# Questions tagged [estimation-theory]

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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 ...
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### 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 , p. 173, in the form of a reference or of a direct argument: Percentile ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
129 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. ...
56 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 ...
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### Lower bound of q pochhammer symbol [closed]

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

I am confused about the definition of the Fisher information and the case when probability is 0. Consider discrete set $\epsilon$ of possible measurement outcomes. Fisher information is defined as: ...
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### A different objective function in liner regression analysis

I'm an undergraduate student who is green in statistics. I have a problem in the chose of objective function when estimating the parameters. Let $Y = \beta^TX + \epsilon$ be the standard liner ...
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### Reconstructing the number of distinct elements from a random projection

Assume we have an unknown sequence $x_1,\ldots, x_n\in \mathcal U$. We get to observe the sequence $h(x_1),h(x_2),\ldots, h(x_n)$, where $h:\mathcal U\to \{1,\ldots, k\}$ is a random function such ...
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### Calculate sample mean confidence interval of noisy logistical distribution

I have $n$ samples which follow a logistic distribution with unknown $u$ and $s$; it is affected by a Gaussian noise with 0 mean. I would like to estimate its average $u$ with a confidence interval (...
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### Cramer Rao bound for relative estimation

I have an observed vector ${\bf y}$ from which I would like to estimate a parameter vector ${\bf c}$ (denote the estimate $\hat{{\bf c}}$). A feature of our estimation problem is that the involved ...
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### 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 ...
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### Uniform mean-square-error estimates

Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...
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### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
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### Elliptic Equation with Wentzell boundary condition

I'm looking for a reference showing how to obtain a priori estimate for solutions to a linear second-order elliptic equation with Wentzell boundary condition in a bounded domain in $H^1$ space. The ...
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### KL divergence Inequality

I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading. Consider two probability measures $P$ and $Q$ both ...
Consider a scalar system with $2K$ outputs and $K+2$ unknowns: $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$. The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and \$...