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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.
22
votes
Accepted
Bias of maximum likelihood estimators for logistic regression
Consider the simple binary logistic regression model, with a binary dependent variable and only a constant and a binary regressor $T$.
$$\Pr(Y_i=1\mid T_i=1) = \Lambda (\alpha + \beta T_i)$$
where $\L …
- 60.7k
10
votes
Accepted
What's the difference between asymptotic unbiasedness and consistency?
In the related post over at math.se, the answerer takes as given that the definition for asymptotic unbiasedness is $\lim_{n\to \infty} E(\hat \theta_n-\theta) = 0$.
Intuitively, I disagree: "unbiase …
- 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
$$E(X_ …
- 60.7k
9
votes
Is this an unbiased estimator for standard deviation of normal distribution?
The proposed estimator is not unbiased, at least if we indeed know the true mean, $\mu$, and if we are dealing with a normal sample as the title says, where the distribution is symmetric and unimodal …
- 60.7k
8
votes
Counterexample for the sufficient condition required for consistency
Glad to see that my (incorrect) answer generated two more, and turned
a dead question into a lively Q&A thread. So it's time to try to offer something worthwhile, I guess).
Consider a serially …
- 60.7k
7
votes
Flaws in Frequentist Inference
It is a bit sad to see printed such carelessly written prose.
Consider the phrase
"For any prior density $g(\mu)$, the posterior density $g(\mu\mid x)=
g(\mu)f_{\mu}(x)/f(x)$ ....depends only on …
- 60.7k
7
votes
Accepted
Unbiased estimate of population standard deviation: is sqrt(2) a superior correction?
Maybe. What it appears that you did, is hit upon the $c_4(N)$ correction factor stated also in this wikipedia article. Specifically:
You propose the estimator
$$\tilde s = \frac 1{\sqrt {N-2^{1/2}}}\ …
- 60.7k
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
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
6
votes
Accepted
Unbiased Estimator for the CDF of a Normal Distribution
As a comment suggested, an unbiased estimator is (one minus) the empirical distribution function
$$\hat P(X_1 > 0) = 1-\hat F_X(0) = 1-\frac 1n \sum_{i=1}^n I\{x_i \leq 0\}$$
where $I\{\}$ is the in …
- 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
4
votes
Accepted
Inference about the true intercept of the model and the OLS being BLUE
Your last sentence is self-conradictory: "BLUE" means "Best Linear Unbiased Estimator" - so you are saying "the OLS estimator is biased but it is still Best Linear Unbiased" -except if you meant to sa …
- 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
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
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