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Botnakov N.
Botnakov N.
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I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator
or https://math.stackexchange.com/questions/1251393/when-is-the-maximum-likelihood-estimator-measurable ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

I think that in $(3.14)$ the set $|\hat{\theta}_n - \theta_0|$ may be not measurable (but I don't know counterexample).

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator
or https://math.stackexchange.com/questions/1251393/when-is-the-maximum-likelihood-estimator-measurable ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator
or https://math.stackexchange.com/questions/1251393/when-is-the-maximum-likelihood-estimator-measurable ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

I think that in $(3.14)$ the set $|\hat{\theta}_n - \theta_0|$ may be not measurable (but I don't know counterexample).

added 108 characters in body
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Botnakov N.
Botnakov N.
  • 141
  • 6

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator  
or https://math.stackexchange.com/questions/1251393/when-is-the-maximum-likelihood-estimator-measurable ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator  ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator
or https://math.stackexchange.com/questions/1251393/when-is-the-maximum-likelihood-estimator-measurable ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here

Source Link
Botnakov N.
Botnakov N.
  • 141
  • 6

Measurability of maximum likelihood estimator under conditions from Lehmann's "Theory of point estimation"

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity conditions (e.g. stats.stackexchange.com/questions/430954/example-of-a-non-measurable-maximum-likelihood-estimator ) MLE is measurable, but it didn't help me to prove that the closest root from the proof is a measurable function.

The problem is as follows. By definition, an estimate is a measurable function. Even the existence of a measurable version of MLE is not so obvious, but here an extremum arises over the set of MLE. It is unlikely that a theorem from a classical book can be wrong, but how to prove it?

The theorem is here:

enter image description here