936:. Fisher consistency and asymptotic consistency are distinct concepts, although both aim to define a desirable property of an estimator. While many estimators are consistent in both senses, neither definition encompasses the other. For example, suppose we take an estimator
797:
389:
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on (0,θ) and we wish to estimate θ. The sample maximum is Fisher consistent, but downwardly biased. Conversely, the sample variance is an unbiased estimate of the population variance, but is not Fisher consistent.
642:
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is a deterministic sequence of nonzero numbers converging to zero. This estimator is asymptotically consistent, but not Fisher consistent for any
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1014:
Philosophical
Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character
975:
estimate of the population mean, but not all Fisher consistent estimates are unbiased. Suppose we observe a sample from a
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A loss function is Fisher consistent if the population minimizer of the risk leads to the Bayes optimal decision rule.
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over all permutations of the data. The resulting estimator will have the same expected value as
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792:{\displaystyle n^{-1}\sum _{i=1}^{n}\sum _{j=1}^{m}p_{j}Z_{j}=n^{-1}\sum _{i=1}^{n}\mu =\mu ,}
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1049:
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384:{\displaystyle T\left(\lim _{n\rightarrow \infty }{\hat {F}}_{n}\right)=\theta .\,}
309:, allowing us to express Fisher consistency as a limit — the estimator is
17:
943:
that is both Fisher consistent and asymptotically consistent, and then form
89:
1034:
637:{\displaystyle n^{-1}\sum _{i=1}^{n}\sum _{j=1}^{m}I(X_{i}=Z_{j})Z_{j},}
428:
taking on each value in the population. Writing our estimator of θ as
1044:
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32:
asserting that if the estimator were calculated using the entire
40:, the true value of the estimated parameter would be obtained.
814:
gives an estimate that is Fisher consistent for a parameter
1010:"On the mathematical foundations of theoretical statistics"
1125:"Natural Increase Refers to Net Population Growth Rates"
399:
Suppose our sample is obtained from a finite population
924:
Relationship to asymptotic consistency and unbiasedness
932:
in statistics usually refers to an estimator that is
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can be applied, the empirical distribution functions
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173:{\displaystyle {\hat {\theta }}=T({\hat {F}}_{n})\,,}
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655:. The population analogue of this expression is
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284:and its variance will be no larger than that of
899:{\displaystyle E\left=0{\text{ at }}b=b_{0},\,}
454:), the population analogue of the estimator is
1158:Statistics 881: Advanced Statistical Learning
8:
99:based on the sample can be represented as a
971:The sample mean is a Fisher consistent and
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516:Suppose the parameter of interest is the
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229:{\displaystyle T(F_{\theta })=\theta \,.}
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413:. We can represent our sample of size
806:Role in maximum likelihood estimation
7:
810:Maximising the likelihood function
265:can be converted into an estimator
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14:
1101:Robust Statistical Methods with R
269:that can be defined in terms of
28:, is a desirable property of an
1074:Cox, D.R., Hinkley D.V. (1974)
802:so we have Fisher consistency.
105:empirical distribution function
1149:Lee, Yoonkyung (Spring 2008).
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916:represents the true value of
183:the estimator is said to be
88:which depends on an unknown
520:ÎĽ and the estimator is the
293:strong law of large numbers
1199:
934:asymptotically consistent
395:Finite population example
1160:. Ohio State University.
256:defined in terms of the
984:Role in decision theory
524:, which can be written
78:cumulative distribution
1076:Theoretical Statistics
1035:10.1098/rsta.1922.0009
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95:. If an estimator of
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1099:; Jan Picek (2006).
1078:, Chapman and Hall,
1020:(594–604): 309–368.
977:uniform distribution
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1086:. (defined on p287)
1026:1922RSPTA.222..309F
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653:indicator function
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493:Fisher consistency
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50:statistical sample
48:Suppose we have a
22:Fisher consistency
1178:Estimation theory
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513:) = θ.
491:). Thus we have
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185:Fisher consistent
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1127:. Archived from
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1097:Jurečková, Jana
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239:As long as the
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36:rather than a
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1183:Ronald Fisher
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1151:"Consistency"
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1131:on 2009-03-13
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1110:1-58488-454-1
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1103:. CRC Press.
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276:by averaging
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26:Ronald Fisher
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19:
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1133:. Retrieved
1129:the original
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1006:Fisher, R.A.
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250:exchangeable
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930:consistency
522:sample mean
67:where each
1172:Categories
1135:2009-01-09
1054:48.1280.02
1045:2440/15172
992:References
472:), where p
101:functional
76:follows a
44:Definition
34:population
18:statistics
928:The term
845:
784:μ
778:μ
758:∑
749:−
701:∑
680:∑
671:−
570:∑
549:∑
540:−
375:θ
355:^
343:∞
340:→
220:θ
209:θ
152:^
130:^
127:θ
90:parameter
30:estimator
1008:(1922).
973:unbiased
957:, where
267:T′
1022:Bibcode
651:is the
506:, ...,
465:, ...,
443:, ...,
406:, ...,
291:If the
103:of the
58:, ...,
1107:
1082:
1060:
1052:
909:where
647:where
38:sample
1154:(PDF)
1062:91208
1058:JSTOR
1105:ISBN
1080:ISBN
248:are
187:if:
1050:JFM
1040:hdl
1030:doi
1018:222
818:if
495:if
333:lim
313:if
16:In
1174::
1156:.
1056:.
1048:.
1038:.
1028:.
1016:.
1012:.
968:.
920:.
842:ln
297:FĚ‚
288:.
271:FĚ‚
113::
108:FĚ‚
20:,
1138:.
1113:.
1064:.
1042::
1032::
1024::
966:n
962:n
959:E
955:n
952:E
948:n
945:T
941:n
938:T
918:b
914:0
911:b
893:,
888:0
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877:b
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