All Questions
Tagged with sufficient-statistics inference
72
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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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Help developing intuition behind sufficient statistics (Casella & Berger) [duplicate]
Migrated from MSE
I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272):
A sufficient statistic captures all of the information ...
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Find minimal sufficient statistic of this random sample with cursed support
Suppose $X_1,X_2,...,X_n$ is a i.i.d random sample with probability mass function $p(x_i,\theta)$ where $x_i \in \{\theta,\theta+1,\theta+2,...\}$ and $\theta \in \mathbb{R}$. I claim that minimal ...
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Sufficient Statistic for Truncated Normal
I am doing exercise 3.18 of "The Bayesian Choice":
Give a sufficient statistic associated with a sample $x_1,...,x_n$
from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x
...
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Usage of Sufficient statistic for a Gamma distribution
I need some help to understand how to utilize sufficient statistic from a data.
Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I ...
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Likelihood principle and inference
I've been reading Casella and Berger's Statistical Inference. In section 6.3 the author stated the likelihood principle: if the likelihood functions from two samples are proportional, then the ...
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Karlin-Rubin theorem: relationship between test statistic having the MLR property vs being sufficient
Let's suppose we are trying to compare two hypotheses for a single parameter $\theta$. The null hypothesis $H_0$ is that $\theta = \theta_0$, and the alternative is that $\theta ≥ \theta_0$.
The ...
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How do I know which statistic is for which parameter when calculating joint sufficient statistics using factorization criteria?
For the normal distribution for example, after factorization we get
$\mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{n\mu^2}{2\sigma^2}\right) \exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=...
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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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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, minimal, complete
Are all complete statistics functions of each other?
For example if I have T and S complete statistics
Can you always write T in terms of S and S in terms of T?
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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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Rao Cramèr Lower Bound problem
Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
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Rao-Blackwellisation using non-sufficient statistics
The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1")
Now, I do understand that the ...
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What are the "Dangers" of using "Non-Sufficient" Statistics?
I was reading one of the answers listed on this previous Stackoverflow question about the importance of sufficient statistics (Generalized Linear Models - What's special about the exponential ...
