Questions tagged [sufficient-statistics]

For questions about sufficient statistics. A statistic is sufficient for a parametric model if the distribution of the data conditioned on the statistic is parameter-free. For more general questions about statistics and estimators, please use "statistical-inference".

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Help understanding sufficient statistic proof

I am having a hard time following this proof, maybe the solutions have jumped some steps but was wondering if someone could help me follow it. The question: let $X_{1},...,X_{n}$ be independent and ...
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How can I prove that $T=\sum_{i=1}^n X_i^6$ is sufficient for $\theta$?

I’m revising for an exam in Mathematical Statistics and I have found the following problem in one of the previous exams (I don’t know the source): Let $X_1,…,X_n$ be a random sample that follows the ...
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Given n iid Pareto distributed random variables, find the UMP one sided test of the first moment

Given $X_1,...,X_n$ ($n\geq 2$) are iid and each have density: $f_X(x) = \frac{c^\theta \theta}{x^{1+\theta}}\mathbb{1}(x> c)$ for known $c$ and $\theta > 1$ then we can easily find the first ...
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Conditional expectation of product of Normal variate given their sum

Given $$X_1,\ldots,X_n\stackrel{\text{iid}}{\sim}\mathcal N(0,1)$$ I would like to compute the conditional expectation $$\mathbb E\Big[\prod_{i=1}^n X_i \Big| X_1+\cdots+X_n=x\Big]$$ for statistical ...
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Tricky exercise on sufficient statistics (undergraduate level)

During my semester, my class and I were subjected to an interesting exercise that challenged my entire understanding of sufficient statistics (It turns out that the median of this exercise was 0). ...
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Showing order statistic is sufficient

Suppose the model $P_θ$ is the class of all continuous distributions; this is called a ‘nonparametric family’, where the unknown parameter $θ$ is the whole distribution function. Let $x_1,...,x_n$ be ...
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An exercise in "Mathematical Statistics Jun Shao" about the completeness of a 'modified' exponetial family

It is not the first time meeting this problem in StackExchange and I have read the answer to it(the original solution is copied at the bottom, also available in Show a statistic is complete but not ...
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How can I formally reason that the data itself is a sufficient statistic? [closed]

Intuitively, it is very clear that the entire data itself is a sufficient statistic for a parameter of interest. Formally, if random variable $X$ represents data, $S(X)$ is a sufficient statistic ...
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I am a bit confused with this exercise, since I never worked with samples of this type. I would appreciate if you can help me. The exercise is as follows: Let $\{Xi\} \sim N(iθ, 1)$ for $i = 1, .... ,... • 21 1 vote 0 answers 57 views Complete and sufficient statistic Let$X_1\dots X_n$be iid observations with pdf$f(x|\theta)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$for$-\infty<x<\infty$and$-\infty<\theta<\infty$. I need to find a complete sufficient ... • 215 2 votes 0 answers 44 views Minimal Sufficient Statistics and MLE for Parameters in a Piecewise Random Variable [duplicate] Problem Setting:$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^{\... • 346 0 votes 0 answers 56 views Sufficient but not Complete statistics for binomial experiment Consider a binomial experiment with probability of success p in which m trials are conducted resulting in R successes. A further set of trials is then conducted until s further successes have ... 0 votes 1 answer 58 views Equally Distributed Data Set Measurement I will be creating my own dataset with scores ranging from 50.00 to 100.00. How will I say that the dataset I chose is equally distributed and unbiased ? Is there a formula to know this? • 213 0 votes 1 answer 137 views Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete. I just proof that T=(X_{(1)},X_{(n)}) is not complete but now I want to show a more general case. To be more specific I want to show that if a function of a sufficient statistic is ancillary, then ... 0 votes 1 answer 95 views finding UMVUE for \theta_x/\theta_y Let Xi ~ Exp(\theta_x), Yj ~ Exp(\theta_y), i = 1; ... ; n1, j = 1;...;n2. Find UMVUE of \theta_x/\theta_y. Since \bar{X} and \bar{Y} are compelete sufficient statistic, by using Lehmann-... • 53 0 votes 1 answer 89 views minimal sufficient statistic Let X ~ Ber(n1; p), Y ~ Ber(n2; p^2), where X and Y are independent. Find a minimal sufficient statistic T and, using a nontrivial function, show that it is not complete. I get confused by having two ... • 53 2 votes 0 answers 40 views Finding minimal sufficient statistics for this family coming from the given probability mass function 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 ... • 2,265 3 votes 1 answer 115 views Is \max\{-X_{(1)},X_{(n)}\} a one dimensional or two dimensional statistic? Is statistic \max\{-X_{(1)},X_{(n)}\} one dimension or two dimension? I was trying to find the minimal sufficient statistic for U(-\theta,\theta) from n i.i.d random variables X_i. The ... • 521 1 vote 0 answers 54 views Self-Study, Minimal Sufficient Statistic, MLE, Beta, Correct Argument The following example is taken from Hogg Introduction to Mathematical Statistics 7e and the exercise is to show that the MLE is a minimal sufficient statistic. I am not 100% sure about my argument. ... • 389 0 votes 1 answer 57 views Self-Study Sufficient Statistics, Pdf with Indicator Function The example is from the Book Hogg Introduction to Mathematical Statistics Page 384, Chapter 7.2 Sufficient Statistics. Please let me know if my argument for the solution is correct, since I used a ... • 389 0 votes 1 answer 57 views Data Processing Inequality for sufficient statistic case Consider a Markov chain X \rightarrow Y \rightarrow Z and assume Z is a sufficient statistic for X (i.e I(X;Y)=I(X;Z)), do we have a case for X, Y and Z where H(Y) > H(Z)? Here is ... • 621 1 vote 1 answer 283 views Is sample variance a complete statistic for variance of a normal distribution if the mean is known? Suppose X \sim N(\mu,\sigma^2). I know that T(X)=(\bar X, S^2) is a complete sufficient statistic for \mu, \sigma^2 if \mu, \sigma^2 are unknown. But if \mu is known, is S^2 still a ... 1 vote 1 answer 529 views Question of the minimal sufficient statistics of beta-distribution The beta distribution with parameters \alpha>0 and \beta>0 has density$$f(x|\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1$$... • 905 1 vote 1 answer 164 views Sufficient statistic by factorization theorem Suppose we have a random sample X_1,\dots,X_n of X, where X has the following pdf:$$f_{\mu,\sigma}(x)=\left(\pi\cdot\sqrt{(x-\mu)(\mu+\sigma-x)}\right)^{-1}$$where x\in(\mu,\mu+\sigma),\mu\... 0 votes 1 answer 91 views Showing a minimal sufficient statistic If we have common density$$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$with x\in(0,1), \theta>0 and \textbf{X}=(X_1,...,X_n) is a random sample. Then how can we show that the ... -1 votes 1 answer 120 views Finding Sufficient Statistics Let X1, . . . , Xn be a random sample from the following pmf. P(X = 0) = θ, P(X = 1) = 2θ, P(X = 2) = 1 − 3θ, 0 < θ < 1/3 Find a non-trivial sufficient statistic. I start like this: L(θ)=L(θ)=∏... • 33 0 votes 1 answer 351 views When does a sufficient statistic not exist by the Factorization Theorem? The Neyman Factorization Theorem states the following: Let f(x_1, ..., x_n; \theta) denote the joint pmf or pdf of X_1, ..., X_n. Then T = t(x_1, ..., x_n) is a sufficient statistic for \theta ... • 1,603 1 vote 0 answers 73 views Minimally Sufficient Statistics Partition Intuition I am trying to understand the intuitive idea of a minimally sufficient statistic. It is my understanding that a statistic T is minimally sufficient for \theta for a family of populations X\sim P_\... • 1,174 1 vote 1 answer 97 views Sufficient statistic for (\theta,j) when X_i\sim f_{\theta,j} Let X_1,X_2,\ldots,X_n be i.i.d random variables with pmf f_{\theta,j}(\cdot) where \theta \in (0,1) and j=1,2. f_{\theta,j}: pmf of Poisson (\theta) when j=1 and f_{\theta,j}: pmf of ... 2 votes 0 answers 45 views Minimally Sufficient Statistics I'm trying to find the minimally sufficient statistic where \{X_i\}_{i=1}^{n} are iid from the following family of populations:$$P=\{U(0,\theta): \theta>0\}$$I looked at the ratio of the ... • 1,174 4 votes 1 answer 993 views Full Rank Exponential Families I am trying to better understand the importance of full rank exponential families of distributions i.e. a family of populations dominated by a \sigma-finite measure such that the radon-nykodym ... • 1,174 1 vote 1 answer 1k views Complete Sufficient Statistic for double parameter exponential I am trying to show that (X_{(1)}, \sum_{i=1}^{n}(X_i-X_{(1)}) are joint complete sufficient for (a,b) where \{X_i\}_{i}^{n}\sim exp(a,b). I know the joint pdf is$$\prod_{i=1}^{n}\frac{1}{b}... • 1,174 1 vote 1 answer 110 views Can someone clear my understanding of sufficient statistics? The definition of sufficient statistics says that the conditional distribution of a sufficient statistic, say$S$, must be independent of the unknown parameter,say$\theta$. Consider the$Ber(\theta)...
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Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ...