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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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FInding a complete and sufficient statistic

I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class: Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
Harry Lofi's user avatar
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Why are complete statistics named "complete"?

I get why sufficient statistics are named "sufficient", but what about "complete" statistics? I have this definition from F.J. Samaniego, Stochastic Modelling and Statistical ...
Alexandre Huat's user avatar
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Structural (causal) interpretation of the completeness condition

Consider two random variables $X,Y$. We say the joint distribution of $P(X,Y)$ is complete w.r.t. $X$ if the following condition holds: For all $y$, $E\{g(x)|y\}=0$ almost everywhere if and only $g(x)...
Mingzhou Liu's user avatar
2 votes
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134 views

Completeness of Gamma family

Let $X_1,...,X_n$ has a Gamma$(\alpha,\alpha)$ distribution. Find the minimal sufficient statistics. Is this a complete family? My attempt: I found the Minimal sufficient statistics is $T(x)=(\...
Cyno Benette's user avatar
5 votes
1 answer
430 views

Show that the sample Mean is not complete

Suppose that $X_1, ..., X_n$ is a sample from a $\mathcal{N}(-\frac12 \sigma^2, \sigma^2)$ density. Show that the statistic $\bar{X}$ = $n^{-1} \sum_{i=1}^n X_i$ is NOT complete. I am struggling to ...
Stats_Rock's user avatar
5 votes
2 answers
269 views

Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?

I can't understand the paragraph in Completeness (statistics) - Wikipedia: We have an identifiable model space parameterised by $\theta$, and a statistic $T$. Then consider the map $f:p_{\theta }\...
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When is sufficiency and completeness of a statistic preserved? [duplicate]

This question has been asked in math stack but no one has replied. I have been given these definitions in my statistical inference class: Let $(X_1,...,X_n)$ be a simple random sampling of $X\...
José's user avatar
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Complete Statistic for a family with finite r-th moment

Consider the family of all continuous distributions with finite $r$-th moment (where $r \geq 1$ is a given integer). We denote this family as, $$\mathscr{P}_r=\left\{f:f \ \text{is a pdf and} \int|x|^...
user671269's user avatar
0 votes
1 answer
210 views

Show minimal sufficient statistic is not complete in normal distribution

Let $Z_i$ for $1 \leq i \leq n$ be a sample from the $N(ap, bp(1-p))$ density, where $a \gt 0, b \gt 0$ are known but $p \in (0,1)$ is an unknown parameter. I have shown that $T = (\sum^n_{i = 1} Z_i, ...
Oscar24680's user avatar
2 votes
1 answer
135 views

Showing incompleteness of density

You observe a sample of 100 independent observations $X_i$ from a population with the density $$ g(x)=C \sqrt{\lambda} \exp \left(-\lambda x^2-\lambda^2 x^4\right), \quad-\infty<x<\infty $$ ...
Stats_Rock's user avatar
1 vote
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Proving Incompleteness of joint sufficient statistic

Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $ -\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
Stats_Rock's user avatar
11 votes
2 answers
658 views

How to show that $\{N(\theta,1):\theta \in \Omega\}$ is not a complete family of distributions when $\Omega$ is finite?

Consider the $\{N(\theta,1):\theta \in \Omega\}$ family of distributions where $\Omega=\{-1,0,1\}$. I am trying to show that this is not a complete family. That is, if $X\sim N(\theta,1)$, I need to ...
StubbornAtom's user avatar
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UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta) $ where $\theta \in[1, \infty)=\Theta$

Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
Debarghya Jana's user avatar
2 votes
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599 views

Verifying the statistics are complete and sufficient for two parameter Pareto distribution

Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$ $\textbf{(i)}$ Show that ...
Aleph Alpha's user avatar
1 vote
1 answer
409 views

Show that the minimal sufficient statistics for the shifted exponential is complete for $n = 2$

If we had $Y_i$, $i = 1, 2, ..., n$ are $iid$ and have the density $$f(y) = \lambda e^{-\lambda(y - \mu)} I_{y > \mu} , ~~~y >0$$ where $\lambda >0,~ \mu >0$ are unknown parameters. I was ...
Stats_Rock's user avatar
2 votes
1 answer
90 views

Sufficiency and completeness of truncated distribution

[From Theory of Point Estimation (Lehmann and Casella, 1999, Exercise 6.37)] Let $P=\{P_\theta:\theta \in \Theta\}$ be a family of probability distributions and assume that $P_\theta$ has pdf $p_\...
WinnieXi's user avatar
1 vote
0 answers
99 views

Finding UMVUE of a parameter in form of probability of discrete random variables

We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$. Their pmf's are: $f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$ $f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
AlgoManiac's user avatar
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1 answer
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How does one test the efficiency and completeness of an estimator using monte-carlo simulation?

How does one test the efficiency and completeness of an estimator using monte-carlo simulation? In particular, I want to use-montecarlo simuation to answer. Maybe the better question is how does one ...
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1 answer
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sufficient, minimal, complete

Are all complete statistics functions of each other? For example if I have T and S complete statistics Can you always write T in terms of S and S in terms of T?
statistic-user's user avatar
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Does this distribution belong to the exponential family? [duplicate]

I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
Yeison Augusto Quiceno Duran's user avatar
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175 views

Complete statistic for discrete distribution

Let $X$ be a discrete random variable with the probability mass function $$P(X=-1)=\theta,\ P(X=0)=1-2\theta,\ P(X=1)=\theta$$ I'm trying to find a complete statistic for $\theta$ based on the single ...
Alex He's user avatar
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UMVUE for P(X > k) in exponential distribution [duplicate]

I have to find UMVUE for $exp(-k*a)$ where X ~ Exponential(a); k is a positive real number. I tried it using Lehmann-Scheffe theorem. Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
Kcd's user avatar
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What is the space that a class of probability distributions spans when T is a complete sufficient statistic?

There are a few good posts/notes (see here, and here) giving high level geometric intuition of a complete statistic ($E_{T}[g(T); \theta] = 0 \Rightarrow P(g(T)=0; \theta) = 1 \text{ almost everywhere}...
Morris Greenberg's user avatar
6 votes
2 answers
123 views

Are These Conjectures Regarding Sufficient Statistics True?

I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true ---...
Shang Zhang's user avatar
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68 views

Sufficient statistic and complete sufficient statistic [duplicate]

I'm trying self-study some inference and now I'm trying to understand how to solve some problems on this topic but I found this basic problem that I'm not being able to solve. Problem: Let $X_{1},...,...
user1trill's user avatar
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1 answer
123 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 ...
legon's user avatar
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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 $g(T)$ ...
The Pointer's user avatar
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6 votes
1 answer
574 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 $\...
The Pointer's user avatar
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1 vote
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133 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 ...
Hijaw's user avatar
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0 answers
221 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, ...
Martund's user avatar
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1 answer
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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 ...
userNoOne's user avatar
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1 answer
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Jointly complete and sufficient statistics for multivariate normal distribution [duplicate]

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 ...
Hamed Said's user avatar
2 votes
0 answers
628 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 ...
Tan's user avatar
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2 votes
1 answer
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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 $\...
Tan's user avatar
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3 votes
1 answer
609 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.
Tan's user avatar
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2 votes
2 answers
382 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 ...
Javier Moreno Sepena's user avatar
1 vote
1 answer
427 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 ...
Chesso's user avatar
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2 votes
1 answer
625 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 ...
Remy's user avatar
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3 votes
1 answer
288 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 ...
zet5000's user avatar
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1 vote
1 answer
450 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, ...
BelwarDissengulp's user avatar
7 votes
1 answer
2k 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 ...
n1234's user avatar
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2 votes
1 answer
282 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 ...
Maverick Meerkat's user avatar
0 votes
1 answer
157 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) = \...
user1916067's user avatar
0 votes
1 answer
2k 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: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
Student's user avatar
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0 answers
50 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 ...
Michael Devin Smith's user avatar
1 vote
1 answer
138 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^*=\...
Michael Devin Smith's user avatar
30 votes
2 answers
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 ...
pineapple's user avatar
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1 vote
1 answer
72 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 $\...
failedstatistician's user avatar
3 votes
1 answer
406 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 ...
Maverick Meerkat's user avatar
2 votes
1 answer
348 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 ...
Ron Snow's user avatar
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