# Questions tagged [sufficient-statistics]

A sufficient statistic is a lower dimensional function of the data which contains all relevant information about a certain parameter in itself.

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### minimal sufficiency for $\sum_{i = 1}^p X_i$ and $\sum_{i = 1}^p X_i^2$

Take the random sample $(X_1, X_2, \dots, X_p)$ from $N(\mu, \mu)$, where $\mu > 0$. I am told that the statistic $\sum_{i = 1}^p X_i^2$ is minimal sufficient for $\mu$. But it then says that the ...
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### minimal sufficient statistics of 1-parameter Gamma distribution

If $x_i \sim Gamma(\alpha, \alpha)$, are the minimal sufficient statistics still $\Pi_i x_i$ and $\sum_i x_i$ (same as when $x_i \sim Gamma(\alpha, \theta)$ where $\alpha \neq \theta$)? My reasoning ...
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### Does a sufficient statistic imply the existence of a conjugate prior?

In the comments on this answer, user Scortchi asks: So iff there's a sufficient statistic of constant dimension, there's a conjugate prior? As far as I know this didn't get a complete answer, so I'm ...
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### For iid $X_1, \dots, X_n \sim N(0,\sigma^2)$, get sufficient statistic $T = \sum_{i=1}^nX_i^2$, how to find unbiased estimator of $\sigma^a$

For $X_1, \dots, X_n \sim N(0,\sigma^2)$, we define a sufficient statistic $T = \sum_{i=1}^nX_i^2$. There is a positive number $a$. My question is how to find unbiased estimator of $\sigma^a$ using ...
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### What is the score function of two parameters?

According to this wikipedia article, score is the derivative of the log-likelihood function. However, I don't understand what if we have two parameters? For example, the logarithm of pdf has the ...
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### Is there any difference between sufficient statistics for known and unknown distribution parameters?

I wonder if there is any difference between sufficient statistics for known and unknown distribution parameters. For example, let us take $Beta(a,b)$ distribution. If we use Fisher-Neyman theorem, we ...
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### If the dimension of a sufficient statistic $T(X)$ equals the dimension of parameter space, $T(X)$ is minimal sufficient?

I came cross an interesting comment saying If the dimension of a sufficient statistic $T(X)$ is the same as that of the parameter space, then $T(X)$ is minimal sufficient. Is this is true? I ...
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### minimal sufficient statistic for $U(\theta, \theta+c)$. $(\theta,c)$ unknown

Suppose $X_1,\cdots,X_n$ are $i.i.d$ from a distribution with p.d.f $$\delta_{(\theta,c)}(x)=\frac{1}{c}\mathbb{1}_{(x\in[\theta,\theta+c])},$$ where $\theta\in\mathbb{R}$ and $c\in\mathbb{R}^+$ ...
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### Sufficient statistic definition in Koller's Probabilistic Graphical Models

In Daphne Koller's Probabilistic Graphical Models, the sufficient statistic is defined as follows (p 721): A function $\tau(\xi)$ from instances of $\chi$ to $\mathbb R^l$ (for some $l$) is a ...
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### 2-dimensional minimal sufficient statistic for $U(-k\theta+k,k\theta+k)$

Find a two dimensional minimal sufficient statistic for $\theta$ from $n$ independent random variables $X_k\sim > U(-k\theta+k,k\theta+k)$, $k\in\{1,\cdots,n\}$ Here is what I've attempted. The ...
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