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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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 ...
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How to eliminate constant to derive the decision rule in terms of the sufficient statistic $\bar{X}$ for normal distribution means hypothesis test?
Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $\mu$, and the standard deviation, $\sigma$, of the population are unknown. We want to ...
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MLE of parameters for a difference of two Exponential IID
Suppose I have $X_1 \sim Exp(\theta_1)$ and $X_2\sim Exp(\theta_2)$. Then it is not difficult to show that $Y = X_1 - X_2$ will have density:
$f_Y(y) = \frac{1}{\theta_1 + \theta_2}e^{-y/\theta_1}\...
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Why sample size is not a part of sufficient statistic?
Following simple example from Wikipedia's definition of sufficient statistic with Bernoulli distribution with parameter $\theta$, where sufficient statistic is a sum of successes
$$T(X_n)=\sum_{i=1}^n ...
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sufficient, minimal, complete
Are all complete statistics functions of each other?
For example if I have T and S complete statistics
Can you always write T in terms of S and S in terms of T?
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Sufficient statistics for bernoulli distribution
Let $Y_1, \ldots, Y_n $ be a random sample of size $n$ where each $Y_i \sim \textrm{Bernoulli}(p), $ and
let $Y = \sum Y_i $ for $i = 1, \ldots, n.$
The estimator is $W= (Y+1)/(n+2). $
Is the ...
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Is this a sufficient statistic for variance?
I have $X_1,\dots,X_n,X_{n+1}\overset{iid}{\sim}F_X(x)$, where $F_X$ has a finite mean $\mu$ and variance $\sigma^2$.
If I calculate $\bar X_n = \dfrac{1}{n}\sum_{i=1}^n$ and $S^2_n = \dfrac{1}{n-1}\...
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How is $\text{Cov}(\bar{Y}, Y_i - \bar{Y}) = \dfrac{1}{n^2} \text{Cov} \left( \sum_{j = 1}^n Y_j, nY_i - \sum_{j = 1}^n Y_j \right)$?
I have this example of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence
$$\...
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How does the result $\dfrac{1}{n^T} \dfrac{T!}{\prod_{i = 1}^n Y_i!}$ tell us what distribution $T(\mathbf{Y})$ is?
This follows on from my question here.
I have the following problem:
Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ ...
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Is there a standard measure of the sufficiency of a statistic?
Given a parametrical model $f_\theta$ and a random sample $X = (X_1, \cdots, X_n)$ from this model,
a statistic $T(X)$ is sufficient if the distribution of $X$ given $T(X)$ doesn't depend on $\theta$.
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Minimal sufficient statistic with parameter-dependent support on density function
I'm having trouble finding a minimal sufficient statistic for this particular population. It has the pdf defined as $$f_\theta (x) = 4\theta^4 x^{-5}$$ if $\theta \leq x$, and 0 otherwise. We take $n$ ...
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