All Questions
Tagged with sufficient-statistics probability
29
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Formal definition of sufficient statistic
Let $(\Omega_X,\mathcal{F}_X)$ and $(\Omega _T,\mathcal{F}_T)$ be measurable spaces. Let $\mathfrak{M}$ be a family of probability measures on $(\Omega_X,\mathcal{F}_X)$. Let $X:\Omega\to \Omega _X$ ...
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Likelihood ratio as minimal sufficient statistics in infinite parameter space
I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first:
Consider a family of density functions $f(x|\theta)$ where the ...
- 31
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37
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Sufficient Statistic for a finite family of Normal distributions
Suppose we have a finite family of normal distributions $P=\{N(0, 1), N(0, 2), N(1, 2), N(2, 2)\}$ and we want to find a sufficient statistic for this family. Intuitively it is clear that as the means ...
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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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643
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Is there a standard measure of the sufficiency of a statistic?
Given a parametrical model $f_\theta$ and a random sample $X = (X_1, \cdots, X_n)$ from this model,
a statistic $T(X)$ is sufficient if the distribution of $X$ given $T(X)$ doesn't depend on $\theta$.
...
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2
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2
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365
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Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution
Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function:
$$ f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R} $$
Find the joint ...
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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(\...
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1
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78
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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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1k
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Understanding the Rao-Blackwell Theorem
I've been reading up a lot on the practical applications of the Rao-Blackwell theorem. I do understand how the Bias and Variance and MSE aspects of the theorem fall in place (i.e. the mathematical ...
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Do sufficient statistics for parameters of interest depend on whether nuisance parameters are known?
The definition of sufficient statistic is as follows:
A statistic $T(X_1,...,X_n)$ is sufficient for parameter $\theta$ if the conditional distribution of $X_1,...,X_n$, given that $T=t$, does not ...
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969
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Sufficient statistic for the distribution of a random sample of Poisson distribution
Let $X_1,...,X_n$ be a random sample from a Poisson distribution with mean $\lambda$ and $T = \sum_{i=1}^n X_i $ . Show that the distribution of $X_1,...,X_n$ given T is independant of $\lambda$ so ...
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3k
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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 ...
- 61
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Sufficient Statistic for Normal Distribution | Mean, Variance & Kurtosis
I have seen multiple times that a normal distribution is fully specified by mean and variance. It is obvious that the third moment is not necessary for a perfect normal distribution as it is 0. I ...
- 870
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When family of DF's $\mathcal{P}$ fail to be dominated by a measure $\mu$
On the topic of minimal sufficient statistics, there is an important theorem which requires the family of probability distributions $\mathcal{P}$ is dominated by some measure $\mu$.
As I understand it,...
- 2,564
14
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Intuitive understanding of the Halmos-Savage theorem
The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...
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