Questions tagged [estimation-theory]

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Quotient of estimators

Say $A$ is a set of a finite number of samples, and $\hat{\mu}_A$ and $\hat{\sigma}_A$ are unbiased estimators (computed over $A$) of $\mu$ and $\sigma$ which are some distinct population statistics. ...
3 votes
1 answer
88 views

Perturbation results for statistical estimators

Suppose I have a continuous random variable whose distribution $f$ is some parametric form (normal, exponential, etc.) that is known to me. If I draw many independent samples $x_i$ from $f$, I can ...
2 votes
2 answers
592 views

An alternative proof of Bayesian Cramer-Rao

My question is: Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality? Let me outline the classical proof and explain why I am interested in this ...
1 vote
0 answers
140 views

conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here, Form of minimax estimator, which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
1 vote
1 answer
322 views

Form of minimax estimator

Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$. Suppose additionally that $\Delta$ is endowed with some norm $||\...
1 vote
0 answers
81 views

Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space

Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
1 vote
1 answer
164 views

Error metric for joint estimation of mean and variance

Background: Let $\mu:\mathbb{R}^n\to\mathbb{R}$ and $\sigma:\mathbb{R}^n\to\mathbb{R}_+$ be two unknown functions, and consider a stochastic model of the form $$ \mathbb{E}[Y\mid\mathbf{x}] = \mu(\...
1 vote
0 answers
55 views

Parameter estimation of a Taylor expansion

Let $a,b$ two real numbers, $\theta$ a real parameter and suppose that you have an analytic function of the form: $$ f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b], $$...
3 votes
3 answers
2k views

Is the min function ever an unbiased estimator for the mean?

Given $n$ i.i.d. variables $X_1$ to $X_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability ...
7 votes
1 answer
353 views

Singular Fisher information matrix and existence of unbiased estimators

I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
0 votes
1 answer
481 views

Derivative of log-likelihood function for Gaussian distribution with parameterized variance

Suppose we have a parameter $\theta \in R^{n}$ that defines some noisy observation $z=\mu(\theta)+\eta, z\in R^{m}$ where the noise follows a Gaussian distribution whose covariance is a function of ...
0 votes
0 answers
51 views

Estimation of Nonzero Coefficients of Binary Cylotomic Polynomials

I am reading Fouvry's paper https://msp.org/ant/2013/7-5/ant-v7-n5-p07-p.pdf . I am still confused on section 4.2 why $P\leq x^{\frac{20}{9}\gamma -\frac{2}{3}}\mathcal{L}^{-16}$ leads to estimate in ...
2 votes
0 answers
84 views

A complex problem involving densities (likelihood functions) and optimization

Consider the following autoregressive process with normal errors: \begin{equation}\label{7YlUV4i8nuO}\tag{I} y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2) \end{equation} We ...
4 votes
2 answers
260 views

Generalization of van der Corput's estimate on oscillatory integrals

Question: Given exponents $0<\alpha<\beta$ and an interval $[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any $\lambda_1,\lambda_2\in\mathbb{R}$, $$\left|\int_a^be(\...
1 vote
0 answers
54 views

Functional approximation with derivatives

I am trying to solve a functional approximation problem. Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
2 votes
0 answers
111 views

Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc

Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$: $$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
2 votes
1 answer
117 views

DKW inequality for $L^1$-norm

Suppose that $X,X_1,X_2,X_3\dots$ is a sequence of $\mathbb{P}$-i.i.d. random variables supported in the interval $[0,1]$. Let $F$ be the cumulative distribution of $X$, i.e. $F(x):=\mathbb{P}[X \le x]...
3 votes
0 answers
70 views

Is the Kalman Filter computationally optimal for Kalman filtering?

Kalman filtering is known to be a recursive process that minimizes mean square error in linear problems. My question is: has anybody shown that this algorithm is computationally optimal, i.e. that you ...
3 votes
1 answer
311 views

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/...
0 votes
0 answers
19 views

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{...
0 votes
0 answers
92 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
0 answers
33 views

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 ...
0 votes
0 answers
128 views

How to estimate sums over arithmetic progressions?

For $x>1$ $$ N(x)=\sum_{0<n<x \\n \equiv 1 \pmod 4\\ n\text{ squarefree}} 1 $$ How to estimate $N(x)$'s order? (Like $N(x) \sim Ax$) Furthermore, for $n=p_1p_2\cdots p_v$, define $\alpha (n)=...
4 votes
1 answer
293 views

What journal(s) do you recommend for submitting a paper on a topic that spans information theory and estimation theory?

I've written a paper that a) demonstrates an equivalence between conditional complexity $K$($Y$|$X$) in information theory and the random component of an effect size estimate $r_{xy}$, and then b) ...
1 vote
1 answer
93 views

Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question. Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
1 vote
1 answer
646 views

Estimating the average of two gaussians' mean

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$. In my setting, $\sigma_1,\sigma_2$ are known ...
2 votes
0 answers
218 views

Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and $$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$ for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
3 votes
0 answers
110 views

Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$ \tilde f = f \star h + n $$ where $\star$ is convolution, $h$ is the point spread function (caused,...
4 votes
1 answer
332 views

Showing that $\sum_{n=0}^\infty (4n+1)q^{\left (\frac{4n+1}{2}\right)^2} - \sum_{n=1}^\infty (4n-1)q^{\left (\frac{4n-1}{2}\right)^2} \geq 0.1$

Recently I came along the following problem concerning a lower bound on the difference of two series: I want to show that for every $q \in [e^{-2},e^{-\frac{1}{2}}]$ we have $$ f(q) := \sum_{n=0}^\...
1 vote
0 answers
80 views

Calculating the mean squared error for an estimate of a large sum

Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...
2 votes
0 answers
64 views

Distribution of unbiased estimator of covariance matrix with missing values

Initial setup Assuming $X_1, ..., X_n \in \mathbb{R}^m$ are iid, sampled from $\mathcal{N}(\mu, V)$, one can define the estimators for the sample mean $\hat{\mu} = \frac{1}{n} := X^T 1_n$, and sample ...
2 votes
0 answers
110 views

L1 error of estimators

I came across the following problem and I have no clue how to approach it. I am looking for help with directions or references. Consider the $\alpha$-stable distribution with unknown true mean $\mu$, ...
0 votes
1 answer
116 views

How to detect, track and map a Markov chain

You are receiving a time series whose elements belong to a finite set. Assume the time series is distributed as a Discrete-Time Markov Chain. You receive one element at each time step. For each time ...
2 votes
0 answers
138 views

'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem. Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
2 votes
1 answer
233 views

Spline Interpolation error of higher degree

It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$. Can I assume that, if one uses polynomials of degree $p$ and ...
1 vote
0 answers
70 views

Percentile interval Lemma

Let $\theta$ be a parameter and $\hat{\theta}$ the plug-in estimate, I need a proof of the following lemma, as given in [1], p. 173, in the form of a reference or of a direct argument: Percentile ...
1 vote
0 answers
70 views

Bootstrap-$t$ confidence intervals

I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need ...
2 votes
1 answer
152 views

Stability estimates on quotients of the form $ \frac{\prod_{j=1}^n a_j}{\prod_{j=1}^n b_j} $

Suppose that $a_j,b_j \in \mathbb C$ are complex numbers, $j=1,\dots,n$, with the property that $|a_j|,|b_j| \geq c > d >0$ where $c,d$ are positive real numbers. I'm interested in the stability ...
2 votes
0 answers
86 views

The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively. $$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$ $$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$ In the ...
3 votes
1 answer
125 views

Design a random variable which has the maximal correlation with another random variable

$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...
-1 votes
1 answer
166 views

How to combine estimator with different variances?

Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$ Further, assume that we know that $Var[X_i]=\sigma_i^2$. In the ...
6 votes
3 answers
632 views

How to estimate the integral involving the distance function

Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral: $$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$ My problem is how ...
1 vote
0 answers
110 views

How to retrieve back the input using Bussgang theorem?

If we have a non-linear function $f$, that is applied to input $x$, we have then the output $y=f(x)$ Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as $y=Bx+ η$, ...
6 votes
1 answer
403 views

Probability of complex eigenvalues

I find this is the best site to post this question, even though I considered cs. It is a Monte Carlo experiment over the set of 10.000 n×n matrices. If a single matrix eigenvalue is complex then ...
3 votes
0 answers
213 views

Proving the exponential decay of Green's function for the lattice $-\Delta+p$

The Green function $G(x,y) =G(x-y)$ of the discrete Klein-Gordon operator $-\Delta+p$ on $\mathbb{Z}^{d}$ is given by: \begin{eqnarray} G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{...
1 vote
0 answers
52 views

Estimation of parameters through multivariate Taylor expansion?

I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters: $M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...
-1 votes
1 answer
137 views

Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

I am really interested in sufficient conditions on $a_i, b_i$ guaranteeing that the linear forms $a_1\phi(n)+b_1,\dots, a_k\phi(n)+b_k$ become simultaneously prime for infinitely many positive ...
2 votes
0 answers
360 views

Extended Kalman Filter and its State Transition Matrix

Sorry for what might be a long post, I want to give background. Initially I had regular Kalman filter, and the state model was defined by Newtonian kinematics, with initial position 0 and speed of 2. ...
1 vote
0 answers
61 views

Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$, where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
-2 votes
2 answers
414 views

Lower bound of q pochhammer symbol [closed]

How one could prove, that q pochhammer symbol $(1,1/n) = \prod_{k = 1}^{\infty}(1-\frac{1}{n^k}) \geq 1 - \frac{1}{n-1}$