All Questions
Tagged with sufficient-statistics estimation
31
questions
2
votes
1
answer
35
views
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
2
votes
0
answers
134
views
Solving the Neyman-Scott problem via Conditional MLE
In section 2.4 of the book Essential Statistical Inference by Boos and Stefanski, the authors discuss the idea conditional likelihoods and illustrate their usefulness by describing how they can be ...
- 1,501
4
votes
1
answer
194
views
Rao-Blackwellisation using non-sufficient statistics
The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1")
Now, I do understand that the ...
- 236
4
votes
1
answer
259
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What are the "Dangers" of using "Non-Sufficient" Statistics?
I was reading one of the answers listed on this previous Stackoverflow question about the importance of sufficient statistics (Generalized Linear Models - What's special about the exponential ...
0
votes
0
answers
139
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Subscripts for Expectations and variances in for estimators [duplicate]
Is there any significance for subscripts to E and Var?
For example, the risk function of an estimator $\delta(\mathbf x)$ of $\theta$ in my book is:
$$
R(\theta,\delta)=E_\theta[L(\theta,\delta(\...
1
vote
1
answer
78
views
Did I correctly apply the factorisation theorem in this example?
Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}(x \in]\theta,\theta+1[)$ and the random variable $\mathbf{X}=(X_1,\ldots,X_n)$ are independently identically distributed ...
- 155
0
votes
0
answers
39
views
What's the maximum likelihood estimation of $\theta$ in this density? [duplicate]
Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$.
Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
- 155
6
votes
1
answer
1k
views
What is the score function of two parameters?
According to this wikipedia article, score is the derivative of the log-likelihood function. However, I don't understand what if we have two parameters? For example, the logarithm of pdf has the ...
3
votes
1
answer
288
views
Complete and Sufficient Statistic for Discrete Distribution
I have a single observation X from the following distribution:
$$𝑃(𝑋=−1)=\dfrac{𝑝}{3},𝑃(𝑋=0)=(1−𝑝),𝑃(𝑋=1)=\dfrac{2𝑝}{3}$$
I'm trying to find a complete and sufficient statistic for p based on ...
- 33
2
votes
1
answer
110
views
How to identify one-one correspondance in Sufficient Statistics?
The correct answer to the given question is (1),(3) and (4). I understood how 3 and 4 are correct but I could not understand how (1) is also a correct answer.
I know that here $\sum_i X_i$ is a ...
- 123
1
vote
1
answer
242
views
nonexistence of a sufficient statistic
Let $X_1,X_2,\dots,X_n$ be a random sample from a $\Gamma(\theta,\theta)$ distribution. Then
$$
\prod_{i=1}^n f(x_i;\theta) = \frac{1}{\Gamma(\theta)^n\theta^n}(\prod_{i=1}^n x_i)^{\theta-1}e^{-\frac{...
- 220
0
votes
1
answer
244
views
Sufficient statistics from exponential distributions with different means [closed]
If $X$ and $Y$ are independent exponential random variables with means $\theta$ and $2\theta$ respectively, then show that $X + 2Y$ is sufficient for $\theta$.
I know how to find sufficient ...
- 663
3
votes
2
answers
1k
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 ...
- 1,387
2
votes
1
answer
131
views
How to prove this Corollary regarding ratios of densities being sufficient
The following Corollary is used in "Theory of Point Estimation" by Lehmann to prove a theorem. However I'm unsure how to prove this Corollary (it's left as a problem, so proof is omitted). The ...
- 2,564
5
votes
1
answer
633
views
Jointly sufficient statistics of a multi-parameter exponential family
Let $f_X$ be a joint density function that comes from an $s$-parameter exponential family with sufficient statistics $(T_1, T_2, \dots, T_s)$ so that the density $f_X$ can be expressed as
$$f_{X|\...
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