Skip to main content

# 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.

465 questions
Filter by
Sorted by
Tagged with
23 votes
7 answers
3k views

### Why does a sufficient statistic contain all the information needed to compute any estimate of the parameter?

I've just started studying statistics and I can't get an intuitive understanding of sufficiency. To be more precise I can't understand how to show that the following two paragraphs are equivalent: ...
3 votes
0 answers
110 views

### Does 𝑓 : 𝑝 𝜃 ↦ 𝑝 𝑇 | 𝜃 being injective imply statistic T is sufficient? [closed]

Wikipedia says ... consider the map $f:p_{\theta }\mapsto p_{T|\theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$. The statistic $T$ is ...
0 votes
0 answers
17 views

3 votes
1 answer
38 views

### How does knowing the sign of the population correlation affect the sufficiency of its statistic?

As noted here, the sufficient statistic for the correlation under bivariate normality is Pearson's $r$, the maximum likelihood estimate of $\rho$. I suppose, however, this does not guarantee that $r$ ...
1 vote
0 answers
47 views

### 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 ...
3 votes
1 answer
137 views

### What are sufficiency and parameterization invariance in the context of Maximum Likelihood estimation?

While looking at properties of MLE, the article mentions 4 things below: sufficiency (complete information about the parameter of interest contained in its MLE estimator) consistency efficiency ...
3 votes
1 answer
362 views

### Is the Sufficiency Principle an axiom?

Sufficiency Principle as defined in Casella: Where Sufficient Statistic is defined as: Question: Is the Sufficiency Principle an axiom? My thoughts and research so far: I'm uncertain if the ...
12 votes
1 answer
2k views

### 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 "...
2 votes
1 answer
37 views

### Why does the sufficient statistic for the bivariate normal not imply a sufficient statistic for the correlation under bivariate normality?

This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven ...
2 votes
0 answers
39 views

### Reference request for the existence of minimal sufficient statistics

I'd like a recent paper or book that shows in what conditions we can guarantee the existence of a minimal sufficient statistic. I know the paper "Sufficiency and Statistical Decision Functions&...
6 votes
3 answers
131 views

### Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?

I am interested in the Pitman-Koopman-Darmois theorem. I'm having a hard time finding a simple rigorous version of this theorem as I struggle finding sources. This helpful post provides three sources ...
0 votes
0 answers
70 views

### 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 $X:\Omega\to \Omega _X$ ...
0 votes
0 answers
45 views

3 votes
1 answer
174 views

### 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 ...
10 votes
2 answers
2k views

### Find the unique MVUE

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388. Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta<x<2\theta,$ ...
1 vote
2 answers
354 views

### Showing that $X_{(1)}$ is sufficient for shifted exponential distribution

If the pdf of a random sample is $f(x)=e^{-(x-θ)}$ where $x \geq θ$, Show that $T=X_{(1)}$ is a sufficient statistic for $θ$. Can one show that $T$ is a sufficient statistic for $θ$ in the following ...
0 votes
0 answers
217 views

### Rao blackwell theorem but the unbiased estimator is a function of the sufficient statistic

The Rao-Blackwell Theorem states the following: Let $T(\mathbf X)$ be a sufficient statistic for the statistical model $(S, \{f_{\theta}: \theta \in \Theta\})$ and $\hat \theta(\mathbf X)$ be and ...
4 votes
2 answers
496 views

### Why is the weak likelihood principle not a theorem?

The weak likelihood principle (WLP) has been summarized as: If a sufficient statistic computed on two different samples has the same value on each sample, then the two samples contain the same ...
0 votes
1 answer
118 views

### Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]

Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$. $(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$? To ...
1 vote
1 answer
2k views

### 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 ...
8 votes
2 answers
3k views

### How to show that a sufficient statistic is NOT minimal sufficient?

My homework problem is to give a counterexample where a certain statistic is not in general minimal sufficient. Irrespective of the details of finding a particular counterexample for this particular ...
2 votes
1 answer
134 views

### Completeness of Gamma family

Let $X_1,...,X_n$ has a Gamma$(\alpha,\alpha)$ distribution. Find the minimal sufficient statistics. Is this a complete family? My attempt: I found the Minimal sufficient statistics is \$T(x)=(\...
0 votes
1 answer
275 views

### Sufficient statistic under transformation

I just read a weird question when I'm learning sufficient statistic myself. It says whether the sufficient statistic of variance and standard deviation are the same under normal distribution with ...