133:
3125:
3121:
3120:
3115:
3107:
3082:
3080:
3079:
3074:
3072:
3067:
3066:
3065:
3047:
3046:
3034:
3033:
3023:
3018:
3017:
3012:
3004:
2981:
2979:
2978:
2973:
2971:
2970:
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2924:
2923:
2907:
2905:
2904:
2899:
2869:
2867:
2866:
2861:
2845:
2843:
2842:
2837:
2835:
2834:
2826:
2816:
2814:
2813:
2808:
2791:
2790:
2782:
2751:
2749:
2748:
2743:
2741:
2740:
2724:
2722:
2721:
2716:
2704:
2702:
2701:
2696:
2694:
2693:
2665:
2663:
2662:
2657:
2643:
2641:
2640:
2635:
2609:
2607:
2606:
2601:
2572:
2549:
2547:
2546:
2541:
2533:
2532:
2508:
2506:
2505:
2500:
2492:
2491:
2473:
2456:
2454:
2453:
2448:
2437:
2436:
2415:
2398:
2396:
2395:
2390:
2379:
2378:
2363:
2343:
2342:
2334:
2318:
2316:
2315:
2310:
2298:
2296:
2295:
2290:
2288:
2287:
2279:
2269:
2267:
2266:
2261:
2253:
2252:
2237:
2226:
2225:
2217:
2207:
2205:
2204:
2199:
2187:
2185:
2184:
2179:
2174:
2173:
2139:
2137:
2136:
2131:
2129:
2128:
2112:
2110:
2109:
2104:
2092:
2090:
2089:
2084:
2060:
2058:
2057:
2052:
2029:
2018:
2017:
2001:
1999:
1998:
1993:
1981:
1979:
1978:
1973:
1971:
1970:
1962:
1944:central tendency
1933:
1931:
1930:
1925:
1914:
1913:
1905:
1887:
1885:
1884:
1879:
1863:
1861:
1860:
1855:
1853:
1852:
1844:
1827:
1825:
1824:
1819:
1808:
1807:
1799:
1775:
1774:
1766:
1743:
1741:
1740:
1735:
1733:
1732:
1724:
1714:
1712:
1711:
1706:
1689:
1688:
1680:
1658:
1656:
1655:
1650:
1648:
1647:
1639:
1629:
1627:
1626:
1621:
1604:
1603:
1595:
1572:
1570:
1569:
1564:
1553:
1552:
1544:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1492:
1491:
1483:
1473:
1471:
1470:
1465:
1453:
1451:
1450:
1445:
1433:
1431:
1430:
1425:
1423:
1422:
1414:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1374:
1373:
1365:
1355:
1353:
1352:
1347:
1336:
1335:
1327:
1309:
1308:
1300:
1284:
1282:
1281:
1276:
1274:
1273:
1265:
1238:
1236:
1235:
1230:
1225:
1224:
1212:
1211:
1203:
1188:
1187:
1179:
1158:
1157:
1149:
1130:
1128:
1127:
1122:
1120:
1119:
1111:
1081:
1079:
1078:
1073:
1059:
1058:
1050:
1028:
1026:
1025:
1020:
1012:
1011:
1003:
979:
978:
970:
952:
951:
943:
919:
918:
910:
882:
880:
879:
874:
872:
871:
863:
849:
847:
846:
841:
817:
815:
814:
809:
801:
800:
776:
775:
767:
746:
745:
737:
715:
713:
712:
707:
705:
704:
696:
670:
668:
667:
662:
647:
645:
644:
639:
619:
618:
610:
582:
580:
579:
574:
572:
571:
563:
549:
547:
546:
541:
509:
507:
506:
501:
499:
498:
490:
480:
478:
477:
472:
461:
460:
452:
442:
440:
439:
434:
416:
414:
413:
408:
396:
394:
393:
388:
377:
376:
368:
346:
344:
343:
338:
336:
335:
327:
317:
315:
314:
309:
296:sample estimates
289:
287:
286:
281:
267:Suppose a fixed
188:
186:
185:
180:
178:
177:
169:
155:
153:
152:
147:
71:point estimators
21:
5460:
5459:
5455:
5454:
5453:
5451:
5450:
5449:
5435:
5434:
5426:
5417:
5400:
5394:
5374:
5368:
5341:
5334:
5314:
5295:
5292:
5290:Further reading
5287:
5286:
5276:
5274:
5270:
5265:
5264:
5260:
5253:
5238:
5237:
5230:
5225:
5221:
5216:
5212:
5201:
5184:
5183:
5179:
5174:
5169:
5125:Particle filter
5077:
5072:
5066:
5060:
4961:
4936:
4922:
4921:
4883:
4873:
4869:
4842:
4823:
4822:
4795:
4770:
4756:
4755:
4731:
4726:
4725:
4704:
4699:
4698:
4652:
4603:
4568:
4567:
4536:
4501:
4500:
4444:
4443:
4435:
4429:
4387:
4386:
4357:
4356:
4349:
4256:
4241:
4240:
4230:
4194:
4193:
4158:
4157:
4142:
4136:
4081:
4076:
4075:
4032:
4031:
4005:
4000:
3999:
3978:
3973:
3972:
3945:
3917:
3891:
3890:
3884:
3799:
3798:
3794:
3793:
3789:
3765:
3764:
3754:
3738:
3732:
3718:
3712:
3707:
3669:
3668:
3643:
3642:
3614:
3613:
3590:
3589:
3561:
3560:
3529:
3446:
3445:
3441:
3415:
3410:
3409:
3378:
3377:
3356:
3351:
3350:
3318:
3292:
3277:
3276:
3254:
3238:
3222:
3185:
3153:
3152:
3105:
3101:
3086:
3085:
3057:
3038:
3025:
3024:
3002:
2993:
2992:
2962:
2957:
2956:
2937:
2936:
2915:
2910:
2909:
2872:
2871:
2852:
2851:
2819:
2818:
2766:
2765:
2732:
2727:
2726:
2707:
2706:
2685:
2680:
2679:
2672:
2645:
2644:
2611:
2610:
2551:
2550:
2524:
2510:
2509:
2483:
2458:
2457:
2428:
2400:
2399:
2370:
2321:
2320:
2301:
2300:
2272:
2271:
2244:
2210:
2209:
2190:
2189:
2165:
2142:
2141:
2120:
2115:
2114:
2095:
2094:
2063:
2062:
2009:
2004:
2003:
1984:
1983:
1955:
1954:
1892:
1891:
1870:
1869:
1837:
1836:
1750:
1749:
1717:
1716:
1664:
1663:
1632:
1631:
1579:
1578:
1528:
1527:
1505:
1504:
1476:
1475:
1456:
1455:
1436:
1435:
1407:
1406:
1387:
1386:
1358:
1357:
1287:
1286:
1258:
1257:
1250:
1216:
1133:
1132:
1104:
1103:
1096:
1034:
1033:
888:
887:
856:
855:
832:
831:
828:
792:
721:
720:
689:
688:
681:
653:
652:
588:
587:
556:
555:
532:
531:
528:
520:
512:random variable
483:
482:
445:
444:
419:
418:
399:
398:
361:
360:
357:random variable
320:
319:
300:
299:
272:
271:
265:
243:The problem of
234:parameter space
219:decision theory
191:random variable
162:
161:
138:
137:
130:semi-parametric
96:
56:population mean
28:
23:
22:
15:
12:
11:
5:
5458:
5456:
5448:
5447:
5437:
5436:
5433:
5432:
5425:
5424:External links
5422:
5421:
5420:
5415:
5398:
5392:
5376:Lehmann, E. L.
5372:
5366:
5339:
5332:
5322:(5 ed.).
5312:
5291:
5288:
5285:
5284:
5258:
5251:
5228:
5219:
5210:
5199:
5176:
5175:
5173:
5170:
5168:
5167:
5162:
5157:
5152:
5150:State observer
5147:
5142:
5137:
5132:
5127:
5122:
5116:
5107:
5101:
5095:
5090:
5085:
5078:
5076:
5073:
5062:Main article:
5059:
5056:
4985:
4984:
4973:
4968:
4964:
4960:
4957:
4954:
4951:
4948:
4943:
4939:
4935:
4932:
4929:
4918:
4917:
4905:
4901:
4898:
4895:
4890:
4886:
4882:
4879:
4876:
4872:
4868:
4864:
4860:
4857:
4854:
4849:
4845:
4841:
4838:
4835:
4831:
4819:
4818:
4807:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4777:
4773:
4769:
4766:
4763:
4738:
4734:
4711:
4707:
4694:
4693:
4679:
4676:
4673:
4670:
4667:
4664:
4659:
4655:
4651:
4648:
4645:
4639:
4636:
4630:
4627:
4624:
4621:
4618:
4615:
4610:
4606:
4602:
4599:
4593:
4590:
4584:
4581:
4578:
4575:
4561:estimator bias
4548:
4543:
4539:
4535:
4532:
4526:
4523:
4517:
4514:
4511:
4508:
4484:
4481:
4478:
4475:
4472:
4466:
4463:
4457:
4454:
4451:
4431:Main article:
4428:
4425:
4400:
4397:
4364:
4347:
4319:
4318:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4283:
4279:
4274:
4271:
4268:
4263:
4259:
4255:
4250:
4228:
4207:
4203:
4192:grows. Using
4175:
4169:
4165:
4138:Main article:
4135:
4132:
4119:
4115:
4111:
4108:
4105:
4102:
4097:
4090:
4087:
4060:
4057:
4051:
4045:
4042:
4012:
4008:
3985:
3981:
3969:
3968:
3957:
3952:
3948:
3944:
3941:
3938:
3935:
3932:
3929:
3924:
3920:
3916:
3913:
3910:
3904:
3901:
3883:
3880:
3849:
3848:
3836:
3833:
3829:
3825:
3822:
3818:
3814:
3811:
3806:
3802:
3797:
3792:
3788:
3783:
3780:
3777:
3773:
3736:
3714:Main article:
3711:
3708:
3706:
3703:
3702:
3701:
3697:
3682:
3679:
3664:standard error
3650:
3627:
3624:
3597:
3574:
3571:
3553:
3541:
3536:
3532:
3528:
3522:
3519:
3513:
3510:
3507:
3504:
3501:
3495:
3492:
3486:
3483:
3480:
3477:
3474:
3468:
3465:
3459:
3456:
3453:
3440:
3437:
3422:
3418:
3395:
3390:
3386:
3363:
3359:
3339:
3338:
3325:
3321:
3317:
3313:
3308:
3303:
3299:
3295:
3291:
3288:
3284:
3274:
3261:
3257:
3250:
3245:
3241:
3234:
3229:
3225:
3221:
3216:
3211:
3208:
3205:
3201:
3194:
3191:
3188:
3184:
3179:
3174:
3169:
3165:
3160:
3143:
3142:
3131:
3128:
3124:
3119:
3113:
3110:
3104:
3100:
3097:
3093:
3083:
3070:
3064:
3060:
3056:
3053:
3050:
3045:
3041:
3037:
3032:
3028:
3021:
3016:
3010:
3007:
3000:
2969:
2965:
2944:
2922:
2918:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2859:
2832:
2829:
2806:
2803:
2800:
2797:
2794:
2788:
2785:
2779:
2776:
2773:
2739:
2735:
2714:
2692:
2688:
2671:
2668:
2655:
2652:
2633:
2630:
2627:
2624:
2621:
2618:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2571:
2567:
2564:
2561:
2558:
2539:
2536:
2531:
2527:
2523:
2520:
2517:
2498:
2495:
2490:
2486:
2482:
2479:
2476:
2472:
2468:
2465:
2446:
2443:
2440:
2435:
2431:
2427:
2424:
2421:
2418:
2414:
2410:
2407:
2388:
2385:
2382:
2377:
2373:
2369:
2366:
2362:
2358:
2355:
2352:
2349:
2346:
2340:
2337:
2331:
2328:
2308:
2285:
2282:
2259:
2256:
2251:
2247:
2243:
2240:
2236:
2232:
2229:
2223:
2220:
2197:
2177:
2172:
2168:
2164:
2161:
2158:
2155:
2152:
2149:
2127:
2123:
2102:
2082:
2079:
2076:
2073:
2070:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2028:
2024:
2021:
2016:
2012:
1991:
1968:
1965:
1923:
1920:
1917:
1911:
1908:
1902:
1899:
1889:if and only if
1877:
1850:
1847:
1835:The estimator
1817:
1814:
1811:
1805:
1802:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1772:
1769:
1763:
1760:
1757:
1746:
1745:
1730:
1727:
1704:
1701:
1698:
1695:
1692:
1686:
1683:
1677:
1674:
1671:
1660:
1645:
1642:
1619:
1616:
1613:
1610:
1607:
1601:
1598:
1592:
1589:
1586:
1562:
1559:
1556:
1550:
1547:
1541:
1538:
1535:
1512:
1489:
1486:
1463:
1443:
1420:
1417:
1394:
1371:
1368:
1345:
1342:
1339:
1333:
1330:
1324:
1321:
1318:
1315:
1312:
1306:
1303:
1297:
1294:
1285:is defined as
1271:
1268:
1249:
1246:
1241:expected value
1228:
1223:
1219:
1215:
1209:
1206:
1200:
1197:
1194:
1191:
1185:
1182:
1176:
1173:
1170:
1167:
1164:
1161:
1155:
1152:
1146:
1143:
1140:
1117:
1114:
1095:
1092:
1084:expected value
1071:
1068:
1065:
1062:
1056:
1053:
1047:
1044:
1041:
1030:
1029:
1018:
1015:
1009:
1006:
1000:
997:
994:
991:
988:
985:
982:
976:
973:
967:
964:
961:
958:
955:
949:
946:
940:
937:
934:
931:
928:
925:
922:
916:
913:
907:
904:
901:
898:
895:
883:is defined as
869:
866:
839:
827:
824:
819:
818:
807:
804:
799:
795:
791:
788:
785:
782:
779:
773:
770:
764:
761:
758:
755:
752:
749:
743:
740:
734:
731:
728:
702:
699:
680:
677:
660:
649:
648:
637:
634:
631:
628:
625:
622:
616:
613:
607:
604:
601:
598:
595:
583:is defined as
569:
566:
539:
527:
524:
519:
516:
496:
493:
470:
467:
464:
458:
455:
432:
429:
426:
406:
386:
383:
380:
374:
371:
333:
330:
307:
279:
264:
261:
227:loss functions
175:
172:
145:
100:point estimate
95:
92:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5457:
5446:
5443:
5442:
5440:
5431:
5428:
5427:
5423:
5418:
5416:0-387-98674-X
5412:
5408:
5404:
5399:
5395:
5393:0-387-98502-6
5389:
5385:
5381:
5377:
5373:
5369:
5363:
5359:
5355:
5351:
5347:
5346:
5340:
5335:
5329:
5325:
5321:
5317:
5316:Jaynes, E. T.
5313:
5309:
5305:
5304:
5299:
5294:
5293:
5289:
5269:
5262:
5259:
5254:
5248:
5244:
5243:
5235:
5233:
5229:
5223:
5220:
5214:
5211:
5206:
5202:
5200:0-534-05101-4
5196:
5192:
5188:
5181:
5178:
5171:
5166:
5163:
5161:
5160:Wiener filter
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5120:
5117:
5115:
5111:
5108:
5105:
5102:
5099:
5096:
5094:
5093:Kalman filter
5091:
5089:
5086:
5083:
5080:
5079:
5074:
5071:
5065:
5057:
5055:
5053:
5049:
5045:
5041:
5036:
5034:
5030:
5026:
5017:
5013:
5011:
5003:
4999:
4993:
4989:
4966:
4962:
4955:
4952:
4949:
4941:
4937:
4930:
4927:
4920:
4919:
4903:
4899:
4896:
4888:
4884:
4877:
4870:
4866:
4858:
4855:
4847:
4843:
4836:
4821:
4820:
4800:
4796:
4789:
4786:
4783:
4775:
4771:
4764:
4761:
4754:
4753:
4752:
4736:
4732:
4709:
4705:
4674:
4668:
4665:
4662:
4657:
4649:
4646:
4637:
4634:
4625:
4616:
4608:
4600:
4597:
4591:
4588:
4576:
4566:
4565:
4564:
4562:
4541:
4533:
4530:
4524:
4521:
4509:
4498:
4482:
4479:
4476:
4473:
4464:
4461:
4452:
4441:
4440:loss function
4434:
4426:
4424:
4421:
4417:
4395:
4385:
4381:
4376:
4362:
4354:
4350:
4343:
4339:
4335:
4331:
4326:
4324:
4305:
4299:
4296:
4293:
4287:
4281:
4277:
4269:
4266:
4261:
4257:
4248:
4239:
4238:
4237:
4235:
4231:
4224:
4205:
4201:
4191:
4173:
4167:
4163:
4155:
4152:approaches a
4151:
4147:
4141:
4133:
4131:
4117:
4113:
4109:
4106:
4103:
4100:
4095:
4088:
4085:
4055:
4049:
4043:
4040:
4028:
4010:
4006:
3983:
3979:
3950:
3946:
3939:
3936:
3933:
3930:
3922:
3918:
3911:
3908:
3902:
3899:
3889:
3888:
3887:
3881:
3879:
3877:
3873:
3869:
3865:
3860:
3858:
3854:
3834:
3831:
3827:
3823:
3820:
3816:
3812:
3809:
3804:
3800:
3795:
3790:
3775:
3763:
3762:
3761:
3757:
3752:
3749:
3743:
3739:
3729:
3727:
3723:
3717:
3709:
3704:
3698:
3680:
3677:
3666:
3665:
3648:
3625:
3622:
3611:
3595:
3572:
3569:
3558:
3554:
3539:
3534:
3520:
3517:
3508:
3502:
3493:
3490:
3481:
3478:
3475:
3466:
3463:
3454:
3451:
3443:
3442:
3438:
3436:
3420:
3416:
3393:
3388:
3384:
3361:
3357:
3348:
3344:
3323:
3319:
3315:
3311:
3306:
3301:
3297:
3293:
3289:
3282:
3275:
3259:
3243:
3239:
3232:
3227:
3223:
3214:
3209:
3206:
3203:
3199:
3192:
3189:
3186:
3182:
3177:
3172:
3167:
3163:
3158:
3151:
3150:
3149:
3147:
3129:
3126:
3122:
3117:
3108:
3102:
3098:
3091:
3084:
3068:
3062:
3058:
3054:
3051:
3048:
3043:
3039:
3035:
3030:
3026:
3019:
3014:
3005:
2998:
2991:
2990:
2989:
2987:
2983:
2967:
2963:
2942:
2920:
2916:
2895:
2892:
2886:
2880:
2857:
2849:
2830:
2827:
2804:
2801:
2798:
2795:
2786:
2783:
2774:
2762:
2760:
2737:
2733:
2712:
2690:
2686:
2676:
2669:
2667:
2653:
2650:
2631:
2628:
2625:
2622:
2619:
2616:
2597:
2594:
2588:
2585:
2582:
2576:
2573:
2569:
2565:
2562:
2559:
2556:
2537:
2534:
2529:
2525:
2521:
2518:
2515:
2496:
2493:
2488:
2484:
2480:
2477:
2474:
2470:
2466:
2463:
2444:
2441:
2433:
2429:
2422:
2419:
2416:
2412:
2408:
2405:
2383:
2380:
2375:
2371:
2367:
2364:
2360:
2356:
2350:
2347:
2338:
2335:
2326:
2306:
2283:
2280:
2257:
2254:
2249:
2245:
2241:
2238:
2234:
2230:
2227:
2221:
2218:
2195:
2170:
2166:
2162:
2159:
2153:
2150:
2147:
2125:
2121:
2100:
2080:
2077:
2074:
2071:
2068:
2045:
2042:
2039:
2033:
2030:
2026:
2022:
2019:
2014:
2010:
1989:
1966:
1963:
1951:
1949:
1945:
1940:
1935:
1921:
1918:
1909:
1906:
1897:
1890:
1875:
1867:
1848:
1845:
1833:
1831:
1812:
1809:
1803:
1800:
1791:
1785:
1782:
1779:
1770:
1767:
1758:
1728:
1725:
1702:
1699:
1696:
1693:
1684:
1681:
1672:
1661:
1643:
1640:
1617:
1614:
1611:
1608:
1599:
1596:
1587:
1576:
1575:
1574:
1560:
1557:
1548:
1545:
1536:
1524:
1510:
1487:
1484:
1461:
1441:
1418:
1415:
1392:
1369:
1366:
1343:
1340:
1331:
1328:
1319:
1313:
1304:
1301:
1292:
1269:
1266:
1255:
1247:
1245:
1242:
1221:
1207:
1204:
1195:
1189:
1183:
1180:
1168:
1162:
1153:
1150:
1141:
1138:
1115:
1112:
1101:
1093:
1091:
1089:
1085:
1063:
1054:
1051:
1042:
1016:
1007:
1004:
995:
989:
983:
974:
971:
965:
956:
947:
944:
935:
929:
923:
914:
911:
905:
899:
893:
886:
885:
884:
867:
864:
853:
837:
825:
823:
805:
797:
789:
786:
780:
771:
768:
756:
750:
741:
738:
729:
726:
719:
718:
717:
700:
697:
686:
678:
676:
674:
658:
635:
632:
629:
623:
614:
611:
605:
599:
593:
586:
585:
584:
567:
564:
553:
537:
525:
523:
517:
515:
513:
494:
491:
465:
456:
453:
430:
427:
424:
404:
381:
372:
369:
358:
354:
350:
331:
328:
305:
297:
293:
277:
270:
262:
260:
258:
254:
250:
246:
241:
239:
235:
230:
228:
224:
223:decision rule
220:
216:
212:
208:
204:
200:
194:
192:
173:
170:
159:
143:
135:
131:
127:
123:
119:
118:
113:
109:
105:
101:
93:
91:
89:
84:
83:
78:
76:
72:
68:
64:
59:
57:
53:
49:
45:
44:observed data
41:
37:
33:
19:
5402:
5379:
5344:
5319:
5301:
5275:. Retrieved
5261:
5241:
5222:
5213:
5205:Google Books
5203:– via
5190:
5180:
5037:
5022:
5009:
5007:
4997:
4986:
4695:
4436:
4377:
4355:centered at
4345:
4341:
4337:
4333:
4329:
4327:
4322:
4320:
4226:
4189:
4149:
4143:
3970:
3885:
3868:scale factor
3863:
3861:
3852:
3850:
3755:
3750:
3741:
3734:
3730:
3719:
3662:
3346:
3342:
3340:
3145:
3144:
2985:
2984:
2847:
2763:
2755:
2093:. Then, for
1952:
1936:
1834:
1747:
1744:is unbiased.
1525:
1474:the bias of
1251:
1240:
1097:
1087:
1031:
851:
829:
820:
682:
672:
650:
529:
521:
352:
295:
294:to a set of
292:sample space
268:
266:
242:
231:
199:unbiasedness
195:
115:
97:
80:
79:
60:
35:
29:
4384:sample mean
3726:sample size
3710:Consistency
3700:algorithms.
3610:square root
2986:Expectation
443:) is then
257:time series
207:consistency
52:sample mean
5277:9 December
5172:References
5155:Testimator
5058:Robustness
4427:Efficiency
4221:to denote
1659:is biased.
417:(i.e. for
351:: thus if
263:Definition
158:circumflex
122:parametric
94:Background
61:There are
32:statistics
5445:Estimator
5308:EMS Press
4963:θ
4956:
4938:θ
4931:
4900:θ
4897:−
4885:θ
4878:
4859:θ
4856:−
4844:θ
4837:
4797:θ
4790:
4772:θ
4765:
4733:θ
4706:θ
4675:θ
4669:
4650:θ
4647:−
4638:^
4635:θ
4626:
4601:θ
4598:−
4592:^
4589:θ
4577:
4534:θ
4531:−
4525:^
4522:θ
4510:
4477:θ
4474:−
4465:^
4462:θ
4453:
4399:¯
4363:θ
4321:for some
4270:θ
4267:−
4089:^
4086:σ
4059:¯
4044:^
4041:μ
4011:θ
3951:θ
3934:θ
3903:^
3900:θ
3824:ε
3813:θ
3810:−
3782:∞
3779:→
3748:parameter
3681:^
3678:θ
3649:θ
3626:^
3623:θ
3596:θ
3573:^
3570:θ
3521:^
3518:θ
3494:^
3491:θ
3482:
3467:^
3464:θ
3455:
3417:σ
3358:σ
3320:σ
3290:
3249:¯
3233:−
3200:∑
3190:−
3130:μ
3112:¯
3099:
3052:⋯
3009:¯
2964:θ
2943:θ
2917:θ
2896:θ
2881:
2858:θ
2831:^
2828:θ
2799:θ
2796:−
2787:^
2784:θ
2775:
2734:θ
2713:θ
2687:θ
2654:θ
2629:−
2620:θ
2595:−
2583:θ
2577:⋅
2563:⋅
2535:−
2522:⋅
2494:−
2478:⋅
2442:−
2420:⋅
2381:−
2368:⋅
2339:^
2336:θ
2307:θ
2284:^
2281:θ
2255:−
2242:⋅
2222:^
2219:θ
2196:θ
2075:θ
2040:θ
2034:⋅
1990:θ
1967:^
1964:θ
1910:^
1907:θ
1876:θ
1849:^
1846:θ
1813:θ
1810:−
1804:^
1801:θ
1792:
1783:θ
1780:−
1771:^
1768:θ
1759:
1729:^
1726:θ
1697:θ
1694:−
1685:^
1682:θ
1673:
1644:^
1641:θ
1615:≠
1612:θ
1609:−
1600:^
1597:θ
1588:
1561:θ
1558:−
1549:^
1546:θ
1537:
1488:^
1485:θ
1462:θ
1419:^
1416:θ
1393:θ
1370:^
1367:θ
1344:θ
1341:−
1332:^
1329:θ
1320:
1305:^
1302:θ
1270:^
1267:θ
1208:^
1205:θ
1196:
1190:−
1184:^
1181:θ
1169:
1154:^
1151:θ
1142:
1116:^
1113:θ
1055:^
1052:θ
1043:
1008:^
1005:θ
996:
990:−
975:^
972:θ
948:^
945:θ
936:
930:−
915:^
912:θ
868:^
865:θ
790:θ
787:−
772:^
769:θ
757:
742:^
739:θ
730:
701:^
698:θ
659:θ
633:θ
630:−
615:^
612:θ
568:^
565:θ
495:^
492:θ
457:^
454:θ
373:^
370:θ
332:^
329:θ
306:θ
278:θ
269:parameter
174:^
171:θ
144:θ
108:parameter
104:statistic
36:estimator
5439:Category
5407:Springer
5384:Springer
5350:Springer
5318:(2007).
5075:See also
4278:→
4202:→
3864:multiple
3855:, if it
3146:Variance
2761:(MVUE).
2670:Unbiased
1100:variance
1094:Variance
117:estimand
48:estimand
40:estimate
4025:is the
2061:, with
1573:and 0:
1082:is the
550:, the "
102:" is a
5413:
5390:
5364:
5330:
5249:
5197:
5121:(MMSE)
5100:(MCMC)
5084:(BLUE)
4681:
4499:(MSE)
3971:Where
3758:> 0
1939:median
1864:is an
1032:where
850:, the
651:where
69:. The
5271:(PDF)
5106:(MAP)
3608:(the
552:error
526:Error
255:of a
110:in a
63:point
34:, an
5411:ISBN
5388:ISBN
5362:ISBN
5328:ISBN
5279:2023
5247:ISBN
5195:ISBN
5025:MVUE
4950:<
4867:<
4784:<
4378:The
4340:the
3998:and
3821:<
3555:The
2078:<
2072:<
1254:bias
1252:The
1248:Bias
1098:The
683:The
132:and
124:and
65:and
5354:doi
4953:MSE
4928:MSE
4787:Var
4762:Var
4666:Var
4330:V/n
4236:if
4232:is
4144:An
3874:by
3772:lim
3744:≥ 0
3667:of
3641:of
3588:of
3479:Var
3452:MSE
1868:of
1662:If
1577:If
1503:is
1434:is
1256:of
1139:Var
1102:of
727:MSE
687:of
30:In
5441::
5409:,
5405:,
5386:.
5360:.
5352:.
5326:.
5306:,
5300:,
5231:^
5189:.
5112:,
5054:.
5050:,
5046:,
5042:,
4375:.
4325:.
4225:,
4130:.
3878:.
3787:Pr
3740:;
3283:2.
3159:1.
3092:2.
2999:1.
2817:,
2666:.
2319::
2208::
1950:.
1832:.
1715:,
1630:,
1523:.
229:.
209:,
205:,
201:,
58:.
5396:.
5370:.
5356::
5338:.
5336:.
5311:.
5281:.
5255:.
5207:.
5010:y
4972:)
4967:2
4959:(
4947:)
4942:1
4934:(
4904:|
4894:)
4889:2
4881:(
4875:E
4871:|
4863:|
4853:)
4848:1
4840:(
4834:E
4830:|
4806:)
4801:2
4793:(
4781:)
4776:1
4768:(
4737:2
4710:1
4678:)
4672:(
4663:+
4658:2
4654:)
4644:)
4629:(
4623:E
4620:(
4617:=
4614:]
4609:2
4605:)
4583:(
4580:[
4574:E
4547:]
4542:2
4538:)
4516:(
4513:[
4507:E
4483:0
4480:=
4471:)
4456:(
4450:E
4396:X
4348:n
4346:t
4338:V
4323:V
4306:,
4303:)
4300:V
4297:,
4294:0
4291:(
4288:N
4282:D
4273:)
4262:n
4258:t
4254:(
4249:n
4229:n
4227:t
4206:D
4190:n
4174:n
4168:/
4164:1
4150:θ
4118:n
4114:/
4110:D
4107:S
4104:S
4101:=
4096:2
4056:X
4050:=
4007:T
3984:n
3980:T
3956:)
3947:T
3943:(
3940:h
3937:=
3931:,
3928:)
3923:n
3919:T
3915:(
3912:h
3909:=
3847:.
3835:1
3832:=
3828:}
3817:|
3805:n
3801:t
3796:|
3791:{
3776:n
3756:ε
3751:θ
3742:n
3737:n
3735:t
3733:{
3696:.
3540:,
3535:2
3531:)
3527:)
3512:(
3509:B
3506:(
3503:+
3500:)
3485:(
3476:=
3473:)
3458:(
3421:2
3394:2
3389:n
3385:S
3362:2
3347:n
3343:n
3324:2
3316:=
3312:]
3307:2
3302:n
3298:S
3294:[
3287:E
3260:2
3256:)
3244:n
3240:X
3228:i
3224:X
3220:(
3215:n
3210:1
3207:=
3204:i
3193:1
3187:n
3183:1
3178:=
3173:2
3168:n
3164:S
3127:=
3123:]
3118:n
3109:X
3103:[
3096:E
3069:n
3063:n
3059:X
3055:+
3049:+
3044:2
3040:X
3036:+
3031:1
3027:X
3020:=
3015:n
3006:X
2968:1
2921:2
2893:=
2890:]
2887:T
2884:[
2878:E
2848:T
2805:0
2802:=
2793:)
2778:(
2772:E
2752:.
2738:1
2691:2
2651:=
2632:2
2626:2
2623:+
2617:=
2598:2
2592:)
2589:2
2586:+
2580:(
2574:4
2570:/
2566:1
2560:4
2557:=
2538:2
2530:1
2526:p
2519:4
2516:=
2497:2
2489:1
2485:p
2481:n
2475:n
2471:/
2467:4
2464:=
2445:2
2439:]
2434:1
2430:N
2426:[
2423:E
2417:n
2413:/
2409:4
2406:=
2387:]
2384:2
2376:1
2372:N
2365:n
2361:/
2357:4
2354:[
2351:E
2348:=
2345:]
2330:[
2327:E
2258:2
2250:1
2246:N
2239:n
2235:/
2231:4
2228:=
2176:)
2171:1
2167:p
2163:,
2160:n
2157:(
2154:n
2151:i
2148:B
2126:1
2122:N
2101:n
2081:1
2069:0
2049:)
2046:2
2043:+
2037:(
2031:4
2027:/
2023:1
2020:=
2015:1
2011:p
1922:0
1919:=
1916:)
1901:(
1898:B
1816:)
1795:(
1789:E
1786:=
1777:)
1762:(
1756:E
1703:0
1700:=
1691:)
1676:(
1670:E
1618:0
1606:)
1591:(
1585:E
1555:)
1540:(
1534:E
1511:b
1442:b
1338:)
1323:(
1317:E
1314:=
1311:)
1296:(
1293:B
1227:]
1222:2
1218:)
1214:]
1199:[
1193:E
1175:(
1172:[
1166:E
1163:=
1160:)
1145:(
1088:d
1070:)
1067:)
1064:X
1061:(
1046:(
1040:E
1017:,
1014:)
999:(
993:E
987:)
984:x
981:(
966:=
963:)
960:)
957:X
954:(
939:(
933:E
927:)
924:x
921:(
906:=
903:)
900:x
897:(
894:d
838:x
806:.
803:]
798:2
794:)
784:)
781:X
778:(
763:(
760:[
754:E
751:=
748:)
733:(
673:e
636:,
627:)
624:x
621:(
606:=
603:)
600:x
597:(
594:e
538:x
469:)
466:x
463:(
431:x
428:=
425:X
405:x
385:)
382:X
379:(
353:X
20:)
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