What property is an arbitrarily chosen estimator unbiased with respect to?
$begingroup$
$renewcommand{vector}[1]{vec{#1}}$If I come up with a completely arbitrary estimator, how do I determine what, if anything, it is an unbiased estimator for?
Suppose I have a random finite sample $vector{w}$ of
length $n$ drawn from an arbitrarily chosen distribution $theta$ .
$vector{w}_1$ (the first value in the sample)✱✱ can be thought of
as an estimator of some property.✱
If we're considering $vector{w}_1$ as an estimator
of the variance, it is an extremely bad estimator since it has nothing to do with the variance, even in principle. The variance is an example of a candidate property that could be targeted by $vector{w}_1$, in order to rule it and almost every other property out, I want to insist that my estimator is unbiased.
$vector{w}_1$ is an unbiased estimator of the mean, $bar{theta}$ .
There are better unbiased estimators of the mean, but this is one of them.
How would I go about determining what property of a distribution $left(vector{w}_1-vector{w}_2right)^2$
is an unbiased estimator for? Its definition is "inspired by" the definition of variance, but I don't have
any reason to suspect that the variance is the thing it's actually estimating unbiasedly.
✱ I'm thinking of a property as just a function from
$mathrm{dist}[mathbb{R}] to mathbb{R} cup {bot}$ , like the population mean, population variance,
or population median. It is irrelevant to the notion of a property how a particular family of distributions is
parameterized. I don't know the actual term for this.
✱✱ $vector{w}_1$ is undefined if $vector{w}$ has length zero. Similarly
$(vector{w}_1 - vector{w}_2)^2$ is undefined if $vector{w}$ has length zero or one. As long as a
statistic is defined for all samples above a particular length, it is allowed to be undefined in some places.
statistics
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
$endgroup$
add a comment |
$begingroup$
$renewcommand{vector}[1]{vec{#1}}$If I come up with a completely arbitrary estimator, how do I determine what, if anything, it is an unbiased estimator for?
Suppose I have a random finite sample $vector{w}$ of
length $n$ drawn from an arbitrarily chosen distribution $theta$ .
$vector{w}_1$ (the first value in the sample)✱✱ can be thought of
as an estimator of some property.✱
If we're considering $vector{w}_1$ as an estimator
of the variance, it is an extremely bad estimator since it has nothing to do with the variance, even in principle. The variance is an example of a candidate property that could be targeted by $vector{w}_1$, in order to rule it and almost every other property out, I want to insist that my estimator is unbiased.
$vector{w}_1$ is an unbiased estimator of the mean, $bar{theta}$ .
There are better unbiased estimators of the mean, but this is one of them.
How would I go about determining what property of a distribution $left(vector{w}_1-vector{w}_2right)^2$
is an unbiased estimator for? Its definition is "inspired by" the definition of variance, but I don't have
any reason to suspect that the variance is the thing it's actually estimating unbiasedly.
✱ I'm thinking of a property as just a function from
$mathrm{dist}[mathbb{R}] to mathbb{R} cup {bot}$ , like the population mean, population variance,
or population median. It is irrelevant to the notion of a property how a particular family of distributions is
parameterized. I don't know the actual term for this.
✱✱ $vector{w}_1$ is undefined if $vector{w}$ has length zero. Similarly
$(vector{w}_1 - vector{w}_2)^2$ is undefined if $vector{w}$ has length zero or one. As long as a
statistic is defined for all samples above a particular length, it is allowed to be undefined in some places.
statistics
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
$endgroup$
add a comment |
$begingroup$
$renewcommand{vector}[1]{vec{#1}}$If I come up with a completely arbitrary estimator, how do I determine what, if anything, it is an unbiased estimator for?
Suppose I have a random finite sample $vector{w}$ of
length $n$ drawn from an arbitrarily chosen distribution $theta$ .
$vector{w}_1$ (the first value in the sample)✱✱ can be thought of
as an estimator of some property.✱
If we're considering $vector{w}_1$ as an estimator
of the variance, it is an extremely bad estimator since it has nothing to do with the variance, even in principle. The variance is an example of a candidate property that could be targeted by $vector{w}_1$, in order to rule it and almost every other property out, I want to insist that my estimator is unbiased.
$vector{w}_1$ is an unbiased estimator of the mean, $bar{theta}$ .
There are better unbiased estimators of the mean, but this is one of them.
How would I go about determining what property of a distribution $left(vector{w}_1-vector{w}_2right)^2$
is an unbiased estimator for? Its definition is "inspired by" the definition of variance, but I don't have
any reason to suspect that the variance is the thing it's actually estimating unbiasedly.
✱ I'm thinking of a property as just a function from
$mathrm{dist}[mathbb{R}] to mathbb{R} cup {bot}$ , like the population mean, population variance,
or population median. It is irrelevant to the notion of a property how a particular family of distributions is
parameterized. I don't know the actual term for this.
✱✱ $vector{w}_1$ is undefined if $vector{w}$ has length zero. Similarly
$(vector{w}_1 - vector{w}_2)^2$ is undefined if $vector{w}$ has length zero or one. As long as a
statistic is defined for all samples above a particular length, it is allowed to be undefined in some places.
statistics
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
$endgroup$
$renewcommand{vector}[1]{vec{#1}}$If I come up with a completely arbitrary estimator, how do I determine what, if anything, it is an unbiased estimator for?
Suppose I have a random finite sample $vector{w}$ of
length $n$ drawn from an arbitrarily chosen distribution $theta$ .
$vector{w}_1$ (the first value in the sample)✱✱ can be thought of
as an estimator of some property.✱
If we're considering $vector{w}_1$ as an estimator
of the variance, it is an extremely bad estimator since it has nothing to do with the variance, even in principle. The variance is an example of a candidate property that could be targeted by $vector{w}_1$, in order to rule it and almost every other property out, I want to insist that my estimator is unbiased.
$vector{w}_1$ is an unbiased estimator of the mean, $bar{theta}$ .
There are better unbiased estimators of the mean, but this is one of them.
How would I go about determining what property of a distribution $left(vector{w}_1-vector{w}_2right)^2$
is an unbiased estimator for? Its definition is "inspired by" the definition of variance, but I don't have
any reason to suspect that the variance is the thing it's actually estimating unbiasedly.
✱ I'm thinking of a property as just a function from
$mathrm{dist}[mathbb{R}] to mathbb{R} cup {bot}$ , like the population mean, population variance,
or population median. It is irrelevant to the notion of a property how a particular family of distributions is
parameterized. I don't know the actual term for this.
✱✱ $vector{w}_1$ is undefined if $vector{w}$ has length zero. Similarly
$(vector{w}_1 - vector{w}_2)^2$ is undefined if $vector{w}$ has length zero or one. As long as a
statistic is defined for all samples above a particular length, it is allowed to be undefined in some places.
statistics
statistics
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
asked Dec 24 '18 at 18:30
Gregory NisbetGregory Nisbet
709512
709512
add a comment |
add a comment |
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