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If I have this linear regression equation: $$y=X\beta+\epsilon$$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ... 0answers 43 views ### Determine a distribution of a gaussian stochastic at different time I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of y=x(t_1)-x(t_2) is also a Gaussian stochastic with ... 0answers 37 views ### paramter estimation (maximum likelihood) of a mixture density I have this mixture distribution f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma)  where \mathcal{LN}(\mu,\sigma) is a lognormal distribution. I now have random ... 1answer 76 views ### Order related to Empirical distribution function and Normal distribution Let X_1,\dots,X_n are i.i.d with distribution function F. Let \hat F_n be its empirical distribution function, i.e.,$$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$where 1_A(x) is the ... 0answers 25 views ### Monte carlo formula to compute the approximation of variance of MLE In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ... 0answers 16 views ### Can we predict the positions of zero entries in a sparse vector given a model I am wondering if we can predict the positions of the zero entries in a n-dimensional sparse vector x given the linear model: y=Ax where y\in\mathbb{R}^m and the matrix ... 1answer 64 views ### Finding the MLE of pareto dist., and trouble interpreting \prod notation properly. I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample X_1,...,X_n. Find the MLE of ... 1answer 119 views ### Find the Method Moment Estimator of parameter \theta Find the MME of parameter \theta in the distribution with the density f(x,\theta)=(\theta +1)x^{-(\theta+2)}, for x>1 and \theta >0. So far I think I have a basic understanding of the ... 0answers 68 views ### How to simplify conditional probability of union of several events I have an output binary scalar, y∈B=[0,1], and an input binary vector x=[x_1, x_2,…x_M] where x_i∈B=[0,1]. I know that the output P(y)=1 depends entirely on the input x. Thus, I want to ... 1answer 26 views ### Estimator of a Random Variable Given a random varable Y where$$ 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 ... 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 ... 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: ... 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 ... 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 ... 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 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 ... 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 ... 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 ...
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 ...