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Results tagged with unbiased-estimator
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user 7224
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.
6
votes
Accepted
Unbiased estimator of $2\left(\frac{E[x^2]}{E[x]^2}-1\right)^{-1}$
The quantity is
$$2\frac{\mathbb E[X]^2}{\text{var}(X)}$$
Given an iid sample $X_1,\ldots,X_n$, the estimator$$X_1X_2$$is an unbiased estimator of the numerator. In the case $\text{var}(X)^{1/2}=\sigm …
- 108k
1
vote
Accepted
Unbiased Estimator for $EX_1EY_1$
Since $X_i$ and $Y_i$ are dependent, $\mathbb{E}[X_iY_i]\ne\mathbb{E}[X_i]\mathbb{E}[Y_i]$. However, since $X_i$ and $Y_j$ are independent when $j\ne i$, $$\mathbb{E}[X_iY_j]=\mathbb{E}[X_i]\mathbb{E} …
- 108k
7
votes
Accepted
An unbiased estimator of σ³
Let us consider a Normal sample. Since$$\sigma^{-2}\sum_{i=1}^n (x_i-\bar{x}_n)^2\sim\chi^2_{n-1}$$which is also a Gamma $\mathcal{Ga}(n-1,1/2)$ variate, the $3/2$ moment of this variate is
$$\int_0^\ …
- 108k
2
votes
Combining importance sampling with optimization - does this yield an unbiased estimate?
The solution remains unbiased as $(\mu^\star,\sigma^\star)$ does not depend on the sample from $q_{\mu^\star,\sigma^\star}(\cdot)$.
The optimisation$$\mu^*,\sigma^* := \arg \max_{\mu,\sigma} \mathbb{E …
- 108k
3
votes
For iid $X_1, \dots, X_n \sim N(0,\sigma^2)$, get sufficient statistic $T = \sum_{i=1}^nX_i^...
By writing the $X_i$'s as $X_i=\sigma \epsilon_i$, where the $\epsilon_i\sim\mathcal N(0,1)$ are standard Gaussians, $$T=\sigma^2 \sum_{i=1}^n\epsilon_i^2$$ writes as $\sigma^2$ times a fixed rv, dist …
- 108k
7
votes
Understanding the relationship between a 'sufficient statistic' and an 'unbiased estimator'
Sufficiency is an essential if rare property: if $S(X)$ is sufficient for model $f_\theta$, considering $S(X)$ for estimation of $\theta$ is sufficient, meaning you need nothing else from $X$. In othe …
- 108k
0
votes
Equivalence of the completeness of the order statistics and the uniqueness of symmetric unbi...
The equality$$\int h(T(x_1,...,x_k)) dF(x_1)\cdots dF(x_k) = \int h^{[n]}(x_1,...,x_k)dF(x_1)\cdots dF(x_k)$$does not hold for any function $h$. For instance, if$$h(x_1,...,x_k)=x_1$$
$$\int h(T(x_1,. …
- 108k
4
votes
Accepted
Finding the MVUE of the center of a circle of unknown location
Here is a homework problem from Mark Schervish's Theory of Statistics that addresses a similar question:
Let $(X_1, Y_1),\dots,(X_n, Y_n)$ be conditionally IID with uniform distribution on the risk …
- 108k
7
votes
Accepted
Importance sampling: unbiased estimator of the normalizing constant
Why the author takes $\mathfrak{Z}=∫φ(x)dx$?
Since $p$ is a density, its integral is equal to $1$. If $\mathfrak{Z}$ is the normalising constant of $\varphi$, it has to satisfy
$$\int p(x)\ …
- 108k
12
votes
For which distributions is there a closed-form unbiased estimator for the standard deviation?
Although this is not directly connected to the question, there is a 1968 paper by Peter Bickel and Erich Lehmann that states that, for a convex family of distributions $F$, there exists an unbiased es …
- 108k
15
votes
Is unbiased maximum likelihood estimator always the best unbiased estimator?
In my opinion, the question is not truly coherent in that the maximisation of a likelihood and unbiasedness do not get along, if only because maximum likelihood estimators are equivariant, ie the tran …
- 108k
9
votes
Accepted
Unbiased estimator of binomial PMF
Since, for a Binomial $\text{B}(n,p)$ variable $X$, and $k\le n$, the factorial moment is given by
$$\mathbb{E}_p[X(X-1)\cdots(X-k+1)] = n(n-1)\cdots(n-k+1)p^k,$$
the $s$ Bernoulli rvs $\lbrace X_i\rb …
- 108k
2
votes
The UMVUE of ratio of parameters for two uniform distributions,
First, one only need look at sufficient statistics:
Second, one need find an unbiased estimator based on a sufficient statistic:
Last, one can call for a completeness argument.
- 108k
4
votes
Accepted
Unbiased estimator of standard deviation
While $\mathbb E_\sigma[c^2]=\sigma^2$,
\begin{align}
\mathbb E_\sigma[|c|] &= \int_0^\infty \sqrt{2/\pi}\, \sigma^{-1} x\, \exp\{-x^2/2\sigma^2\}\,\text{d}x\tag{symmetry}\\
&= \sigma\int_0^\infty \s …
- 108k
7
votes
Accepted
Optimal importance sampling with ratio estimator
This is an interesting [and very far from "stupid"] question that actually bothered me for a while! We cover it in Monte Carlo Statistical Methods (Section 3.3.2, pages 95-96). The crux of it is that, …
- 108k