# Questions tagged [complete-statistics]

A complete statistic T (in some statistical model) is such that for all functions g, if E g(T)=0 for all parameter values, then g is identically zero.

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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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### Does this distribution belong to the exponential family? [duplicate]

I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
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### Complete statistic for discrete distribution

Let $X$ be a discrete random variable with the probability mass function $$P(X=-1)=\theta,\ P(X=0)=1-2\theta,\ P(X=1)=\theta$$ I'm trying to find a complete statistic for $\theta$ based on the single ...
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### If $T$ is a complete sufficient statistic, then $Cov(T, U)=0$ for all unbiased $U$ [duplicate]

I want to prove the following- Show that if $T$ is complete sufficient for $θ$, then $Cov_θ(T, U) = 0$ for all $θ ∈ Θ$ and for all $U$ satisfying $E_θ(U) = 0$ for all $θ ∈ Θ$. I think in essence it ...
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### UMVUE for P(X > k) in exponential distribution [duplicate]

I have to find UMVUE for $exp(-k*a)$ where X ~ Exponential(a); k is a positive real number. I tried it using Lehmann-Scheffe theorem. Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
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### How is a sufficient complete estimator minimal sufficient [duplicate]

the question is quite theoretical But I am finding it extremely hard to understand it without cramming the derivation ,although I understand the basic concept of it . Can someone explain how a ...
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### Equivalence of the completeness of the order statistics and the uniqueness of symmetric unbiased estimators

I am reading A.J. Lee's 1990 book "U-statistics: Theory and Practice". There is an equation on page 6 that I cannot explain why it holds, and I hope somebody could help me. Here is the ...
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### Completeness calculation: Where does the $\sum_{i = 0}^n g(i)$ in $E_p[g(T)] = \sum_{i = 0}^n g(i) {n\choose{i}} p^i (1 - p)^{n - i}$ come from?

I have the following definition of completeness: Definition Let $Y_1, \dots, Y_n$ be an i.i.d. $f_\theta (y)$, where $\theta \in \Theta$. A statistic $T(\mathbf{Y})$ is complete if no function $g(T)$ ...
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### Example of curved exponential family with $T$ being a complete statistic?

Is there any example of curved exponential family with $T$ being a complete statistic? Here $T$ is the sufficient statistic.
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### Completeness of a statistic - Open ball

I was studying the slides of the course in statistics, but there is a theorem that is not clear for me. This chapter was about finding a complete statistic, and it explains that it can be found with ...
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### Sufficient Statisitics and Discrete Distributions

I am trying to master minimal/complete sufficient statistics, however I am having trouble when the distributions are discrete and involve indicator functions. Here is my 3 part question: Let $X$ be a ...
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### Minimal Sufficient Statistic for Bivariate Binomial

Find a minimal sufficient statistic for $p$ where $Y\sim\mathsf{Binom}(n,p)$ and $Z\sim\mathsf{Binom}\left(n,p^2\right)$ are independent random variables. Determine if this statistic is complete. If ...
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### Complete and Sufficient Statistic for Discrete Distribution

I have a single observation X from the following distribution: $$𝑃(𝑋=−1)=\dfrac{𝑝}{3},𝑃(𝑋=0)=(1−𝑝),𝑃(𝑋=1)=\dfrac{2𝑝}{3}$$ I'm trying to find a complete and sufficient statistic for p based on ...
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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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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 $\... 3 votes 1 answer 266 views ### Rao-Blackwell part of the Lehmann-Scheffe theorem I'm trying to understand the proof of this theorem. An unbiased estimator$T$, that is a function of a complete statistic$S$, is unique, i.e. there can't be other unbiased estimators that are ... 1 vote 1 answer 189 views ### Show that$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 ... 0 votes 1 answer 113 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 1 answer 30 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 ... 1 vote 1 answer 125 views ### Is the sample mean complete sufficient for the Expectation when X \sim F where F is some symmetric distribution? Let X_1,X_2... X_n be iid \sim F where F is any symmetric continuous distribution and let \mid E(X)\mid<\infty. Is \bar{x} complete sufficient for E(x)=\int{xf(x)dx}? Assume that all ... 2 votes 1 answer 55 views ### What is the broadest context where \bar{X} is complete sufficient for \mathbb E(X)? \bar{X} is complete sufficient for \mathbb E(X) if X is Normal with known standard deviation \sigma. Are there broader contexts? Like for exponential families in general or more general than ... 5 votes 1 answer 616 views ### Relationship between completeness and sufficiency hopefully this isn't a duplicate of another question (at least I didn't find one). Here is a question I have about completeness and sufficiency: Problem: Suppose T(x) is complete sufficient for \... 3 votes 1 answer 295 views ### Proving X is a complete statistic to find a UMVUE I'm learning about Stein's phenomenon. This standard problem is considered: Let X_1, \dots, X_p be independent random variables with X_i \sim N(\theta_i, 1) for i = 1, \dots, p. Let \theta = (\... 2 votes 1 answer 284 views ### Complete statistics for f_X(x) = e^{-(x - \mu)} I_{\mu, \infty}(x) I am studying parametric statistical inference. One of the self study I have to find a sufficient, minimal and complete statistic for the \mu parameter of the following p.d.f.$$ f_X(x \mid \mu) = e^... 0 votes 1 answer 81 views ### Verifying whether$X$is a complete statistic The pmf of$X$is as follows:$X = -1 \rightarrow p(x)= \thetaX = 0 \rightarrow p(x)= \theta^2X = 1 \rightarrow p(x)= 1-\theta-\theta^2$I know that to show whether$X$is complete it is ... 2 votes 1 answer 107 views ### Problem on sufficient statistics Let the distribution of$X_1,X_2,...X_n$depend on two parameters$a, b$such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ... 1 vote 0 answers 80 views ### Prove that the Pitman estimator is itself complete sufficient It's a homework question, but I just have no idea about it ... Let be random variables according to a distribution having joint density ,where is a location parameter. Assume that there exists a ... 1 vote 1 answer 558 views ### Use the Lehmann-Scheffé theorem to deduce that$\overline{X}$is an UMVUE estimator for$\theta$Let$X_1,X_2,\ldots,X_n$be a random sample whose distribution is$X\sim\operatorname{Bernoulli}(\theta)$. (a) Prove that$\sum_{i=1}^n X_i$is complete. (b) Use the Lehmann-Scheffé to deduce that$\... 172 views

### Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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### Checking if a minimal sufficient statistic is complete

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[-\theta, 2\theta]$ with $\theta \in \mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete. I found ...
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### Poisson distribution and completeness, what happens when one point removed from parameter space?

Long time ago (early 1980's) my professor showed me a paper (I think it was Teachers' Corner or something similar) about the Poisson distribution and completeness. Showed that if only one point was ...
Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$ I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
### Why is $g(x)=0$ a.s. if $p(x | \theta)$ is a density and $\int_0^\infty g(x) p(x | \theta) dx = E[g(x)] = 0$ for all $\theta$?
I am trying to prove that a statistic $T$ is complete, where $T \sim \text{Gamma}(n, e^\theta)$. I am stuck at deducing that $g(T)$ must be zero almost surely, just by looking at the expectation:  E[...