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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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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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 &\...
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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 ...
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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 ...
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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 ...
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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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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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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, ...
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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 ...
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1answer
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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_\...
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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 ...
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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 ...
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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; \...
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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})...
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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 ...
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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. $N(0,\sigma^2), \...
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$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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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} ...
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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 $\mathbf{Y}$ ...
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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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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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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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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
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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
93 views

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 ...
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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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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 ...
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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
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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 ...
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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
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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 ...
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1answer
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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)"?
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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 ...

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