Questions tagged [estimation]

Question 1

Let $X$ be a $d$-dimensional random vector drawn from a Gaussian mixture $$ X \sim \sum_{k=1}^K \pi_k \, \mathcal{N}_d(\mu_k, \Sigma_k), $$ and let $$ Y = X + N, \quad N \sim \mathcal{N}_d(0, \Sigma_N)...

Question 2

I am studying random signals and noise (a course for EE students, but mathematical and formal), and have a question about the definition of an estimator (in the context of estimating a random variable ...

Question 3

Theorem (Cramér-Rao inequality). Consider a sample from a parametric model satisfying regularity conditions. Let $\theta^*$ be an unbiased estimator of $\tau(\theta)$. Then for any $\theta \in \Theta$,...

Question 4

Consider a sequence of i.i.d. random variables $X_1,\, \dots,\, X_n$ whose mean is denoted as $x_0$ and variance $\sigma^2 < \infty$. From the Strong Law of Large Numbers, the empirical mean $\bar ...

Question 5

I've always learned than when determining minimum sample size for a given margin of error, you should use a prior estimate of $\hat{p}$ if one is available, and the worst case scenario of assuming $\...

Question 6

Let $\ell\ge 2$ be a fixed integer. For each $k\ge1$ define the affine map $$ \theta_{k}(s)=\Bigl(\frac{k}{\ell}+1\Bigr)s-\frac{1}{\ell} =\frac{k+\ell}{\ell}s-\frac{1}{\ell}. $$ Question. Does there ...

Question 7

Question is in the title. Given that $\delta:=\delta(\mathbf X_n)$ is MVUE (minimum variance unbiased estimator) of a scalar parameter $\theta$, we are asked to show that for all natural numbers $k$, $...

Question 8

I am not a mathematician, just a curious computer science student. I came up with the following problem while thinking about sampling, and I’m sure there must be related questions out there, but I ...

Question 9

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Suppose $(u_n)$ is a sequence of functions satisfying uniform a priori Schauder estimates: \begin{equation} |u_n|_{C^{2,\alpha}(\Omega)} \leq C \...

Question 10

This is something I have been trying to understand. Consider the following local level linear state space model: Observation equation: $y_t = \mu_t + \epsilon_t$ where $\epsilon_t \sim N(0, \sigma_\...

Question 11

How to solve the diophantine equation $$(a^3 - a - 1)b = c^3 - c - 1$$ for integers $a,b,c>1$ ? Is the expected number of solutions ,denoted $f(c)$, for $a_n$ such that $(a_n^3 - a_n - 1)b_n = c_n^...

Question 12

Does it make sense to study statistical models in which the maximum likelihood estimator (MLE) $ \hat{\theta}_n $ exists only for infinitely many $ n $, but not necessarily for all $ n $? Suppose, for ...

Question 13

Let $0<r<1$ be a real number. Suppose $f$ is a continuous function on $[0,1]$, and let $$B_n(f)=\sum_{k=0}^nf\left(\frac kn\right)\binom{n}{k}x^k(1-x)^{n-k}$$ denote the Bernstein polynomial of ...

Question 14

Assume we have $f\in C^\infty(\mathbb{R})$ with $f(x)=0$ for any $x\leq 0$ and $f(x)=1$ for any $x\geq 1$. What's the best possible bound for $|f''|$? I know from the lecture notes that we can choose $...

Question 15

Let $X_1, X_2,...,X_n$ be $iid$ continuous uniform $\mathcal{U} (0,\theta)$ and let $T=Max(X_i)$ Show that the family of distributions of T is complete. Step I: Find the CDF (using independence ...

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Questions tagged [estimation]

Reference for the expected MSE of the optimal estimator in a Gaussian mixture channel

definition of estimator in random variable estimation

Uniqueness of the function $\tau(\theta)$ in the Cramér-Rao inequality

Transfer of strong consistency

Using $\hat{p}$ or $0.5$ when determining minimum sample size.

Uniform bound for a family of integrals with affine rescaling inside the integrand

Show that powers of an MVUE is an MVUE

How many samples until the percentile estimate stops wobbling?

Existence of a subsequence that converges uniformly

Why is the Kalman filter so popular?

Diophantine $(a^3 - a - 1)b = c^3 - c - 1$

Does it make sense to consider situations where the MLE exists only for infinitely many $n$?

Error estimation between an analytic function $f$ and its Bernstein polynomial

Bounds for derivatives of smooth function

Proving completeness for a statistic [duplicate]

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