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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Conjugate priors outside exponential family
The usual exception I have come across regarding non-existence of conjugate prior outside the exponential family is the uniform distribution on $(0,\theta)$ (i.e. $U(0,\theta)$) where $\theta$ has a ...
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How to determine data size is statistically efficient?
I have a question about the data size for probability of default model.
For each consumer, I have a binary bit to indicate whether the client goes default or not (1 is default and 0 is current). And I ...
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Are the two definitions of a sufficient statistic equivalent? [duplicate]
I am confused about the fundamental definition of a sufficient statistic. I found two different definitions and I wounder if they are equal.
with
data $X$
sufficient statistics $t$
parameter $\theta$
...
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0answers
19 views
Can use Factorization criterion for proving given estimator is not Sufficient?
I was learning about the Sufficiency of Estimators and Factorization criteria. Now, I noticed that whenever we prove a given estimator is not sufficient, we use a counterexample with concrete values. ...
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Why inferences on odds ratio can be based on the conditional distribution of X given X+Y
Consider two independent Binomial variables with parameters $(n_1,p_1)$ and $(n_2,p_2)$, say $X$ and $Y$. The odds ratio is then defined by $\phi = \frac{p_1(1-p_2)}{(1-p_1)p_2}$. The minimal ...
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39 views
UMVUE of Bernoulli random variables
Let $X_1, X_2..... X_n$ be a random sample from a Bernoulli population with parameter $p$.
A sufficient statistic is $\sum_{i=1}^{n}X_i$. If we define
$$
U(X_1,X_2,\ldots,X_n)= \begin{cases}1/2n &\...
3
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2answers
55 views
Sufficient statistic of Bernoulli Trial
Let $X_1$ and $X_2$ be iid random variables from a $Bernoulli(p)$ distribution. Verify if the statistic $X_1+2X_2$ is sufficient for $p$.
I calculated and found out $X_1+X_2$ as a sufficient statistic ...
3
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1answer
49 views
Likelihood Ratio and Sufficient Statistics
I am not very experienced with statistics, so I apologize if this is an incredibly basic question. A book I am reading (Examples and Problems in Mathematical Statistics - Zacks) makes the following ...
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Showing a minimal sufficient statistic [duplicate]
If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$
with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample.
Then how can we show that the ...
3
votes
1answer
62 views
Sufficient statistic for poisson
Possion have mean and variance of the same value, and obviously the mean of samples is a sufficient statistic
Is the variance of the sample a sufficient statistic as well?
1) If not, how do I prove ...
4
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1answer
69 views
Can this statistic be shown not to be sufficient for $\theta$?
This problem comes from Casella and Berger, who do not rigorously demonstrate (in their solution key) that the statistic is not sufficient.
Let $X_1,\dots,X_n$ be a random sample from a population ...
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30 views
If likelihood is statistically sufficient, how could some cases have no sufficient statistics?
Almost all elementary texts clarify that in some cases minimal sufficient statistics might not exist. However, it seems that the likelihood itself induces a partition that essentially provides a ...
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1answer
57 views
Sufficient Statistic for Normal Distribution | Mean, Variance & Kurtosis
I have seen multiple times that a normal distribution is fully specified by mean and variance. It is obvious that the third moment is not necessary for a perfect normal distribution as it is 0. I ...
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0answers
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Showing Sufficiency of a Statistic [duplicate]
Suppose that $X_1, X_2, \cdot \cdot \cdot \ , X_n$ is a random sample from a continuous distribution with pdf $f_X(x;\theta) = \theta x^{\theta-1}$, for $\ 0\leq x \leq 1$. Show that $W=\prod_{i=1}^n ...
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Conditional distribution of complete sufficient statistics being ancillary of $\alpha$
Regarding the distribution and statistics as described here, I need to show that the conditional distribution of $\overline{X}$ given $X^*=x^*$ does not depend on $\alpha$. I remember my professor ...
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1answer
51 views
Gamma distribution: ratio of 2 CSS not containing $\beta$
Let $X_1,...,X_n$ be iid and follow $Gamma(\alpha, \beta)$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $\overline{X}$ and $X^*=\...
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UMVUE of functions of parameters from 2 normal samples
So the problem I have asks to find the UMVUE of $\sigma^2$ and of $\left( \epsilon - \eta \right)^2$, where $\sigma$,$\epsilon$, and $\eta$ are parameters of normal distributions as described here, ...
0
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1answer
46 views
Poisson sufficient statistics problem
I have the following problem:
Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ distribution has the probability function ...
1
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1answer
41 views
Definition of $k$-parameter exponential family
I am currently studying the concept of sufficient statistics in mathematical statistics. The following definition is presented:
Definition: $k$-parameter exponential family
Let $\mathbf{Y} \sim f_\...
0
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1answer
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Variance of sum calculation in example illustrating completeness for minimally sufficient statistic
I have an example where it is said that $$\sum_{i = 1}^n Y_i \sim N(n \mu, n a^2 \mu^2)$$ and
$$\begin{align} E \left[ \left( \sum_{i = 1}^n Y_i \right)^2 \right] &= \text{Var} \left( \sum_{i = 1}^...
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Properties of sufficiency
In book "Theory of point estimation" second ed. (E.L. Lehmann, G. Casella), page 33, it is said that when we know the realization of sufficient statistic $T(\mathbf{x})=t$ and throw out the rest of ...
0
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1answer
103 views
Joint distribution simplification in minimal sufficient statistics example
My notes introduce the concept of minimal sufficient statistics as follows:
Definition
A sufficient statistic $T(\mathbf{Y})$ is called a minimal sufficient statistic if it is a function of any other ...
1
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1answer
170 views
Sufficient statistics in the uniform distribution case
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \...
3
votes
1answer
174 views
Sufficient statistics are not unique?
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
2
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1answer
93 views
How to find confidence interval for Uniform([a,1])?
Let $ U_1, \dots, U_n $ be a random sample of uniform distribution
over $ [a,1] $. Construct a confidence interval for $ a $ with $ 1-\alpha = 0.95 $.
I managed to show that $ T = \min\{U_i\} $ is ...
0
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1answer
42 views
Two-dimensional sufficient statistic for OLS model
Suppose that the random variables $Y_1,...,Y_n$ satisfy $$Y_i=\beta x_i + \epsilon_i, i=1,...,n$$ where $x_1,...,x_n$ are fixed constants, and $\epsilon_1,...,\epsilon_n$ are i.i.d. $N(0,\sigma^2), \...
0
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1answer
39 views
$Y_1 + \dots + Y_n \sim N(n\mu, n\sigma^2)$ implies that $\bar{Y} \sim N(\mu, \sigma^2/n)$?
I have this example of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence
$$\...
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Bayesian Linear Regression and the Exponential Family
In a straight forward linear regression model, assuming a fixed input $\mathbf{x}$, and additive noise with unit variance we can write:
\begin{equation}
p(y\mid \mathbf{x,w})=\frac{1}{\sqrt{2\pi}\...
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2answers
141 views
Finding the form $g(T(\mathbf{y}), \lambda) \times h(\mathbf{y})$ for sufficiency statistic examples
I'm studying some notes that present examples of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y}...
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1answer
72 views
Conditional probability, statistic and sufficient statistic
In statistical model $(\mathcal{X}, \{P_\theta\mid\theta\in\Theta\})$ statistic $T=T(\mathbf{X})$ (where $\mathbf{X}$ marks random sample) is said to be sufficient for $\theta$, when conditional ...
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Poisson sufficiency statistic example
I'm studying some notes that present examples of sufficiency:
Let $Y_1, \dots, Y_n$ be a i.i.d. $\text{Pois}(\lambda)$. Then
$$\begin{align} L(\lambda; \mathbf{y}) &= \prod_{i = 1}^n e^{-\lambda} ...
0
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1answer
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Explanation of “sufficient statistic” example
I just started learning what a sufficient statistic is:
Definition
A statistic $T(\mathbf{Y})$ is sufficient for an unknown parameter $\theta$ if the conditional distribution of the data $\mathbf{Y}$ ...
3
votes
2answers
49 views
Understanding sufficient statistics geometrically
Consider the distribution $\mathcal{P} = \mathcal{N}(\mu, 1)$, where the variance is known but the mean is unknown. Let $X_1,X_2\sim P$ i.i.d. In this case $T = X_1+X_2$ is a sufficient statistic.
I ...
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Sufficient statistic for a logistic distribution
I'm using the statistical inference book from Casella and Berger. More specifically I'm interested in knowing how to get the sufficient statistic of a logistic distribution in the form of $$f(x|\theta)...
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2answers
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Why do we not care about completeness, sufficiency of an estimator as much anymore?
When we begin to learn Statistics, we learn about seemingly important class of estimators that satisfy the properties sufficiency and completeness. However, when I read recent articles in Statistics I ...
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2answers
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Sufficient Statistics - Relating the Intuition with the Mathematical Definition
I believe the heuristic definition of a Sufficient Statistic makes sense to me - when you take a sample in order to make an inference about the parameter related to the probability distribution, and ...
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1answer
51 views
Statistics Theory Question
Casella& Berger Theorem 6.2.28: If a minimal sufficient statistics exists, any complete statistics is minimal sufficient.
So let's suppose $X_1...X_n$ are iid $Bernoulli(p)$ $p\in (0,1)$, then $\...
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0answers
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What's wrong with this proof that the sample sum is sufficient for $\theta$ in $U(0,\theta)$?
So let's say $X_i ~ U(0, \theta)$, and let's consider the two-sample sample sum, $t = \bar{X_2} = (X_1 + X_2)/2$.
So we want to show that $p(x|t) = p(x,t)/p(t) = p(x)/p(t)$ is independent of $\theta$....
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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 ...
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1answer
46 views
How can I find a complete, minimal sufficient statistic from a $Beta(\sigma,\sigma)$ distribution?
Let $X_1,\cdots,X_n$ be a random sample from $Beta(\sigma,\sigma)$, where $\sigma > 0$ is unknown. Is the minimal sufficient statistic for $\sigma$ complete?
My work
I found the minimal ...
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1answer
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Show that $T(\mathbf{X})=(\sum X_i, \sum X_i^2)$ is not complete
Let $X_1, \cdots X_n \stackrel{\text{iid}}{\sim} N(\alpha \sigma, \sigma^2)$, where $\alpha$ is known, and $\sigma > 0$ is unknown. Show that the family of distributions of $$T(\mathbf{X})=(\sum ...
3
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1answer
70 views
How can I show that $\sum X_i$ is not a sufficient statistic for $\theta$?
Let $X_1,\ldots, X_n \sim f(x\mid \theta)=\frac{x}{\theta}e^{-x^2/(2\theta)}, x > 0$ independently. $\theta > 0$ is unknown.
How can I show that $\sum X_i$ is not a sufficient statistic for $\...
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0answers
25 views
Is it ok to write $\Bbb E[X|T(X)]$? [duplicate]
Suppose that we observe the discrete random variable $X = (X_1, . . . , X_n)$ with state space
$S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs
to a known ...
0
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0answers
71 views
Is it correct to write $\Bbb E[X]$ or $\Bbb E_{\theta}[X]$?
Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space
S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs
to a known ...
4
votes
1answer
108 views
What is the difference between $T(x)$ and $T(X)$?
Suppose that we observe the discrete random variable $X = (X_1, . . . , X_n) $ with state space
$S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs
to a known ...
1
vote
1answer
89 views
Proving that $T(X)$ is a sufficient statistic for $\theta$
I have that $X=(X_1,...,X_n)$ is a rv consisting of $n$ iid exponential rv's where $\theta$ is the parameter (and thus mean $1\over \theta$) .
I have to prove that $T(X)=\sum^n_{i=1}X_i$ is a ...
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1answer
39 views
How to derive a pdf of Complete Sufficient Statistic of exponential family
While studying Mathematical statistics through "Introduction to Mathematical Statistics 7th" (by Hogg and Craig), I've been stuck in the Theorem above.
The answer of the exercise 7.5.8 is not given in ...
0
votes
1answer
20 views
replication of minimal sufficient statistic
Suppose we have a minimal sufficient statistic for observations $X_1, ...,X_n$ that are i.i.d from distribution $f(X|\theta)$, namely $T(X) = (T_1,...,T_k)$ which is a $k$ dimensional statistics. Now ...
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Dimension of minimal sufficient statistic
Is this true that "dimension of every minimal sufficient statistic is less than any sufficient statistic (minimal or not)"?
2
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1answer
53 views
How to identify one-one correspondance in Sufficient Statistics?
The correct answer to the given question is (1),(3) and (4). I understood how 3 and 4 are correct but I could not understand how (1) is also a correct answer.
I know that here $\sum_i X_i$ is a ...
