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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Prove that $T$ is a complete statistic and find a UMVUE for $p$
While preparing for my prelims, I came across this problem:
Let $X_1, X_2,\cdots, X_n$ be a sequence of Bernoulli trials, $n \geq 4.$ It is given that, $X_1,X_2,X_3 \stackrel{\text{i.i.d.}}{\sim} Ber(\...
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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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When are Bayes estimators injective as a function of sufficient statistics?
I know that Bayes estimators can be written only as a function of sufficient statistics. When are those functions injectives? That is, when can I say that, given a bayes estimator $\delta (\cdot)$ and ...
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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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Sufficient statistic as iso-surfaces in the distribution density. Is it possible to generalise to multiple parameters?
For continuous distributions, there is a geometric intuition behind sufficient statistics that regards a multivariate probability density as several iso-surfaces.
This works at least for cases where a ...
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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 $r$ ...
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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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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 ...
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Does $f : p_\theta\mapsto p_{T\,\mid\,\theta}$ being injective imply statistic $T $ is sufficient?
Wikipedia says
... consider the map $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$. The ...
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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 ...
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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&...
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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 $X:\Omega\to \Omega _X$ ...
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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 $$ f (x|\theta) \propto \exp(-(x
...
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Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?
I am interested in the Pitman-Koopman-Darmois theorem.
I'm having a hard time finding a simple rigorous version of this theorem as I struggle finding sources.
This helpful post provides three sources ...
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How does reparametrization of the Fisher information matrix change the variance expression for the sufficient statistics?
If I have an exponential family distribution of the form $$p_{\theta}(x) = e^{\theta^T\cdot t(x) - \psi(\theta)},$$ where $\theta$ is a vector of parameters, $t(x)$ is a vector of sufficient ...
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Usage of Sufficient statistic for a Gamma distribution
I need some help to understand how to utilize sufficient statistic from a data.
Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I ...
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Find a two dimensional sufficient statistic for $\theta$
Let $\{X_i\}_{i=1}^n$ be conditional independent given $\theta$ with distribution
$$p_{X_i | \theta} (x |\theta) = \frac{1}{2i\theta}, \ -i\theta<x<i\theta.$$
Find a two dimensional sufficient ...
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Likelihood principle and inference
I've been reading Casella and Berger's Statistical Inference. In section 6.3 the author stated the likelihood principle: if the likelihood functions from two samples are proportional, then the ...
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FInding a complete and sufficient statistic
I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class:
Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
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Karlin-Rubin theorem: relationship between test statistic having the MLR property vs being sufficient
Let's suppose we are trying to compare two hypotheses for a single parameter $\theta$. The null hypothesis $H_0$ is that $\theta = \theta_0$, and the alternative is that $\theta ≥ \theta_0$.
The ...
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Sufficient statistic for the family of PERT distributions?
A beta distribution is one of the form
$$
\text{constant}\times x^{\alpha-1} (1-x)^{\beta-1} \, dx \quad \text{ for } 0<x<1.
$$
According to this Wikipedia article, the family of "PERT ...
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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 ...
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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)=(\...
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Likelihood ratio as minimal sufficient statistics in infinite parameter space
I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first:
Consider a family of density functions $f(x|\theta)$ where the ...
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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 ...
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Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?
I can't understand the paragraph in Completeness (statistics) - Wikipedia:
We have an identifiable model space parameterised by $\theta$, and a statistic $T$. Then consider the map $f:p_{\theta }\...
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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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Sufficient Statistic for a family of distributions consisting of Poisson family and Bernoulli family
Suppose $(X_1, . . . ,X_n)$ is an i.i.d. sample from the distribution $f_{\theta,k}(x)$, where $\theta \in (0, 1)$ and $k = 1, 2$. Assume that $$f_{\theta, k}(x)=\begin{cases} \text{Poisson($\theta)$},...
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Sufficient Statistic for a finite family of Normal distributions
Suppose we have a finite family of normal distributions $P=\{N(0, 1), N(0, 2), N(1, 2), N(2, 2)\}$ and we want to find a sufficient statistic for this family. Intuitively it is clear that as the means ...
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Show minimal sufficient statistic is not complete in normal distribution
Let $Z_i$ for $1 \leq i \leq n$ be a sample from the $N(ap, bp(1-p))$ density, where $a \gt 0, b \gt 0$ are known but $p \in (0,1)$ is an unknown parameter.
I have shown that $T = (\sum^n_{i = 1} Z_i, ...
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Proving Incompleteness of joint sufficient statistic
Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $ -\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
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Minimal sufficient statistic: a measurability issue in a well-known theorem
Given a statistical model $\{\mathbb{P}_\theta\,|\,\theta\in\Theta\}$ on $(\Omega,\mathscr{F})$, and given a real-valued random variable $X$, we say a real-valued random variable $T=T(X)$ is a ...
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A lemma concerning the distribution of sufficient statistic from exponential family
I understand Lemma 8 in Chapter 1 from Lehmann's Testing Statistical Hypotheses [or Lemma 2.7.2 in Lehmann and Romano] as follows:
If the pdf of an exponential family is $$p_{\theta}(x)=\exp\bigg\{\...
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Prove covariance between sufficient statistic and logarithm of base measure in exponential family is equal to zero
Exponential family form is
$$f_X(x) = h(x)\exp(\eta(\theta)\cdot T(x) - A(\theta))$$
I know
$$\operatorname{Cov}(T(x), \log(h(x)) = 0.$$
But how can I prove it?
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Show that $T=\sum_{i=1}^n X_i$ is a sufficient statistic for $p$ [duplicate]
I try to use the definition of sufficient statistic to prove that
Suppose that $X_1,\dots, X_n$ is an iid random sample from $X\sim \mathrm{Bernoulli}(p)$. Show that $T=\sum_{i=1}^n X_i$ is a ...
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Verifying the statistics are complete and sufficient for two parameter Pareto distribution
Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution
with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$
$\textbf{(i)}$ Show that ...
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Why is median not a sufficient statistic? [duplicate]
Suppose a random sample of $n$ variables from $N(\mu,1)$, $n$ odd. The sample median is $M=X_{(n+1)/2}$, the order statistic of the middle of the distribution.
How to prove that sample median is not a ...
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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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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 ...
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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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Is $ T = X_1 +5 X_2 $ sufficient estimator of $p$? [duplicate]
If $ X_1 $ and $ X_2$ are $\textrm{Ber}(p)$ random variables, examine the sufficiency of $ T_1 = X_1 + 5 X_2 $ for $ p .$
I have no idea on how to proceed, I tried to use the conditional ...
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Concrete example of what Sufficient Statistics is [closed]
Having read articles to try to understand Sufficient Statistics.
Sufficient statistics for layman
A sufficient statistic summarizes all the information contained in a sample so that you would make ...
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The equivalence between two sufficient statistics for the same parameter $\theta$
Exercise. Let $X_1,\cdots,X_{n}$ be i.i.d.r.v.'s from $N(\theta,1),$ where $\theta$ is unknown.Show the statistic $T(\mathbf{X})=\sum_{i=1}^{n}X_{i}/n=\bar{X} $ is sufficient for $\theta$.
The answer ...
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Sufficiency and completeness of truncated distribution
[From Theory of Point Estimation (Lehmann and Casella, 1999, Exercise 6.37)]
Let $P=\{P_\theta:\theta \in \Theta\}$ be a family of probability
distributions and assume that $P_\theta$ has pdf $p_\...
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Can the dimension of a (potentially) sufficient statistic exceed the dimension of the parameter it estimates?
I understand that if the dimension of a sufficient statistic exceeds that of the parameter it estimates, then that particular sufficient statistic won't be minimal. Now, in the following case, I ...
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How to prove that this statistic is not sufficient? [duplicate]
Problem.
Given $X_1,X_2,X_3$ a random sample from the Bernoulli distribution with success $\theta$, show that the statistic $T= X_1+2X_2+3X_3$ is not sufficient.
My attempt
When I try to apply the ...
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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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Finding UMVUE of a parameter in form of probability of discrete random variables
We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$.
Their pmf's are:
$f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$
$f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
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How to prove any one-to-one function of minimal sufficient statistic is minimal sufficient?
So I want to prove that any one-to-one function of minimal sufficient statistic is also minimal sufficient. Here is my proof:
Let $T$ be a minimal sufficient statistic and $f$ is a one-to-one function ...
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Are there any (exponential) families without a minimal sufficient statistic?
Bahadur's theorem says that if a minimal sufficient statistic exists, then a complete sufficient statistic is also minimal sufficient.
Are there any (homogenous, identifiable) families with a complete ...
