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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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, ...
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Showing incompleteness of density
You observe a sample of 100 independent observations $X_i$ from a population with the density
$$
g(x)=C \sqrt{\lambda} \exp \left(-\lambda x^2-\lambda^2 x^4\right), \quad-\infty<x<\infty
$$
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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_\...
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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}\...
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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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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?
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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 ...
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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 ...
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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 ...
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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}...
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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 ---...
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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},...,...
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How is a sufficient complete estimator minimal sufficient [duplicate]
the question is quite theoretical
But I am finding it extremely hard to understand it without cramming the derivation ,although I understand the basic concept of it .
Can someone explain how a ...
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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 ...
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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 $g(T)$ ...
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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 $\...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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
$\...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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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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) = \...
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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:
$f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
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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 ...
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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^*=\...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 $\...
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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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