# Tagged Questions

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component.

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### Properties of an MLE based on likelihood constructed from both PDF and CDF

For continuous RV the likelihood function is (typically) given by a product of PDFs, i.e. $$L(\theta; x_1,x_2, ..., x_n) = \prod_{i=1}^n f(x_i\mid \theta)$$ However, in survival analysis with ...
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### A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\...
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### Estimate number of distinct items

I have a large array of $n$ integers, some of which may be repeated, and I want to estimate how many distinct integers are in the array. Say the number of distinct integers is $N$. I can sample with ...
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### Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
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### Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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### Estimation of Linear Projection

Given a linear system: $Y=AX+W$ Where: $X$ is the input signal of size $N \times K$ $Y$ is the output signal of size $M \times K$ $A$ is a projection of size $M\times N$; with $M >> N$ $W$ ...
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### Polluting an image with Gaussian anisotropic noise and estimate the covariance matrix

Assume that we have a $d\times d$ grey-scale image represented as a vector $$\mathbf{x}=(x_1,\ldots,x_D)^T\in[0,255]^D,$$ where $D=d\times d$. We would like to import some noise concerning the ...
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### 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 ...
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### Higher Order Estimation Errors

I well know estimation measure is the so called minimum mean square error (MMSE) defined as: \begin{align} E[|W-\hat{W}(V)|^2] \end{align} where $W$ is a random variable (that we want to estimate) and ...
I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, $i=1(1)n$...