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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questions
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Finding minimum variance unbiased estimator when Cramer Rao theorem doesn't apply
Let $r$ be the number of successes in $n$ Bernoulli trails with the unknown probability $p$ of success. Obtain the minimum variance unbiased estimator of $p-p^2$.
My work:
$r\sim Binomial(n,p)$
$T(r)=...
0
votes
1answer
57 views
Find best unbiased estimator for $\theta$ when $X_i\sim U(-\theta,\theta)$
I am having an issue finding a best unbiased estimator for $\theta$. Any help is appreciated.
Let $X_1, ..., X_n$ be a random sample from a population with pdf:
$f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
0
votes
0answers
29 views
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 ...
1
vote
1answer
50 views
Gamma distribution: ratio of 2 CSS not containing $\beta$
Let $X_1,...,X_n$ be iid and follow $Gamma(\alpha, \beta)$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $\overline{X}$ and $X^*=\...
0
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0answers
23 views
UMVUE of functions of parameters from 2 normal samples
So the problem I have asks to find the UMVUE of $\sigma^2$ and of $\left( \epsilon - \eta \right)^2$, where $\sigma$,$\epsilon$, and $\eta$ are parameters of normal distributions as described here, ...
29
votes
2answers
3k views
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 ...
1
vote
1answer
51 views
Statistics Theory Question
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 $\...
2
votes
1answer
45 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
1answer
93 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 ...
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1answer
38 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
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1answer
20 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
1answer
49 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
1answer
51 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 ...
2
votes
1answer
89 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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0answers
35 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 ...
2
votes
1answer
119 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
1answer
80 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
1answer
68 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
votes
1answer
67 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
0answers
47 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 ...
0
votes
1answer
163 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
votes
1answer
141 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 $b(\theta)$...
3
votes
2answers
353 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 ...
5
votes
1answer
102 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
votes
1answer
39 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 ...
0
votes
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:
$$ E[...
3
votes
0answers
134 views
What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?
My textbook gives the following theorem for exponential families:
Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
0
votes
1answer
106 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 ...
1
vote
0answers
321 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
votes
1answer
115 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 $\...
2
votes
0answers
106 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 \\
...
5
votes
0answers
532 views
The relationship between UMVUE and complete sufficient statistic [duplicate]
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
489 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 ...
1
vote
0answers
77 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
votes
0answers
294 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 ...
1
vote
2answers
629 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 ...
0
votes
1answer
47 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
786 views
Best unbiased estimator for a location family
Mainly for pedagogical reasons, I'm considering the "simple" one dimensional model:
$$x=\theta+\epsilon$$
where $\epsilon$ has a known distribution $p$ that is independent of $\theta$. This ...
3
votes
1answer
451 views
Doubt regarding the function of complete statistic
If a statistic is complete, then any function of that statistic is complete. Is it true in general? For example, I know that for a gamma distribution $G(2,\theta)$, Sample mean will be a complete ...
4
votes
1answer
3k views
What is exponential family criterion to test the sufficiency and completeness of an estimator?
I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families:
Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...
0
votes
1answer
992 views
Complete Statistic in Gamma distribution
Let we have $n$ independent and identical random variables from gamma distribution with parameters $2$ and $\theta$, ie $G(2,\theta)$ where $n$ is greater than or equal to $2$. Check whether the ...
0
votes
0answers
257 views
Show that Sufficient statistic is complete
I have the following Gamma$(2,\theta)$ distribution:
$$f_\theta(x) = \theta^2x e^{-\theta x}\mathbb{1}_{[0,+\infty[}(x) $$
I am asked the two following questions:
...
5
votes
1answer
130 views
Comparing estimators of equal risk
I'm attending a course in mathematical statistics and it seems the lecturer tacitly assumes that given estimators $T_1,T_2 : \Omega \to \Lambda$ of a parameter $g : \Theta \to \Lambda$, a loss ...
4
votes
2answers
608 views
Complete sufficient statistic and unbiased estimator
I am now studying complete sufficient statistic. My question is: Is there any relationship between the existence of complete sufficient statistic and the existence of unbiased estimator?
I know that ...
4
votes
0answers
61 views
When does $\forall p: E_p[f] = E_p[g]$ imply $f = g$?
Let's say
$$E_p[f(X)] = E_p[g(X)]$$
for all $p \in S$, where $S$ might be some parametric family of densities, for instance. Under which assumptions on $S$ does this imply $f = g$?
I am reading Efron ...
3
votes
1answer
456 views
Completeness of order statistics for normal setup
Suppose we have iid $X_1,\cdots, X_n\sim N(\mu,\sigma^2)$ where $\mu$ is unknown.
Let $X_{(1)}, X_{(2)},\cdots,X_{(n)}$be the order statistics.
Is the order statistics $\textbf{complete sufficient}$ ...
3
votes
1answer
640 views
UMVUE of location parameter (shifted exponential)
Let $X_1,...,X_n$ be a sample from a distribution with pdf, $f_X(x) = e^{-x + \theta}, x \geq \theta$. Let $x_0 \geq \theta$ be given. I'm trying to find the UMVUE of $f_X(x_0) = e^{-x_0 + \theta}$. I ...
2
votes
1answer
613 views
How to find sufficient complete statistic for the density $f(x\mid\theta)=e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$?
For the pdf $$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 ...
4
votes
3answers
2k views
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
votes
1answer
1k 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 ...
