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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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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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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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 ...
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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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48 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 $\...
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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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155 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 ...
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54 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 ...
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1answer
26 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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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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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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59 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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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}}...
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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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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 \\ ...
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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 ...
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1answer
265 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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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 ...
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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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463 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
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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 $\beta>0$. ...
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651 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 ...
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1answer
331 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 ...
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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 definition from casella and Berger about the exponential family sufficiency and completeness: Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...
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1answer
732 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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238 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: ...
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1answer
117 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 ...
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2answers
455 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 ...
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1answer
348 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 $\textbf{complete sufficient}$ ...
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1answer
483 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
368 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?
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775 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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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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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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296 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 ...
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1answer
425 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
186 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 ...
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60 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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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 ...
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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 ...
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394 views

Sufficient Statistic for non-exponential family distribution

Question: Let $X_1,X_2,\ldots,X_n$ be an iid sample from $N(\theta , 4 \theta^2 )$. I want to show that this model is not a member of the exponential family and to find a sufficient statistic for $\...
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Is there a way to visualize what “Minimal sufficiency” and “Completeness” of a statistic means? [duplicate]

As defined (for example, in Wikipedia): Completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of ...
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3answers
9k views

Meaning of completeness of a statistic? [duplicate]

From Wikipedia: The statistic $s$ is said to be complete for the distribution of $X$ if for every measurable function $g$ (which must be independent of parameter $θ$) the following implication ...
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Intuition Behind Completeness [duplicate]

The definition for completeness is that if a statistic $s(x)$ is complete, we have that for every measurable $g$, $$E_\theta(g(s(x))) = 0\,, \ \forall\,\theta\ \Rightarrow\ g(s) = 0 \text{ a.s.}$$ I'...
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Completeness of tests based on a monotone likelihood ratio family

In "Mathematical statistics: basic ideas and selected topics, Volume 1" by Bickle and Doksum, Theoreom 4.3.2. Suppose $\{P_\theta : \theta \in \Theta \},\Theta \subset \mathbb{R}$, is a monotone ...