# 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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### using MI in medical paper [duplicate]

Hi there I don't know much about statics so I'd like to pick up your minds about the result of this paper https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(21)01431-8/fulltext#...
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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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### How to prove or disprove that a complete sufficient statistic exists?

We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
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### Completeness of normal sample values

I have a small query. We know that sample values are always sufficient. Can we say the same for completeness property? Let us say I have $X_1,X_2,...,X_n$ following $N(0, \sigma^2)$. Hence, if we take ...
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### Jointly complete and sufficient statistics for multivariate normal distribution [duplicate]

Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ). *Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient ...
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### UMVUE of two-parameter exponential family distribution

Suppose $\{X_{i}\}_{i=1}^n\overset{i.i.d}{\sim}X$, where $X$ has density $$f_{X}(x)=\frac{1}{b}\exp\left\{\frac{x-a}{b}\right\},x>a$$ What is the UMVUE of $\mathbb{P}(X_1<u)$? Here is what I've ...
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### Whether the minimal sufficient statistic is complete for a translated exponential distribution

Let $X_1, X_2..., X_n$ follows iid negative exponential distribution with pdf $$f(x) = \frac{1}{\theta^2} \: e^{-\frac{(x-\theta)}{\theta^2}} \: \: I_{(x>\theta)}$$ I have to show whether the ...
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### Reparametrization and its effect on sufficient/complete/minimal statistics

Suppose $X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2), X_3 \sim Pois(\lambda_1+\lambda_2)$. Separately I can find a sufficient, complete and minimal statistic for each of them. But considering ...
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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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### 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 ...
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### Verifying whether $X$ is a complete statistic

The pmf of $X$ is as follows: $X = -1 \rightarrow p(x)= \theta$ $X = 0 \rightarrow p(x)= \theta^2$ $X = 1 \rightarrow p(x)= 1-\theta-\theta^2$ I know that to show whether $X$ is complete it is ...
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### 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 ...