All Questions
Tagged with sufficient-statistics maximum-likelihood
27
questions
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
1
vote
1
answer
101
views
MLE of parameters for a difference of two Exponential IID
Suppose I have $X_1 \sim Exp(\theta_1)$ and $X_2\sim Exp(\theta_2)$. Then it is not difficult to show that $Y = X_1 - X_2$ will have density:
$f_Y(y) = \frac{1}{\theta_1 + \theta_2}e^{-y/\theta_1}\...
- 87
0
votes
0
answers
52
views
Rao Cramèr Lower Bound problem
Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
- 31
3
votes
1
answer
115
views
Sufficiency for Truncated Geometric
Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics:
Let
$$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
0
votes
0
answers
39
views
What's the maximum likelihood estimation of $\theta$ in this density? [duplicate]
Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$.
Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
- 155
1
vote
0
answers
115
views
MLE and Minimal Sufficiency of Parameters in a Piecewise Random Variable [closed]
Problem Setting:
$X_i$ is i.i.d. from a piecewise distribution which is
$$
f_{\theta_1, \theta_2}(x) = \frac{1}{\theta_1+\theta_2}e^{-\frac{x}{\theta_1}}I_{[x>0]} + \frac{1}{\theta_1+\theta_2}e^{\...
- 265
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 ...
1
vote
0
answers
273
views
Why do we need to emphasize sufficient statistics in generalized linear models? [closed]
In generalized linear models, $$p(y;\eta)=b(y)exp(\eta^TT(y)-a(\eta)) \\ \eta=\theta^T x$$we assume $x$ is the input variable and $y$ is the output and our target is to get the distribution of input ...
2
votes
1
answer
765
views
Is a maximum likelihood estimator in an exponential family always sufficient?
An exponential family (under natural parameterization) is such that $p(X|\eta)=h(X)\exp\{\eta^\top T(X)-A(\eta)\}$, where $X$ is the data, $\eta$ is the natural parameter, and $h,T,A$ are some ...
- 31
0
votes
0
answers
59
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Showing Sufficiency of a Statistic [duplicate]
Suppose that $X_1, X_2, \cdot \cdot \cdot \ , X_n$ is a random sample from a continuous distribution with pdf $f_X(x;\theta) = \theta x^{\theta-1}$, for $\ 0\leq x \leq 1$. Show that $W=\prod_{i=1}^n ...
- 153
0
votes
0
answers
512
views
Sufficient Statistic and MLE
Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient
statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE
for $\theta$ is $(1-\frac{1}{n}T)^2$.
I am having a lot of ...
- 571
4
votes
1
answer
4k
views
Invariance property
I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators.
As far as I know,
Invariance property of ...
- 71
7
votes
3
answers
300
views
Likelihood function when $X\sim U(0,\theta)$
Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(0,\theta)$. Derive the likelihood function given the sample $x_1, ..., x_n$.
Answer
The likelihood function is:
\begin{...
- 571
5
votes
1
answer
2k
views
Sufficient statistic when $X\sim U(\theta,2 \theta)$
Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(\theta,2 \theta)$. Find a sufficient statistic for $\theta$, and compute $\widehat{\theta}_{MLE}$.
Answer
The joint ...
- 571
1
vote
1
answer
942
views
Maximum likelihood estimation for $\alpha$ with population pdf $f(x;\alpha)=\frac{2}{\alpha^2}(\alpha-x)I_{(0,\alpha)}(x)$
A sample of size two is taken from the distribution $ f(x;\alpha)=\frac{2}{\alpha^2}(\alpha-x)I_{(0,\alpha)}(x)$. We need to find the maximum likelihood estimator for $\alpha$.
The likelihood function ...
- 605