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Results tagged with unbiased-estimator
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user 28746
Refers to an estimator of a population parameter that "hits the true value" on average. That is, a function of the observed data $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$ if $E(\hat{\theta}) = \theta$. The simplest example of an unbiased estimator is the sample mean as an estimator of the population mean.
2
votes
Accepted
Law of Iterated Expectations in Practice
This appears to be a trivial application of the Law of Iterated Expectations, because $X$ is mean-independent of the choice of the sample that contains realizations of it (i.e. the true mean of $X$ do …
- 60.7k
3
votes
Simple OLS with two samples
The unbiasedness poperty of the OLS estimator in the linear regression model is a finite-sample property, and it is based on a specific assumption of the model being correct -that the regressors are " …
- 60.7k
1
vote
Does efficiency imply unbiased and consistency?
In an attempt to... un-unanswer this question:
If one can write the derivative of the log-likelihood as the OP states, then this estimator equals the true value always, irrespective of the realized s …
- 60.7k
7
votes
Accepted
Unbiased estimate of population standard deviation: is sqrt(2) a superior correction?
Maybe. What it appears that you did, is hit upon the $c_4(N)$ correction factor stated also in this wikipedia article. Specifically:
You propose the estimator
$$\tilde s = \frac 1{\sqrt {N-2^{1/2}}}\ …
- 60.7k
3
votes
Accepted
Variance of Mean of Samples from Unknown Distribution
The question you think it might be "similar", is not, because it is concerned mainly with the distribution/variance of the sample variance itself.
You cannot know the variance of the sample mean (i …
- 60.7k
1
vote
Accepted
Unbiased estimators - how to show unbiasedness?
What I always liked about this situation is its very illuminating intuition. We have, given unocorrelatedness,
$$\text{Var} [\hat \theta (\alpha))] = \alpha^2 \sigma^2_1 + (1-\alpha)^2 \sigma^2_2$$
…
- 60.7k
7
votes
Accepted
Obtaining an estimator via Rao-Blackwell theorem
We have
$$F_X(x) = \int_{\theta}^{x}e^{\theta -t} dt = -e^{\theta}e^{-t}\Big|^{x}_{\theta} = 1 - e^{\theta -x} $$
Since $F_{X_{(1)}}(x_{(1)}) = 1 -[1-F_X(x_{(1)})]^{n}$, the density function of the …
- 60.7k
4
votes
More than one unbiased estimator for a single unknown parameter?
As an example, from a i.i.d. sample of (finite) size $n$, where the common mean is $\mu \neq 0$ we can have an infinite (and not even countably) number of unbiased estimators of the form
$$\hat \mu(a …
- 60.7k
10
votes
For which distributions is there a closed-form unbiased estimator for the standard deviation?
A probably well known case, but a case nevertheless.
Consider a continuous uniform distribution $U(0,\theta)$. Given an i.i.d. sample, the maximum order statistic, $X_{(n)}$ has expected value
$$E(X_ …
- 60.7k
9
votes
Is this an unbiased estimator for standard deviation of normal distribution?
The proposed estimator is not unbiased, at least if we indeed know the true mean, $\mu$, and if we are dealing with a normal sample as the title says, where the distribution is symmetric and unimodal …
- 60.7k
1
vote
Accepted
Is the median confidence value an unbiased MLE?
From an i.i.d. sample of Rayleigh r.v.'s we obtain the MLE/Method of Moments estimator
$$\hat \sigma^2_{MLE} = \frac{1}{2N}\sum_{i=1}^n x_i^2 \implies 2\hat \sigma^2_{MLE} = \frac{1}{n}\sum_{i=1}^n x …
- 60.7k
4
votes
Why is it important that estimators are unbiased and consistent?
From a frequentist perspective,
Unbiasedness is important mainly with experimental data where the experiment can be repeated and we control the regressor matrix. Then we can actually obtain many es …
- 60.7k
7
votes
Flaws in Frequentist Inference
It is a bit sad to see printed such carelessly written prose.
Consider the phrase
"For any prior density $g(\mu)$, the posterior density $g(\mu\mid x)=
g(\mu)f_{\mu}(x)/f(x)$ ....depends only on …
- 60.7k
6
votes
Accepted
Distribution of $\bar{X^2} $ when $X\sim N \left( \theta, \sigma^2 \right) $
Since we are looking at a sample mean we have that
$$\bar X_n \sim_{approx} N \left(\theta ,\frac{\sigma^2}{n} \right)$$
which holds for "large but finite $n$" -since if $n\rightarrow \infty$ the sa …
- 60.7k
3
votes
Is it possible to have an estimator that is unbiased and bounded?
I will present conditions under which an unbiased estimator remains unbiased, even after it is bounded. But I am not sure that they amount to something interesting or useful.
Let an estimator $\hat …
- 60.7k