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.

Filter by
Sorted by
Tagged with
1
vote
0answers
11 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
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
votes
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(ยต,ฮฃ). *Show that the sample mean xฬ„ and Sample covariance matrix S are jointly complete and sufficient ...
2
votes
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
vote
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
votes
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
vote
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: $$๐‘ƒ(๐‘‹=โˆ’1)=\dfrac{๐‘}{3},๐‘ƒ(๐‘‹=0)=(1โˆ’๐‘),๐‘ƒ(๐‘‹=1)=\dfrac{2๐‘}{3}$$ 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
votes
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 ...
0
votes
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
votes
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
votes
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
votes
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
votes
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, ...
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
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
vote
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
votes
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
vote
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 ...
4
votes
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
votes
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 ...
1
vote
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-Scheffรฉ 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
votes
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
vote
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
votes
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 $\beta>0$. ...