Questions tagged [estimation-theory]
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111 questions
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Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
Assume $\{h_j\}_{j\in \mathcal{N}}$ are independent Gamma random variables, each with potentially different distributions and parameters. I am looking for an upper bound for $\mathbb{E}\left[\max_{j \...
- 41
1
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1
answer
56
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How to study the convergence of the sample mode for arbitrary probability spaces
(This is not the problem I actually care about, but an analogy with similar issues to the problem I'm actually considering.)
Consider a probability space with i.i.d. random variables $X_i$ producing ...
- 277
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0
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42
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An integral estimate involving Bergman kernel
Let $V$ be the normalized volume measure on $\mathbb D^2$ and $k : \mathbb D \times \mathbb D \longrightarrow \mathbb C$ be the Bergman kernel on $\mathbb D^2$ given by $$k(z,w) = \frac {1} {\left (1 -...
- 41
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2
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Calderón–Zygmund/$L^p$ estimates for the linear heat equation
Let $C_r$ denote the open cylinder
$$
C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\}
$$
and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation
$$
\...
- 233
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0
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31
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What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
2
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1
answer
128
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Almost sure convergence of double averages of IID random variables
Let $ \{X_i\}_{i=1}^{P} $ and $ \{Y_j\}_{j=1}^{Q} $ be two sequences of independent and identically distributed (i.i.d.) random variables. $X_i$ and $Y_j$ are independent between all pairs of $i$ and $...
- 433
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1
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170
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Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression
Consider the multivariate regression model
$$Y = XB + E$$
where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...
1
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0
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148
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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 ...
- 6,541
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1
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341
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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 $||\...
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104
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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 ...
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0
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66
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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:
$$
f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b],
$$...
- 393
0
votes
1
answer
937
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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 ...
- 75
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567
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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 ...
- 75
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51
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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 ...
- 231
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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 ...
- 13
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2
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305
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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
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...
- 1,701
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0
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59
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Functional approximation with derivatives
I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
2
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0
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122
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Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc
Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
- 321
2
votes
1
answer
142
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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]...
- 123
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113
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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 ...
- 33
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1
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377
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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/...
- 809
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0
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21
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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{...
- 101
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106
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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 ...
- 101
1
vote
0
answers
34
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 ...
- 153
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132
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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)=...
- 1
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1
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414
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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) ...
- 183
1
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1
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101
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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 $X,Y$....
- 618
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1
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871
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 ...
- 618
2
votes
0
answers
225
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;...
- 233
4
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1
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339
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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}^\...
- 437
1
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0
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81
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 ...
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0
answers
78
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 ...
- 255
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votes
0
answers
130
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$, ...
- 173
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1
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121
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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 ...
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139
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'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);...
- 115
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1
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269
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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 ...
- 23
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0
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75
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
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0
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108
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
154
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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 ...
- 173
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0
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90
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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)$$
In the ...
- 61
3
votes
1
answer
139
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Design a random variable which has the maximal correlation with another random variable
$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...
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3
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698
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How to estimate the integral involving the distance function
Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how ...
- 561
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votes
1
answer
204
views
How to combine estimator with different variances?
Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...
- 127
1
vote
0
answers
154
views
How to retrieve back the input using Bussgang theorem?
If we have a non-linear function $f$, that is applied to input $x$,
we have then the output $y=f(x)$
Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as
$y=Bx+ η$,
...
- 39
6
votes
1
answer
433
views
Probability of complex eigenvalues
I find this is the best site to post this question, even though I considered cs.
It is a Monte Carlo experiment over the set of 10.000 n×n matrices.
If a single matrix eigenvalue is complex then ...
- 171
3
votes
0
answers
113
views
Image restoration quality general lower bounds
A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe
$$
\tilde f = f \star h + n
$$
where $\star$ is convolution, $h$ is the point spread function (caused,...
- 1,069
3
votes
0
answers
265
views
Proving the exponential decay of Green's function for the lattice $-\Delta+p$
The Green function $G(x,y) =G(x-y)$ of the discrete Klein-Gordon operator $-\Delta+p$ on $\mathbb{Z}^{d}$ is given by:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{...
- 1,305
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0
answers
56
views
Estimation of parameters through multivariate Taylor expansion?
I do have a function $$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,\; ...
- 686
-1
votes
1
answer
144
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
385
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. ...
