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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.
2
votes
Accepted
Why weighted importance sampling is a biased estimator?
Sorry if I was "jumping over a few steps", the argument seems simple enough to me: if $N$ and $D$ are (not necessarily independent) random variables such that $\mathbb E[N]=\nu$ and $\mathbb E[D]=\del …
- 108k
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
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
3
votes
Accepted
Find UMVUE of difference of parameters of two exponential distribution random variables
Changing the question in two different ways allows for some answers:
If $\theta_x$ and $\theta_y$ are rate rather than scale parameters,
$$
\frac{n-1}{n} \frac{\sum_{i=1}^n (1-\Delta_i)}{\sum_{i=1} Z …
- 108k
0
votes
Equivalence of the completeness of the order statistics and the uniqueness of symmetric unbi...
The equality$$\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
14
votes
Why isn't this estimator unbiased?
As indicated in my comment (and then in later answers), the error in the reasoning leading to the apparent paradox is to treat the selected or surviving $X_i$'s that we should denote differently, e.g. …
- 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
1
vote
Finding UMVUE of function of poisson parameter
While the conditional expectation is correct, the series is not properly identified:
\begin{array}
{}\displaystyle\sum_{t=2}^{\infty}&\displaystyle\frac{t(t-1)(n-1)^{t-2}e^{-n\lambda}(n\lambda)^t}{2n^ …
- 108k
2
votes
Completeness of a statistic - Open ball
When formally defining a probability density as
$$f_\theta(x)=c(\theta)h(x)\exp\left\{\sum_{j=1}^k q_j(\theta)t_j(x)\right\}\tag{1}$$
it is always possible to add useless terms in the expression, as f …
- 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$$\mu^*,\sigma^* := \arg \max_{\mu,\sigma} \mathbb{E …
- 108k
7
votes
Accepted
Finding UMVUE for a function of a Bernoulli parameter
Except when $k=1$, given a finite sequence of i.i.d. Bernoulli
$\mathcal B(θ)$ random variables $X_1,X_2,\ldots,X_m$, there exists no
unbiased estimator of $(1−θ)^{1/k}$, when $k$ is a positive …
- 108k
4
votes
Accepted
Unbiased estimator of standard deviation
While $\mathbb E_\sigma[c^2]=\sigma^2$,
\begin{align}
\mathbb E_\sigma[|c|] &= \int_0^\infty \sqrt{2/\pi}\, \sigma^{-1} x\, \exp\{-x^2/2\sigma^2\}\,\text{d}x\tag{symmetry}\\
&= \sigma\int_0^\infty \s …
- 108k
7
votes
Are unbiased efficient estimators stochastically dominant over other (median) unbiased estim...
Here is an experiment in a non-standard case, the location Cauchy problem, where non-standard means that there is no uniformly best unbiased estimator. Let us consider $(X_1,\ldots,X_N)$ a sample from …
- 108k
9
votes
Accepted
Unbiased estimator of binomial PMF
Since, for a Binomial $\text{B}(n,p)$ variable $X$, and $k\le n$, the factorial moment is given by
$$\mathbb{E}_p[X(X-1)\cdots(X-k+1)] = n(n-1)\cdots(n-k+1)p^k,$$
the $s$ Bernoulli rvs $\lbrace X_i\rb …
- 108k
5
votes
Unbiased Estimator for $\log\left[\int p(x\mid z)p(z) \, dz\right]$
Path sampling is a way to evaluate the log integral by an unbiased estimator. Let us introduce a temperature index $0\le t\le 1$ and a sequence of conditional functions $p_t(x|z)$ such that
$$p_0(x|z) …
- 108k