Tagged Questions
1
vote
1answer
57 views
Maximum likelihood estimators, hypergeometric and binomial
I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
0
votes
0answers
20 views
Variability in estimations over a non-ergodic/non-regular Markov process
Imagine we have a non-ergodic/non-regular Markov Process with with $n$ states.
Among these $n$ states, there are $k$ absorbing states.
For each of the $n-k$ non-absorbing states, it is not possible ...
1
vote
0answers
30 views
Cramer-Rao bound for $\chi^2$ distribution parameter estimates.
I've stuck in unpleasant problem with noncentral $\chi^2$ distribution.
I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
0
votes
0answers
47 views
An estimate of the minimum and the second-smallest element in a set.
Suppose I have $N$ independent random variable $(x_1,x_2,x_3,\ldots,x_N)$ with pdf's $(w_1(x_1,z_1),w_2(x_2,z_2),\ldots,w_N(x_N,z_N))$ parameterizes by $N$ non random unknown parameters $\{z\}_1^N$. ...
2
votes
0answers
45 views
Cramer-Rao bounds for noncentral $\chi^2$ distribution parameters estimates
In many practical problems I came across the necessity to estimate the parameters of noncentral $\chi^2$ distribution (for example the degrees of freedom or non-centrality parameter etc.). Found some ...
0
votes
1answer
74 views
Empirical Bayes estimator for a Beta-Binomial parameters
Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
1
vote
1answer
80 views
The distribution when combining two samples together?
Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
0
votes
1answer
112 views
how can I get minimum error probability for this decision problem?
I have the decision problem for 4 hypotheses as follows:
$$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$
where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$
In ...
1
vote
1answer
52 views
biasedness/unbiasedness of an MLE.
To show whether an MLE I just found is biased/unbiased, would I need to find the expectation of the answer? Plus would I do this by integrating $\text{MLE} \cdot \text{pdf}$.
My MLE is $ ...
0
votes
2answers
338 views
Exponential Distribution Maximum Likelihood
I found the following question in a past exam paper and I would like to ask how to solve it as I can't find anything in the notes related to it:
...
1
vote
0answers
99 views
Parameter estimation for a distribution by minimizing its conditional entropy
Let $X$ be a discrete random variable with Laplacian distribution with mean $0$ and scale $\lambda$, as
$$
p(X) = \frac{1}{2\lambda} \exp\left(-\frac{|x|}{2\lambda}\right), \\
X \in ...
1
vote
2answers
60 views
Filter to obtain MMSE of data from Gaussian vector
Data sampled at two time instances giving bivariate Gaussian vector $X=(X_1,X_2)^T$ with
$f(x_1,x_2)=\exp(-(x_1^2+1.8x_1x_2+x_2^2)/0.38)/2\pi \sqrt{0.19}$
Data measured in noisy environment with ...
7
votes
1answer
1k views
Minimum variance unbiased estimator for scale parameter of a certain gamma distribution
Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f.,
$$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of ...
2
votes
2answers
428 views
Some expectation values for a Gamma distribution
Assuming I have a Gamma distributed random Variable $x \sim Gamma( \alpha, \beta )$. Now I like to have the following two expectation values (integrals):
$E \left[ x \ln x \right]$
$E \left[ \ln ...
