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.
83
questions
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1answer
23 views
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 ...
0
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0answers
27 views
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 $g(T)$ ...
5
votes
1answer
160 views
Trying to make sense of claims regarding Rao-Blackwell and Lehmann-Scheffé for sufficient/complete statistics
I am currently trying to learn the two related concepts of the Rao-Blackwell theorem and the Lehmann-Scheffé theorem.
Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\...
0
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0answers
34 views
Complete sufficient statistic of non-identical distribution: $X_i \sim EXP(i\theta)$
Problem
Suppose that $X_1, \dots, X_n$ are independent $\mathrm{EXP}(i\theta)$ random variables. Find a complete sufficient statistic for $\theta$.
My Attempt
Since pdf of $x_i$ is
\begin{equation}
...
1
vote
2answers
53 views
If a statistic $T(X)=\Sigma_{i=1}^n X_i$ is sufficient does that imply the mean is also sufficient?
I've been working on some problems, the question asked me if the mean of a sample is a sufficient statistic for poisson distribution. I've already proved that $T(X)=\Sigma_{i=1}^n X_i$ is a sufficient ...
0
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0answers
17 views
Complete sufficient statistics for bivariate observations
We have observations $(X_i,Y_i), 1\le i\le n$ from a family of distributions $\mathcal F$, consisting of all absolutely continuous bivariate distributions. I wish to show that $(X_{(i)},Y_{\text{...
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0answers
16 views
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, ...
0
votes
1answer
36 views
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 ...
0
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1answer
61 views
Jointly complete and sufficient statistics for multivariate normal distribution
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 ...
2
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0answers
83 views
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 ...
1
vote
1answer
34 views
If a distribution contains a non-trivial unbiased statistic of zero, then this distribution does not have a complete statistic?
If a distribution contains a non-trivial unbiased statistic of zero, then this distribution does not have a complete statistic?
Here is what it means
Suppose we have a family of distribution
$\...
3
votes
1answer
75 views
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.
2
votes
2answers
77 views
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 ...
0
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1answer
54 views
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 ...
1
vote
1answer
73 views
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 ...
2
votes
1answer
43 views
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 ...
0
votes
1answer
61 views
Completeness of a statistic in a truncated distribution
Suppose a random sample $x_1,\dots, x_n$ (i.i.d.) from a random variable $X$ defined over $(\Omega,\mathcal{F},P)$ whose probability density function is $f(x_1,\dots, x_n;\theta)$ and $T(x_1,\dots, ...
7
votes
1answer
517 views
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 ...
2
votes
1answer
65 views
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 ...
0
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0answers
69 views
Complete Statistics of Negative Exponential Distribtion when both location and scale parameter are unknown
Let, $X_1, \ldots, X_n$ be iid having a negative exponential distribution with common pdf $$\dfrac{1}{\sigma} \exp\{ -(x-\mu)/\sigma \} I(x>\mu); \mu \in R, \sigma \in R^+$$
where $\mu$ and $\sigma$...
0
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1answer
98 views
Checking for the completeness for the M.S of $f(x) = \frac12 exp(-|x-\theta|)$
The minimum sufficient statistics for $f(x) = \frac12 exp(-|x-\theta|)$ for $-\infty < \theta < +\infty$ is $ T(X) = \{X_{(1)},X_{(n)} \}$. I want to show that the above is complete.
$f(x) = \...
0
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1answer
382 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
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0answers
33 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
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1answer
87 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^*=\...
30
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
57 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 $\...
3
votes
1answer
152 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
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1answer
148 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
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1answer
57 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
1answer
27 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
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1answer
73 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
52 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 ...
4
votes
1answer
233 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
1answer
208 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
122 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
75 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
87 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
63 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
1answer
354 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
157 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
807 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
165 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
77 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:
$$ E[...
4
votes
0answers
258 views
What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?
My textbook gives the following theorem for exponential families:
Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
1
vote
1answer
244 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
540 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}}...
3
votes
1answer
166 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
114 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
votes
0answers
542 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 ...
