All Questions
Tagged with sufficient-statistics estimators
20
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Show that $T=\sum_{i=1}^n X_i$ is a sufficient statistic for $p$ [duplicate]
I try to use the definition of sufficient statistic to prove that
Suppose that $X_1,\dots, X_n$ is an iid random sample from $X\sim \mathrm{Bernoulli}(p)$. Show that $T=\sum_{i=1}^n X_i$ is a ...
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Extending Minimal sufficient statistics to arbitrary dimension
I am wondering if the following reasoning is correct regarding minimal sufficiency and dimension. Given $X_1,\dots,X_n$ i.i.d. $N(\mu,1)$, we know that the sample mean $S = \bar{X}$ is a minimal ...
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Sufficient statistics for bernoulli distribution
Let $Y_1, \ldots, Y_n $ be a random sample of size $n$ where each $Y_i \sim \textrm{Bernoulli}(p), $ and
let $Y = \sum Y_i $ for $i = 1, \ldots, n.$
The estimator is $W= (Y+1)/(n+2). $
Is the ...
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Distribution of unbiased estimator given the sufficient statistic
Let T be a sufficient statistic for parameter a, and W be an unbiased estimator of a, then will the distribution of W|T always be independent of parameter a?
I understand that T being sufficient for ...
- 107
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Subscripts for Expectations and variances in for estimators [duplicate]
Is there any significance for subscripts to E and Var?
For example, the risk function of an estimator $\delta(\mathbf x)$ of $\theta$ in my book is:
$$
R(\theta,\delta)=E_\theta[L(\theta,\delta(\...
5
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320
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How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?
Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family
$$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
- 2,096
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Does an estimator need to be unbiased in order to be sufficient?
I am reviewing some theoretical statistics content, and I was wondering if an estimator need to be unbiased in order to be sufficient? Is there any way to prove this? Thanks!
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Did I correctly apply the factorisation theorem in this example?
Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}(x \in]\theta,\theta+1[)$ and the random variable $\mathbf{X}=(X_1,\ldots,X_n)$ are independently identically distributed ...
- 155
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What's the maximum likelihood estimation of $\theta$ in this density? [duplicate]
Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$.
Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
- 155
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Proving sufficiency of a statistic using the expectation
I am blocked trying to solve the following question. I would appreciate if someone could give me a hint.
Let $X_1,\ldots,X_n$ be $n$ independent random variables following a continuous uniform ...
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Sufficient estimator for Bernoulli distribution using the likelihood function theorem for sufficiency
Let $(X_1,X_2)$ be a random sample of two iid random variables, $X_1\sim Ber(\theta),\theta\in (0,1)$.
Use the following theorem to show that $\hat{\theta}=X_1+2X_2$ is sufficient.
Likelihood theorem ...
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1
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242
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nonexistence of a sufficient statistic
Let $X_1,X_2,\dots,X_n$ be a random sample from a $\Gamma(\theta,\theta)$ distribution. Then
$$
\prod_{i=1}^n f(x_i;\theta) = \frac{1}{\Gamma(\theta)^n\theta^n}(\prod_{i=1}^n x_i)^{\theta-1}e^{-\frac{...
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"Magical" variance reduction problem
I recently came across this toy problem:
You have two sticks of unknown lengths $a>b$ and a measuring device with constant variance $1$ that you can only use twice. How can you construct ...
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Invariance property
I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators.
As far as I know,
Invariance property of ...
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Efficient Estimator from Insufficient Statistic
Suppose that I have a statistic $T(X)$, and I know for sure that it is not sufficient to estimate a parameter $\theta$.
Is it still possible to have an estimator $\hat\theta(T(X))$ that is efficient (...
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