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Efficiency of estimators and UMVUE
Clash Royale CLAN TAG#URR8PPP
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(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not? Thanks!
statistics statistical-inference parameter-estimation
asked Aug 6 at 9:29
Lucas
232
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(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not? Thanks!
statistics statistical-inference parameter-estimation
asked Aug 6 at 9:29
Lucas
232
Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not? Thanks!
statistics statistical-inference parameter-estimation
asked Aug 6 at 9:29
Lucas
232
(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not? Thanks!
statistics statistical-inference parameter-estimation
asked Aug 6 at 9:29
Lucas
232
asked Aug 6 at 9:29
Lucas
232
asked Aug 6 at 9:29
Lucas
232
asked Aug 6 at 9:29
Lucas
232
232
Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17
add a comment |Â
Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17
Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17
add a comment |Â
1 Answer
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You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $theta$, then it is the UMVUE.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
- Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?
- Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.
answered Aug 8 at 16:05
V. Vancak
9,8302926
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $theta$, then it is the UMVUE.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
- Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?
- Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.
answered Aug 8 at 16:05
V. Vancak
9,8302926
add a comment |Â
up vote
0
down vote
accepted
You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $theta$, then it is the UMVUE.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
- Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?
- Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.
answered Aug 8 at 16:05
V. Vancak
9,8302926
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $theta$, then it is the UMVUE.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
- Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?
- Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.
answered Aug 8 at 16:05
V. Vancak
9,8302926
You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $theta$, then it is the UMVUE.
If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?
- Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).
Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?
- Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.
answered Aug 8 at 16:05
V. Vancak
9,8302926
answered Aug 8 at 16:05
V. Vancak
9,8302926
answered Aug 8 at 16:05
V. Vancak
9,8302926
answered Aug 8 at 16:05
V. Vancak
9,8302926
9,8302926
add a comment |Â
add a comment |Â
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Is this a question about statistics or about logic?
– Taroccoesbrocco
Aug 6 at 9:31
More than one estimator can be efficient for estimating some parameter of interest, and we can then find their relative efficiencies for comparison. Maybe you are referring to asymptotic efficiency here. As for (2), the estimator has to be unbiased for the parameter besides attaining CRLB to be UMVUE. Plus, UMVUE does not always attain CRLB.
– StubbornAtom
Aug 6 at 12:07
What we can say is that if UMVUE of some $theta$ exists, then it will be the most efficient estimator of $theta$ within the unbiased class.
– StubbornAtom
Aug 6 at 12:17