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.
78
questions
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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, ...
0
votes
1answer
32 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
22 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
54 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
28 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
51 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.
1
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2answers
63 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
47 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
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1answer
62 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
30 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
48 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
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1answer
368 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
44 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 ...
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0answers
44 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
93 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
233 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
31 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
69 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
36 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, ...
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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
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1answer
56 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
80 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
131 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
51 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
22 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
62 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 ...
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1answer
181 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
178 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
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1answer
106 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
71 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
80 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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0answers
57 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
281 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
154 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
676 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
138 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
70 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:
$$ E[...
4
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0answers
217 views
What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?
My textbook gives the following theorem for exponential families:
Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
1
vote
1answer
190 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
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0answers
465 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
145 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
111 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
537 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
2answers
827 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
106 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
330 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
783 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
55 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 $\beta>0$.
...
