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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1answer
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Which sufficient statistic is for which parameter
Suppose I have n samples $x_1,x_2,...x_n$ sampled from a $\mathcal{N}(\mu,\sigma^2)$. I need to find the sufficient statistics for $\mu,\sigma^2$. I write the likelihood
$$ \mathcal{L} = (2 \pi \sigma^...
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Complete sufficient statistic of non-identical distribution: $X_i \sim EXP(i\theta)$
Problem
Suppose that $X_1, \dots, X_n$ are independent $\mathrm{EXP}(i\theta)$ random variables. Find a complete sufficient statistic for $\theta$.
My Attempt
Since pdf of $x_i$ is
\begin{equation}
...
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Simple sufficient statistic
Given a sample $\{X_1, ..., X_n \}$ with the following pdf
$$ f(x; \theta) = \begin{cases} 2\theta^{-2} x, & \mbox{for } 0 \leq x \leq \theta \\ 0, & \mbox{otherwise } \end{cases}$$
find a ...
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If a statistic $T(X)=\Sigma_{i=1}^n X_i$ is sufficient does that imply the mean is also sufficient?
I've been working on some problems, the question asked me if the mean of a sample is a sufficient statistic for poisson distribution. I've already proved that $T(X)=\Sigma_{i=1}^n X_i$ is a sufficient ...
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1answer
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Did I correctly apply the factorisation theorem in this example?
Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}(x \in]\theta,\theta+1[)$ and the random variable $\mathbf{X}=(X_1,\ldots,X_n)$ are independently identically distributed ...
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Proving that two statistics are equivalent in sufficient statistics sense
I'm trying to understand an example from Mathematical Statistics by Bickel and Doksum, 2nd edition.
In Example 1.5.4 (continued) part, I'm having hard time figuring out why expressing $t_2$ by $L_X(0,...
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What's the maximum likelihood estimation of $\theta$ in this density?
Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$.
Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
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Show that the maximum of $x_1,…,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. (From definition)
Problem
Show that the maximum of $x_1,...,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$.
Background
This question has been asked before, but most answers tackle the ...
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How to prove that the t distribution doesn't belong to the exponential family?
Or in other words, is there anyway prove that the t distribution doesn't belong to the exponential family without going through all that calculation? Since the density has the gamma function in it ...
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Sufficient statistic for a given distribution from exponential form
Given a particular form, i can verify whether it is sufficient statistic or not using $\frac{p_\theta(x_1,x_2...x_n)}{p_\theta(T(x_1,x_2...x_n))}$ is independendent of $\theta$ then i can say $T(\bar ...
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1answer
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Understanding the Rao-Blackwell Theorem
I've been reading up a lot on the practical applications of the Rao-Blackwell theorem. I do understand how the Bias and Variance and MSE aspects of the theorem fall in place (i.e. the mathematical ...
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MLE and Minimal Sufficiency of Parameters in a Piecewise Random Variable [closed]
Problem Setting:
$X_i$ is i.i.d. from a piecewise distribution which is
$$
f_{\theta_1, \theta_2}(x) = \frac{1}{\theta_1+\theta_2}e^{-\frac{x}{\theta_1}}I_{[x>0]} + \frac{1}{\theta_1+\theta_2}e^{\...
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232 views
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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1answer
43 views
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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1answer
163 views
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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Complete sufficient statistics for bivariate observations
We have observations $(X_i,Y_i), 1\le i\le n$ from a family of distributions $\mathcal F$, consisting of all absolutely continuous bivariate distributions. I wish to show that $(X_{(i)},Y_{\text{...
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MSE of randomized decision in Normal distribution
Suppose a sample $\bf{X}$$=(X_1,...,X_n)$ is from $X\sim N(\theta,1)$. The sample mean $T(\bf{X}$$)=\bar{X}$ is sufficient to the population mean $\theta$.
For $\delta(\bf{X}$$)=X_1$, the decision $\...
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How to prove or disprove that a complete sufficient statistic exists?
We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
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Is there a general way to find the distribution of a sufficient statistic?
Let $f(\boldsymbol{x}|\theta)$ denote the joint pdf of a sample $\boldsymbol{X}$, and suppose $T(\boldsymbol{X})$ is a sufficient statistic for $\theta$. My question is: is there a general way to find ...
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1answer
60 views
Why is $T(X) = X_{1} + … + X_{n}$ a sufficient statistics for Poisson $\lambda$ instead of $\frac{1}{n}\sum{X_{i}}$
From Wikipedia:
If $X_{1},\dots, X_{n}$ are independent and have a Poisson distribution with parameter $\lambda$, then the sum $T(X) = X_{1} + ... + X_{n}$ is a sufficient statistic for $\lambda$.
...
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1answer
34 views
Completeness of normal sample values
I have a small query. We know that sample values are always sufficient. Can we say the same for completeness property?
Let us say I have $X_1,X_2,...,X_n$ following $N(0, \sigma^2)$. Hence, if we take ...
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1answer
61 views
Minimal Sufficient Statistic for the distribution $U(-\theta, \theta)$ [duplicate]
I need to find the sufficient statistic for the parameter $\theta$ for a uniform distribution $U(-\theta, \theta)$ for a sample of size $n$.
The joint density of the sample can be written as:
$f(x_1, ...
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1answer
41 views
Minimal Sufficient Statistic of normal with known variance
I have a problem in which $X_1, X_2, .., X_n$ follows $N(\theta,1)$ and I am required to compute the minimal sufficient statistic for $\theta$.
I can see from exponential family criterion, $T_1(x) = \...
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Doubt in the sufficient statistic for the Location Family [duplicate]
There is a problem in the exercise $6.8$ of Casella and Berger which says that the order statistics $T(X) = (X_{(1)},X_{(2)},X_{(3)}..,X_{(n)})$ will be the sufficient and no further reduction ...
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1answer
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Doubt Regarding the Sufficient Statistic Problem
Let $X_1, X_2, ..., X_n$ be random variables from the following densities:
$$
f(x_i|\theta) = \frac{1}{2i\theta} \text{ for } -i(\theta-1) < x_i < i(\theta+1)
$$ and otherwise the density is ...
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Expectation-Maximization (EM) algorithm with known means
I am trying to fit a mixture of 1D Gaussian distributions to some data. Can the EM algorithm be used in the case of known mean (all the mean values are equal to zero) and fit only the variances of the ...
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1answer
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Proving sufficiency of a statistic using the expectation
I am blocked trying to solve the following question. I would appreciate if someone could give me a hint.
Let $X_1,\ldots,X_n$ be $n$ independent random variables following a continuous uniform ...
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Why is the EM algorithm well suited for exponential families?
I've been brushing up on the EM algorithm, and while I feel like I understand the basics, I keep seeing the claim made (e.g. here, here, among several others) that EM works particularly well for ...
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1answer
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Jointly complete and sufficient statistics for multivariate normal distribution
Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ).
*Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient ...
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Finding the minimal sufficient statistics for this family
Let $X_i\big|_{i = 1...n}$ be random sample from the PMF:
$P(X_i = 0) = \frac{1-\theta}2;\;P(X_i = 1) = \frac12 ; P(X_i = 2) = \frac\theta2$ where $\theta\in(0,1)$. Find the minimal sufficient ...
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$T$ be minimal sufficient statistics. If $T = H(U)$ where $U$ has the same dimension as that of $T$, is $U$ minimal sufficient? [duplicate]
Suppose $T$ is minimal sufficient statistics for a family of
distribution. If $T = H(U)$ where $U$ has the same dimension as that
of $T$, then does it imply that $U$ is also minimal sufficient?
I ...
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2answers
107 views
Do sufficient statistics for parameters of interest depend on whether nuisance parameters are known?
The definition of sufficient statistic is as follows:
A statistic $T(X_1,...,X_n)$ is sufficient for parameter $\theta$ if the conditional distribution of $X_1,...,X_n$, given that $T=t$, does not ...
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1answer
67 views
Minimal sufficient statistics for 2-parameter exponential distribution
Suppose $X_1, \ldots, X_n$ is a random sample with pdf $$f_{X_i}(x_i \mid \alpha, \beta) = \beta^{-1} \exp \left(\frac{-(x_i-\alpha)}{\beta}\right) I(x_i \geq \alpha)$$
for all $i = 1, 2, \ldots, n; \ ...
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Is the Pearson Product Moment Correlation Coefficient (PPMCC) a sufficient statistic?
The PPMCC can be defined as the following:
$$\rho_{X,Y} = \frac{\mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]}{\sqrt{\mathbb{E}[X^2] - (\mathbb{E}[X])^2}\sqrt{\mathbb{E}[Y^2] - (\mathbb{E}[Y])^2}}...
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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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1answer
108 views
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 $c\in\mathbb{R}^+$ ...
2
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1answer
107 views
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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1answer
126 views
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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0answers
64 views
Disprove Sufficiency [duplicate]
I am working on a Problem from Casella Berger.
It reads the following:
Let $X_1, ...., X_n$ be a random sample from a population with pdf
$$ f(x)= \theta x^{\theta -1}$$
with $ 0 <x<1, \theta>...
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Proving two statistics are equivalent
I was trying this problem from the book - Theory of Point Estimation by Lehmann and Casella.
I could do part $a$ and part $b$ but couldn't do part $c$. Any help is appreciated.
Just to clarify $T(x)$ ...
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1answer
152 views
Sufficient statistic for the distribution of a random sample of Poisson distribution
Let $X_1,...,X_n$ be a random sample from a Poisson distribution with mean $\lambda$ and $T = \sum_{i=1}^n X_i $ . Show that the distribution of $X_1,...,X_n$ given T is independant of $\lambda$ so ...
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1answer
83 views
Sufficient estimator for Bernoulli distribution using the likelihood function theorem for sufficiency
Let $(X_1,X_2)$ be a random sample of two iid random variables, $X_1\sim Ber(\theta),\theta\in (0,1)$.
Use the following theorem to show that $\hat{\theta}=X_1+2X_2$ is sufficient.
Likelihood theorem ...
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1answer
48 views
Sufficient Statisitics and Discrete Distributions
I am trying to master minimal/complete sufficient statistics, however I am having trouble when the distributions are discrete and involve indicator functions. Here is my 3 part question:
Let $X$ be a ...
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2answers
181 views
Distribution of the sufficient statistic in the exponential family?
Suppose $\boldsymbol X$ belongs to the exponential family,
$$ f_X\!\left(\,\mathbf{x} ; \boldsymbol \theta\,\right) = h(\mathbf{x}) \, \exp\!\Big(\,\boldsymbol\eta({\boldsymbol \theta}) \cdot \mathbf{...
1
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1answer
68 views
Minimal Sufficient Statistic for Bivariate Binomial
Find a minimal sufficient statistic for $p$ where
$Y\sim\mathsf{Binom}(n,p)$ and $Z\sim\mathsf{Binom}\left(n,p^2\right)$
are independent random variables. Determine if this statistic is
complete. If ...
2
votes
1answer
43 views
Discrete Sufficient Statistics
Let $X$ be a random variable from the following distribution
$$f(x;\theta) = \left\{\begin{array}{ccc} \theta & , & x = -1 \\ (1 - \theta)^2\theta^x & , & x = 0,1,2,\ldots\end{array}\...
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0answers
30 views
Finding the Sufficient Statistic from a PMF
Let X be a random variable with the following PMF
$
f(x;\theta)=\frac{1}{4}, x=1,2
$
$
f(x;\theta)=\frac{1+\theta}{4}, x=3
$
$
f(x;\theta)=\frac{1-\theta}{4}, x=4
$
and $0\leq\theta\leq1$
Find a ...
