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.

Filter by
Sorted by
Tagged with
0
votes
0answers
16 views

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 ...
0
votes
0answers
9 views

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 ...
0
votes
1answer
32 views

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^...
0
votes
0answers
25 views

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} ...
0
votes
0answers
28 views

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 ...
1
vote
2answers
47 views

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 ...
1
vote
1answer
45 views

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 ...
0
votes
0answers
37 views

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,...
0
votes
0answers
23 views

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 $...
1
vote
2answers
146 views

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 ...
0
votes
0answers
26 views

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 ...
0
votes
0answers
22 views

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 ...
2
votes
1answer
149 views

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 ...
1
vote
0answers
27 views

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^{\...
8
votes
1answer
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 ...
2
votes
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 ...
6
votes
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 ...
0
votes
0answers
20 views

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 ...
0
votes
0answers
15 views

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{...
0
votes
0answers
47 views

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 $\...
1
vote
0answers
15 views

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, ...
0
votes
0answers
13 views

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 ...
3
votes
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 $\lambda$. ...
0
votes
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 ...
4
votes
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, ...
0
votes
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) = \...
0
votes
0answers
18 views

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 ...
0
votes
1answer
39 views

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 ...
0
votes
0answers
13 views

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 ...
1
vote
1answer
22 views

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 ...
4
votes
0answers
36 views

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 ...
0
votes
1answer
36 views

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 ...
0
votes
0answers
25 views

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 ...
0
votes
0answers
10 views

$T$ 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 ...
3
votes
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 ...
1
vote
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; \ ...
0
votes
0answers
34 views

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}}...
2
votes
2answers
92 views

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 ...
3
votes
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
votes
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 ...
6
votes
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 ...
0
votes
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>...
0
votes
0answers
21 views

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 $T(x)$ ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
4
votes
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
vote
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}\...
0
votes
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 ...

1
2 3 4 5
8