# Questions tagged [estimation-theory]

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### A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture

[This question is looking at the paper Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/...
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### Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
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### sample complexity of hypothesis testing with non-uniform prior

Given two hypothesis $$\mathcal{H}_0:\; x_i\underset{iid}{\sim} p_0(x), i=1,\cdots,n\\ \mathcal{H}_1:\; x_i\underset{iid}{\sim} p_1(x), i=1,\cdots,n$$ with priors $p$ and $1-p$ ($p<1/2$) ...
33 views

### Maximum likelihood estimator for power law with negative exponent

Background I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration): Now I ...
1 vote
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### Correlating two matrices $A,B$ with stochastic dependency structure imposed by cross-validation

Consider a labelled data set $$D = \{(x_1, y_1),...,(x_n, y_n)\}$$ on which we want to evaluate a machine learning algorithm using $k$-fold cross validation with $m$ different random seeds. This ...
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### Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
84 views

### Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
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### Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
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### Optimal linear measurement operator

Let $x\in R^n$ be an unknown vector. Suppose I am allowed to choose any $A\in R^{m\times n}$, under the constraint that each row of $A$ has $\ell_2$ norm at most $1$. Then I carry out a "measurement", ...
472 views

### Functional Taylor expansion for differential entropy

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is $h(f)=-\int f(x)\log f(x) dx$. In the literature of differential entropy estimation, ...
148 views

### Finding a similarities and differences of sent of matrices

Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices? Regards,
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### Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio? $$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
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### Proving bounds on analytic functions using only the Taylor expansion

I wonder if there is a general method for obtaining bounds on an analytic function using only its Taylor expansion (not using its special properties such as satisfying a good differential equation, ...
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### maximum likelihood estimation of X is better than that of f(X)?

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...