All Questions
Tagged with sufficient-statistics complete-statistics
69
questions
2
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1
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35
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Prove that $T$ is a complete statistic and find a UMVUE for $p$
While preparing for my prelims, I came across this problem:
Let $X_1, X_2,\cdots, X_n$ be a sequence of Bernoulli trials, $n \geq 4.$ It is given that, $X_1,X_2,X_3 \stackrel{\text{i.i.d.}}{\sim} Ber(\...
- 21
1
vote
1
answer
43
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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-\...
- 345
2
votes
1
answer
134
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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)=(\...
- 31
5
votes
2
answers
282
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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 }\...
- 210
0
votes
1
answer
211
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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, ...
- 63
1
vote
1
answer
59
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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 ...
- 133
2
votes
1
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600
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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 ...
- 203
2
votes
1
answer
90
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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_\...
- 31
1
vote
0
answers
100
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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}\...
- 37
0
votes
1
answer
26
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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?
1
vote
0
answers
54
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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 ...
0
votes
1
answer
294
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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 ...
- 107
2
votes
0
answers
114
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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}...
- 254
6
votes
2
answers
123
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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 ---...
- 491
0
votes
0
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68
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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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