# 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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 ...