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Results tagged with unbiased-estimator
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user 173082
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.
1
vote
CLT and Biased Estimators
Your explanation of bias is correct, so long as you understand that bias is always measured with respect to a parameter you are estimating. Without further assumptions, it is not generally true that …
- 133k
1
vote
Accepted
Show that estimator $\bar{X}-1$ is unbiased estimator if $X_1, X_2, ..., X_n$ are random sam...
First of all, the expectation in question cannot be equal to $\bar{X}$, since that is a random variable. (Your present working confuses the estimator with the parameter of interest.) From the specif …
- 133k
1
vote
Accepted
Exogeneity Assumption within or across observations
Short answer: You need to assume zero covariance across all pairs of individuals. If you are seeking to obtain unbiasedness via a covariance assumption (as opposed to the more direct linearity assump …
- 133k
1
vote
Accepted
Unbiased estimator for $e^\lambda$ in Poisson distribution
Hint: Given observed data $x_1,...,x_n$, consider the estimator of the form:
$$\widehat{e^{-\lambda}} = \sum_{k=0}^\infty w_k \Bigg[ \frac{1}{n} \sum_{i=1}^n \mathbb{I}(x_i = k) \Bigg],$$
where $w_0 …
- 133k
6
votes
Accepted
Intuition behind a 0% central/equal-tailed confidence interval
You are close
For a continuous distribution, the 0% equal-tail CI occurs at the point corresponding to the median of the true distribution of the pivotal quantity that is used in constructing the CI. …
- 133k
9
votes
Proving OLS unbiasedness without conditional zero error expectation?
For this question we can make use of a simple decomposition of the OLS estimator:
$$\begin{equation} \begin{aligned}
\hat{\boldsymbol{\beta}} = (\mathbf{X}^\text{T} \mathbf{X})^{-1} \mathbf{X}^\text{ …
- 133k
6
votes
Is it wrong to say that a Riemann sum is an unbiased estimate of an integral?
Any constant is a biased estimator of any different constant
Since you are using a deterministic procedure here, your Riemann sum depends only on $n$, so it is a sequence of constants. Applying the c …
- 133k
6
votes
Accepted
Finding unbiased estimator for Truncated Poisson Distribution
Your answer looks correct to me, but it can be further simplified. You should have:
$$\begin{align}
\mathbb{E}(T)
&= 2 \Bigg[ p_X(1) + p_X(3) + p_X(5) + \cdots \Bigg] \\[6pt]
&= 2 \Bigg[ \frac{e^{-\t …
- 133k
4
votes
Accepted
Variance of an Unbiased Estimator for $\sigma^2$
The easiest way to do this problem is by using vector algebra, re-expressing the estimator as a quadratic form in vector notation:
$$Q = \frac{1}{2(n-1)} \mathbf{X}^\text{T} \mathbf{\Delta} \mathbf{X …
- 133k
3
votes
Proof that $g(p)$ unbiasedly estimable only if it is a polynomial (Binomial Distribution)
The left-hand-side of $(1.2)$ is a weighted sum of polynomials with degree $n$, which is itself a polynomial of degree (no greater than) $n$. If you want to see this more clearly then you can use the …
- 133k
3
votes
Dividing by degrees of freedom
Bessel's correction is adopted to correct for bias in using the sample variance as an estimator of the true variance. The bias in the uncorrected statistic occurs because the sample mean is closer to …
- 133k
10
votes
Does the biased estimator always have less variance than unbiased one?
There are many, many, many different possible estimators in estimation problems. In general there are multiple unbiased estimators and multiple biased estimators, and their variances need to be consi …
- 133k
1
vote
Accepted
Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lam...
...in this setting how would one proceed to compute $1/\lambda$ from $\hat \theta$?
You can't compute $1/\lambda$ because $\lambda$ is the unknown parameter. That is why we estimate it. We can …
- 133k
4
votes
Accepted
Proof of (weak) consistency for an unbiased estimator
The standard method of proving (weak) consistency is to use Chebychev's inequality and apply the triangle inequality to deal with the bias in the estimator. From the triangle inequality, you have:
$ …
- 133k
5
votes
Why doesn't homoskedacticity bias an estimator?
Consider a heteroskedastic linear regression with model form:
$$\boldsymbol{Y} = \boldsymbol{x} \boldsymbol{\beta} + \boldsymbol{\varepsilon} \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \te …
- 133k