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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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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1answer
15 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 ...
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1answer
43 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 ...
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1answer
47 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 ...
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1answer
50 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 $\...
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23 views
A few doubts on UMVUE uniqueness and strategies to find them
I am taking a course in mathematical statistics and I have a few doubts on some aspects of UMVUEs. The first question is: why are UMVUE, if they exist, unique?
I understand that they are unique if I ...
1
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1answer
90 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 = (\...
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1answer
68 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^...
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1answer
61 views
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 ...
2
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1answer
60 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 ...
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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 ...
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1answer
94 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 $\...
2
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1answer
126 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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2answers
216 views
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 ...
4
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1answer
70 views
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 ...
2
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1answer
34 views
Proving completeness of highest-order statistic using Leibnitz' Rule
Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$
I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
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0answers
28 views
Why is $g(x)=0$ a.s. if $p(x | \theta)$ is a density and $\int_0^\infty g(x) p(x | \theta) dx = E[g(x)] = 0$ for all $\theta$?
I am trying to prove that a statistic $T$ is complete, where $T \sim \text{Gamma}(n, e^\theta)$. I am stuck at deducing that $g(T)$ must be zero almost surely, just by looking at the expectation:
$$ E[...
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0answers
100 views
What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?
My textbook gives the following theorem for exponential families:
Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
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1answer
79 views
Minimal sufficient statistic whose dimension is less than dimension of parameter
Consider following example:
Suppose $ X\sim N(0, \sigma^2) $,
consider a random sample of size one from this population.
Clearly $X$ is sufficient statistic but $ |X| $ is minimal sufficient ...
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0answers
253 views
Complete and sufficient statistics of Laplace Distribution [duplicate]
Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,
$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
2
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1answer
93 views
Comparing variances of two unbiased estimators
This question is from a Ph.D Qualifying Exam for Mathematical Statistics. Main reference is Casella & Berger's Statistical Inference.
Let $W_1$ and $W_2$ be unbiased estimators of a parameter $\...
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0answers
103 views
Sufficiency and completeness of distribution
Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf
$p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\
...
4
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0answers
438 views
The relationship between UMVUE and complete sufficient statistic
Let $X_1,...X_n$ $U(-\theta , \theta)$
I want to find the UMVUE of $\theta$ if it is exists.
My answer is , there is no UMVUE in this case.
Because there is no complete sufficient statistic that ...
4
votes
1answer
350 views
Finding complete sufficient statistic
Let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such.
I know that $T=(X_{(1)}, X_{(n)} )$ is a ...
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0answers
67 views
Proving that X is not complete
So, the question is the following:
"Be X a single sample from $Beta(\theta,\theta )$, is X a sufficient statistic? Is it complete?"
So, since it's the entire sample, it's easy to say that it is ...
2
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0answers
254 views
Complete and Sufficient Statistics from Two Samples
Let $X_1,...,X_n$ be a random sample of size $n$ from $N(\epsilon,\sigma^2)$ and let $Y_1,...,Y_m$ be a random sample of size $m$ from $N(\eta,2\sigma^2)$.
Find a vector of complete and ...
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2answers
544 views
UMVUE for $\theta$ where $X \sim Unif\{1 ,\ldots, \theta\}$
Say we have $X \sim Unif\{1, \ldots , \theta\}$ and we want to find the uniformly minimum variance unbiased estimator for $\theta$.
My first assumption was $X_{(n)}$. Which I managed to show is ...
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1answer
44 views
Doubts regarding the Completeness of an estimator [duplicate]
Let $X_{1},...,X_{n}$ be random variable from the probability density function:
$f(x|\beta)=\frac{\beta^{\alpha}}{\Gamma{\alpha}} x^{\alpha-1}e^{-\beta x}$ where $\alpha$ is known and $\beta>0$.
...
3
votes
2answers
719 views
Best unbiased estimator for a location family
Mainly for pedagogical reasons, I'm considering the "simple" one dimensional model:
$$x=\theta+\epsilon$$
where $\epsilon$ has a known distribution $p$ that is independent of $\theta$. This ...
3
votes
1answer
378 views
Doubt regarding the function of complete statistic
If a statistic is complete, then any function of that statistic is complete. Is it true in general? For example, I know that for a gamma distribution $G(2,\theta)$, Sample mean will be a complete ...
3
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1answer
2k views
What is exponential family criterion to test the sufficiency and completeness of an estimator?
I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families:
Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...
0
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1answer
824 views
Complete Statistic in Gamma distribution
Let we have $n$ independent and identical random variables from gamma distribution with parameters $2$ and $\theta$, ie $G(2,\theta)$ where $n$ is greater than or equal to $2$. Check whether the ...
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0answers
250 views
Show that Sufficient statistic is complete
I have the following Gamma$(2,\theta)$ distribution:
$$f_\theta(x) = \theta^2x e^{-\theta x}\mathbb{1}_{[0,+\infty[}(x) $$
I am asked the two following questions:
...
4
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1answer
122 views
Comparing estimators of equal risk
I'm attending a course in mathematical statistics and it seems the lecturer tacitly assumes that given estimators $T_1,T_2 : \Omega \to \Lambda$ of a parameter $g : \Theta \to \Lambda$, a loss ...
4
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2answers
506 views
Complete sufficient statistic and unbiased estimator
I am now studying complete sufficient statistic. My question is: Is there any relationship between the existence of complete sufficient statistic and the existence of unbiased estimator?
I know that ...
4
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0answers
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When does $\forall p: E_p[f] = E_p[g]$ imply $f = g$?
Let's say
$$E_p[f(X)] = E_p[g(X)]$$
for all $p \in S$, where $S$ might be some parametric family of densities, for instance. Under which assumptions on $S$ does this imply $f = g$?
I am reading Efron ...
3
votes
1answer
376 views
Completeness of order statistics for normal setup
Suppose we have iid $X_1,\cdots, X_n\sim N(\mu,\sigma^2)$ where $\mu$ is unknown.
Let $X_{(1)}, X_{(2)},\cdots,X_{(n)}$be the order statistics.
Is the order statistics $\textbf{complete sufficient}$ ...
3
votes
1answer
553 views
UMVUE of location parameter (shifted exponential)
Let $X_1,...,X_n$ be a sample from a distribution with pdf, $f_X(x) = e^{-x + \theta}, x \geq \theta$. Let $x_0 \geq \theta$ be given. I'm trying to find the UMVUE of $f_X(x_0) = e^{-x_0 + \theta}$. I ...
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1answer
448 views
How to find sufficient complete statistic for the density $f(x\mid\theta)=e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$?
For the pdf $$f(x\mid\theta)=e^{-(x-\theta)}\exp(-e^{-(x-\theta)})\,,\qquad-\infty<x<\infty,-\infty<\theta<\infty$$, let $X_1,\cdots,X_n$ be i.i.d observations. How to find the sufficient ...
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3answers
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Is a minimal sufficient statistic also a complete statistic
I know that if a statistic is both sufficient and complete then it must also be minimal sufficient.
But on the other hand, could I say a minimal sufficient statistic must also be a complete statistic?
4
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1answer
878 views
What is the difference between complete statistics and complete family of distributions?
I fail to understand when we call a family of distribution is complete and when a statistic is complete. What is the difference between both?, Is there a relation between them? Please provide examples ...
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0answers
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How do I interpret the completeness in Statistics in practice? [duplicate]
I'm aware of the mathematical definition of complete statistics, but I'm pretty confused about what does the completeness really mean. It would be better to have a practical example from you guys.
...
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0answers
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Check that a statistic is complete
I have a question regarding completeness of a statistic. So the problem is:
$n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and ...
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0answers
354 views
In trying to prove a statistic is NOT complete, can the counter-example function be a function of the data and the parameters of the model?
Suppose that $X_1, \ldots, X_n$ are iid data from a family of distributions with parameter $\theta \in \Theta$ and that $T(\boldsymbol{X})$ is a sufficient statistic. Now suppose that we are trying to ...
3
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1answer
457 views
Is Complete Statistic Uncorrelated with Ancillary Statistic
By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be ...
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Jointly Complete Sufficient Statistics: Uniform(a, b)
Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that ...
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1answer
214 views
If $Y_1, \ldots, Y_n$ are iid from $Pois(\lambda)$, how to show that the sample mean is a complete sufficient statistic?
If $Y_1, \ldots, Y_n$ are iid from $Pois(\lambda)$, then I know that the sum of iid poisson random variables has the $Pois(n\lambda)$ distribution. However, I am trying to show that $\bar{Y}$ is a ...
0
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1answer
63 views
Consider $f_θ(x) = \exp(θ − x)\, \mathbb{I}_{x>θ}$ show that $\inf(X)$ is complete and sufficient
Consider a sequence of variable defined with the following density $$
f_θ(x) = \exp(θ − x)\ \mathbb{I}_{x>θ} $$
What is the ML estimate?
Show that some statistics of the indicator ...
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2answers
3k views
What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it?
In classical statistics, there is a definition that a statistic $T$ of a set of data $y_1, \ldots, y_n$ is defined to be complete for a parameter $\theta$ it is impossible to form an unbiased ...
5
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1answer
1k views
Basu's Theorem Proof
I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below:
Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
