1
vote
1answer
22 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
0
votes
0answers
43 views

Determine a distribution of a gaussian stochastic at different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
2
votes
0answers
37 views

paramter estimation (maximum likelihood) of a mixture density

I have this mixture distribution $f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma) $ where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution. I now have random ...
1
vote
1answer
76 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
0
votes
0answers
25 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
0
votes
0answers
16 views

Can we predict the positions of zero entries in a sparse vector given a model

I am wondering if we can predict the positions of the zero entries in a $n$-dimensional sparse vector $x$ given the linear model: $y=Ax$ where $y\in\mathbb{R}^m$ and the matrix ...
1
vote
1answer
64 views

Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
1
vote
1answer
119 views

Find the Method Moment Estimator of parameter $\theta$

Find the MME of parameter $\theta$ in the distribution with the density $f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for $x>1$ and $\theta >0$. So far I think I have a basic understanding of the ...
0
votes
0answers
68 views

How to simplify conditional probability of union of several events

I have an output binary scalar, $y∈B=[0,1]$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=[0,1]$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
0
votes
1answer
26 views

Estimator of a Random Variable

Given a random varable $Y$ where $$ f_Y(y) = \begin{cases}e^{-(y-k)} \quad x>k\\0\quad \text{otherwise}\end{cases} $$ Given $n$ observations of $Y$. Is the sample mean $\bar{Y}$ an unbiased ...
0
votes
0answers
49 views

Computing an estimator for a piecewise distribution?

Suppose I have a random variable $X$ that follows a distribution with a piecewise function $f(x|\theta)$. What is the correct way to compute an estimator $\theta$ for this function? Should the ...
1
vote
2answers
100 views

Let $(X_1…X_n)$ have an exp($\lambda$) distribution. Prove that $\frac{1}{\frac{1}{n}\sum{X_i}}$ is not a unbiased estimator of $\lambda$

the main problem is that i have no clue on calculating $E(\frac{1}{x})$ let $U = \frac{1}{\frac{1}{n}\sum{X_i}}$ then, $E(U) = n*E(\frac{1}{\sum{X_i}})$. I think that i'm supposed to calculate: ...
0
votes
1answer
59 views

is any upper bound for mean square error of an unbiased estimator?

There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of ...
1
vote
1answer
120 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
0
votes
1answer
193 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
0
votes
0answers
41 views

Assigning prior to $\gamma$ in composite power function $P(t) = max[\lambda t^{-\beta}, \gamma]$

I want to estimate the parameters $\lambda, \beta$ and $\gamma$ using a bayesian approach and an MCMC sampler. With the exception of $t$ all variables are random variables between $0$ and $1$. $t$ is ...
1
vote
0answers
23 views

ML Estimation for number of animals in a park. Hypothesis Testing.

A park of area $S=10 000 km^2$ was surveyed for bears, and out of $n$ disjoint regions of equal area $s=1km^2$, there were $n_k$ regions with $k=0,1,....,N$ bears. On each of these regions, the amount ...
14
votes
3answers
3k views

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
0
votes
1answer
279 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
79 views

Proving that the sum of Good-Turing estimators is $1$

I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step: $$\sum_j \theta[j] = \sum_r ...
0
votes
2answers
682 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: ...
0
votes
1answer
111 views

Proof of convergence of a sum of mean-consistent estimators

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these ...
2
votes
0answers
93 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...