# Questions tagged [sufficient-statistics]

For questions about sufficient statistics. A statistic is sufficient for a parametric model if the distribution of the data conditioned on the statistic is parameter-free. For more general questions about statistics and estimators, please use "statistical-inference".

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### Find fisher information matrix for minimum estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
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### Fisher-Neyman Factorisation Theorem and sufficient statistic misunderstanding

Fisher Neyman Factorisation Theorem states that for a statistical model for $X$ with PDF / PMF $f_{\theta}$, then $T(X)$ is a sufficient statistic for $\theta$ if and only if there exists nonnegative ...
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### Find fisher information matrix for minimizes function (Mathematical Statistics)

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
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### Prove the sufficiency of geometric distribution. [closed]

Let X$_{1}$ and X$_{2}$ be two independent geometric random variables with common parameter $\theta \in$(0,1). Now I have to show the statistic of T=X$_{1}$+X$_{2}$ is sufficient. I know that the ...
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### Proving that an auxiliary parameter is not independent of a sufficient parameter [closed]

Hi! How can I show that a parameter is sufficient and that an ancillary parameter is not independent of the sufficient parameter?
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### Variational Inference for Item Response Models Estimation

I should work on a project about Variational Inference for Item Response Models Estimation. This is my university project but I don´t find relevant information around this topic so I would be very ...
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### Showing a minimal sufficient statistic

If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$ with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample. Then how can we show that the ...
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### Finding Sufficient Statistics

Let X1, . . . , Xn be a random sample from the following pmf. P(X = 0) = θ, P(X = 1) = 2θ, P(X = 2) = 1 − 3θ, 0 < θ < 1/3 Find a non-trivial sufficient statistic. I start like this: L(θ)=L(θ)=∏...
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### When does a sufficient statistic not exist by the Factorization Theorem?

The Neyman Factorization Theorem states the following: Let $f(x_1, ..., x_n; \theta)$ denote the joint pmf or pdf of $X_1, ..., X_n$. Then $T = t(x_1, ..., x_n)$ is a sufficient statistic for $\theta$ ...
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### Minimal Sufficient Statistics and Sufficient Statistic

The procedure I have to take is to first identify that both U(X) are unbiased for theta, and that U2(X) = E[U1(X)|T2(X)]. My question is How would that relate to U2 having a smaller variance than U1,...