All Questions
Tagged with sufficient-statistics conditional-probability
23
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Birnbaum's Theorem: Strong belief in a model $\implies$ the likelihood function must be used as a data reduction device?
Working through understanding section 6.3.2 (pg. 292-294) in Casella and Berger's Statistical Inference (2nd-ed).
The following definitions and principles are given:
Definition (Experiment): An ...
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1
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106
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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 $X:\Omega\to \Omega _X$ ...
- 133
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Solving the Neyman-Scott problem via Conditional MLE
In section 2.4 of the book Essential Statistical Inference by Boos and Stefanski, the authors discuss the idea conditional likelihoods and illustrate their usefulness by describing how they can be ...
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Prove that the sum is sufficient using using the definition of sufficiency
If $X_1,\ldots,X_n$ is an IID random sample, with $X_i\sim\,\text{Ber}(\theta)$, prove that $Y = \sum_i X_i$ is sufficient using the definition of sufficiency (not the factorization criterion).
Now ...
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Rao–Blackwellization of Metropolis–Hastings
I am trying to achieve a Rao–Blackwellization of Metropolis–Hastings algorithm. In the paper by Robert et al. 2018, the following is given.
\begin{align}
ℑ=&\frac{1}{T}\sum_{t=1}^Th(\theta^{(t)})=\...
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What is "Likelihood Principle"?
While I was studying "Bayesian Inference", I happen to encounter the term, "Likelihood Principle" but I don't really get the meaning of it. I assume it is connected to "...
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Conditional distribution of complete sufficient statistics being ancillary of $\alpha$
Regarding the distribution and statistics as described here, I need to show that the conditional distribution of $\overline{X}$ given $X^*=x^*$ does not depend on $\alpha$. I remember my professor ...
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Poisson sufficient statistics problem
I have the following problem:
Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ distribution has the probability function ...
- 2,096
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1
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294
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Joint distribution simplification in minimal sufficient statistics proof
My notes introduce the concept of minimal sufficient statistics as follows:
Definition
A sufficient statistic $T(\mathbf{Y})$ is called a minimal sufficient statistic if it is a function of any other ...
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2
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Conditional probability, statistic and sufficient statistic
In statistical model $(\mathcal{X}, \{P_\theta\mid\theta\in\Theta\})$ statistic $T=T(\mathbf{X})$ (where $\mathbf{X}$ marks random sample) is said to be sufficient for $\theta$, when conditional ...
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Showing that a sum of Bernoulli random variables (that is, a binomial random variable) is a sufficient statistic
I just started learning what a sufficient statistic is:
Definition
A statistic $T(\mathbf{Y})$ is sufficient for an unknown parameter $\theta$ if the conditional distribution of the data $\mathbf{Y}$ ...
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3
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1k
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Intutitive meaning behind the formal definition of sufficient statistic?
According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter.
What I am ...
2
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2
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Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem
Question:
Given, $X\sim N(0,\sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $\sigma^2$.
My Attempt:
This problem is very easy if we use Fisher–Neyman ...
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2k
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Probability conditional on a parameter?
This is a definition of the sufficient statistic from Wikipedia.
A statistic $t = T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $...
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How does conditional expectation relate to sufficiency? [closed]
In what follows, I will disregard all "measure-theoretic niceties about conditioning on measure-zero sets", as my professor calls it. I just want to know if the following general idea, or ...
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