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Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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the behavior of the Fourier series partial sum

Consider the Fourier series partial sum $S_n(x)$ as $n \to \infty$ where $S_n(x)$ is the partial sum of the Fourier series representation of a function $f(x)$ defined on a $2\pi$ periodic interval. ...
ZhouYang's user avatar
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44 views

Estimation of Sum of Binomial Coefficients that doesn't Start from Zero

I am having some trouble showing the following estimation: Let $\mu \in (0, 1)$ be an absolute constant. There exists an absolute constant $\eta \in (0, 1)$ small enough such that the number of ...
Partial T's user avatar
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12 views

Average Rank versus Ranked Average in Parameter Estimation

I have the following problem: In a cricket tournament, the eleven batsmen of a team play 100 matches before the final. The runs scored by each are available. Determine the average rank of the batsmen ...
Starlight's user avatar
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1 answer
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the sum of $O_p$ --$ O_p(s^2\frac{\log d}{n}+s\sqrt{\frac{\log d}{n}}) $

I read papers in the area of inference for high-dimensional graphical models and these papers always state the convergence rate of the estimator. Using $O_p$ is a good choice. Maybe I made some ...
mathhahaha's user avatar
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42 views

Is there a version of the delta method when taking the derivative of an estimator with respect to time for stochastic processes?

Suppose that $X$ is a stochastic process and $f$ some measurable function. Consider the random variable $S_t=\int_0^tf(X_s)\mathrm{d}s$ and an estimator $\hat{S}_t^n=\sum_{i=1}^{\lfloor t/n\rfloor}f(...
Daan's user avatar
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1 answer
49 views

How might one estimate the number of samples in a uniform distribution given only the sample range?

Let's say we have a fair roulette wheel with 2^256 segments, each printed with a unique integer in the range 0 to 2^256 - 1. Let's say that the wheel is hidden from us and that Trevor spins it ...
Lee's user avatar
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4 votes
2 answers
97 views

Estimates for incomplete Gamma functions

I want to show that $$\int_{0}^{\sqrt{n}} \exp(-s^4)s^{n+1} ds - \int_{\sqrt{n}}^{\infty} \exp(-s^4)s^{n+1} ds \geq 0\, \quad \text{for all }n\in\mathbb{N}.$$ These integrals can in fact be written ...
Nils's user avatar
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0 answers
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A vectorial estimate I can't understand.

Let $w\in (0, \kappa], p < 1, p\neq 0, \alpha \in (0, 1), \beta(1-p) -Kp>0, \sigma\in \mathbb{R}^{n\times n}, $invertible, and $ \mu, \eta \in \mathbb{R}^n$. I want to estimate $|\mathbf u|$, ...
oxedex's user avatar
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3 votes
6 answers
245 views

What is a good way to compute $\Gamma(1/3)$ on a standard pocket calculator?

This question is inspired by this one. The earlier question asks how to calculate a certain integral efficiently with a standard pocket calculator. A fine answer by Travis Willse gives a good result ...
Oscar Lanzi's user avatar
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UMVUE of $\mathbb{P}[X=x_0]$ where $X_1,\cdots,X_n$ is Poisson

I came up with this problem and tried to solve it myself. Please check my solution, I am a bit unsure it is correct because it says the UMVUE for the probability where $x_0>\text{sum of ...
harrydiv321's user avatar
4 votes
4 answers
142 views

Evaluate or estimate $\int_{8}^{27}\frac{\mathrm{d}x}{\sqrt{x}-\sqrt[3]{x}}$ without a calculator in a fast way

In a fast way and without a calculator, I need to evaluate the following integral or just estimate it in a way that will lead to the correct answer. $$\begin{align}\\& \text{What is the value of}\...
Hussain-Alqatari's user avatar
4 votes
0 answers
141 views

Observer for LTI system with linear inequality constraints for state

I have searched quite a bit about this topic but only found methods that consider equality constraints. Consider LTI system $$ \begin{align} \frac{d}{dt}x&=Ax+Bu\\ y&=Cx+Du \end{align} $$ with ...
user3137490's user avatar
1 vote
0 answers
72 views

A question about Wiener filter based on Linear Estimation by Kailath

In my linear estimation class based on the textbook Linear Estimation by Kailath, we went through the process of finding LLSE of $\hat{x}(t+\lambda)$ for fixed $\lambda$ given $\{y(\tau)|-\infty<\...
monad's user avatar
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7 votes
1 answer
219 views

How to bound $\mu (\{ x \le X : |ψ(x) − x| \ge εx^{1/2} (\log x)^2 \})$ from above?

From the following estimate $$ \int_0^X |ψ(x) − x|^{2k} dx ≪ (ck^2)^k X^{k+1} \tag{1}$$ where $c$ is an absolute constant, I want to prove the following estimate $$ µ ( \{ x \le X : |ψ(x) − x| \ge ...
Ali's user avatar
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35 views

Estimates of the derivatives of $\Xi(s)$

The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$. This is a problem from my homework: since we can write it ...
Fresh's user avatar
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1 vote
1 answer
53 views

Hölder's condition and Fourier transform

For Fourier Transform of function $f\in L_1(\mathbb{R})$ there are such constants $\alpha\in(0,1), C>0$ such that $|\hat{f}(y)|\le\frac{C}{(1+|y|)^{\alpha+1}} \forall y \in \mathbb{R}$. prove that ...
Jane Doe's user avatar
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1 vote
2 answers
87 views

Accurate estimate of $\sum_{k=0}^\infty z^{k^2+ck}$

I encountered a problem, where I am interested in determining (or estimating) a series of the form $$S = \sum_{k=0} ^\infty z^{k^2+ck},$$ while $z\in (0,1)$ and $c>0$. The most simple estimate I ...
Salfalur's user avatar
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0 answers
17 views

Discrete family of probability distributions and complete sufficient statistics

I need to figure out atleast two families of discrete probability distributions, one where complete sufficient statistics exist and another where it doesnt. I was able to find Power series family of ...
BTM's user avatar
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3 votes
1 answer
104 views

Approximation of $\sigma(n)$ sum.

Investigating: $$\epsilon(n)=\frac{(\pi -3) e^{2 \pi n}}{24 \pi }-\sum _{k=1}^n \sigma(k) e^{2 \pi (n-k)}$$ where $\sigma(n)$ is a divisors sum of $n$. Using long calculations (can not share here ...
Gevorg Hmayakyan's user avatar
5 votes
2 answers
145 views

Asymptotic Behaviour of the Remainder of Certain Alternating Series

Let $a,b >0$ be real constants. Empirical observation (as in: asking WolframAlpha) suggests $$ \lim_{n\to \infty} n \cdot \sum_{k=0}^\infty (\frac{1}{n+ak} - \frac{1}{n+b+ak}) = \frac{b}{a} \tag{$...
Torsten Schoeneberg's user avatar
1 vote
0 answers
30 views

Estimating the parameters of an ellipse (part 2)

This post is a follow up of this previous one. I would like to clarify why the angle estimator works and how to estimate the axes length. Unfortunately, I still have some trouble with this problem. I ...
matteogost's user avatar
1 vote
0 answers
32 views

Estimates on Holder Norms and $C$^k norms

I'm dealing with some estimates. There are some very useful estimates on Holder norms in a paper by Hormander. But I can't relate them immediately to the usual $C^k$ norms. The estimates appear in ...
Master.AKA's user avatar
  • 1,005
10 votes
6 answers
232 views

How to estimate $10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$

I heard an interesting interview question recently, which was as follows: Estimate the value of $X = 10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$. You have 30 seconds to Compute ...
Christopher Miller's user avatar
0 votes
2 answers
40 views

How does the Kalman Filter incorporate unseen states during measurement?

I'm curious how/if the kalman filter is able to infer states which aren't directly measurable during the measurement process. Suppose our system has states $(x_t, v_t)$ representing position and ...
maxical's user avatar
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1 answer
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Find the unbiased estimator for the parameter $\sigma$. [closed]

TASK: Let $(X_1, X_2)$ is a random i.i.d. sample from $N(0, \sigma ^2)$ distribution. Find the unbiased estimator for the parameter $\sigma$. SOLUTION: The estimator $\theta$ is unbiased if $E\theta = ...
Lopirony's user avatar
1 vote
0 answers
13 views

Etimating an error of a model without a precise calculation

Let a set of $N$ pairs of matched measured points $p_i, c_i$ with some noise. Using an optimization method, we optimize a model $m$ that estimates $\hat{c_i}$ from $p_i$: $$\hat{c_i} = m(p_i)$$ ...
havakok's user avatar
  • 1,169
4 votes
0 answers
59 views

Attenuation estimation of the solution of the two-dimensional wave equation Cauchy problem

This is the equation given, $$\begin{array}{l} u_{tt}=a^{2}\left(u_{x x}+u_{y y}\right), \\ \left\{\begin{array}{l} \left.u\right|_{t=0}=\varphi(x, y), \\ \left.u_{t}\right|_{t=0}=\psi(x, y) . \end{...
Zydragon's user avatar
0 votes
1 answer
32 views

Kernel, variance and estimation

In the paper by Terrell 1990, in the Theorem 1 below on the page 471, I would like to derive the formulas for $g(x)$ and $h(x)$ and perhaps also why $\beta(k+2,k+2)$ minimizes that integral given.Why ...
user122424's user avatar
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0 votes
1 answer
43 views

Derive apriori estimate for solution of PDE

I have the following PDE: Let $G = (a, b)$ and $J = (0, 1)$. Consider the Dirichlet problem with general coefficient functions $\alpha(x)$, $\beta(x)$, $\gamma(x)$ \begin{equation} \begin{cases} \...
julian2000P's user avatar
1 vote
1 answer
50 views

Show that estimate is unbiased

TASK: Suppose $X_i (i=1,2,3,…,n)$ are i.i.d random variables with PMF: $f(x, \theta ) = exp(\theta-x), x>\theta$. Is the estimate unbiased: $\mu = \frac{1}{n} + $min$(X_i)$? Answer: First, we find ...
Miganyshi's user avatar
  • 125
0 votes
1 answer
35 views

Estimate on the number of solutions of congruences

Let $F$ be an irreducible, integral polynomial. Is it true that$$|\{\nu:F(\nu)\equiv 0\mbox{ mod } n,\ 0\le\nu<n\}|\ll n^{\epsilon}$$as $n\rightarrow+\infty$? How can one show it?
CarloReed's user avatar
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0 answers
31 views

Estimation of operator norm of inverse matrix

Let $A$ be an $n*n$ real-symetric matrix, B be a $p*n$ real matrix and $\eta$ be a real number. I want to know if we can give an upper bound of the following inverse matrix $G={\begin{pmatrix} A-i\eta ...
Rixinner's user avatar
1 vote
0 answers
28 views

Show that $\dfrac{d^{m-1}}{ds^{m-1}} \Big(\dfrac{\zeta'(s)}{\zeta(s)} \Big) =- \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\log t)$.

Eq. 2.1 in Levinson's paper states $$\dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \dfrac{1}{(s-\rho)^m} = - \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\...
Ali's user avatar
  • 281
0 votes
0 answers
28 views

Definition of confidence interval - intuition

In the context of the confidence interval for parameter $\theta$ with confidence level $1-\alpha$ I was always dealing with such formulation, $P(\theta \in \mathbb{T}_n)=1-\alpha$, with the "=&...
gaghan's user avatar
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0 votes
0 answers
26 views

L^p bounds for function on double annulus

The problem is Consider the open disc $B_r=B(0,r)\subset \mathbb{R}^2$, $\phi=\max\{|x|-1,0\}$ is the distance from $B_1$. For $u\in C^1(\overline{B_3-B_1})$. Show that $\exists$ $c$ such that (i)$1\...
vegetabledoge's user avatar
0 votes
2 answers
66 views

Can I use the BFGS algorithm in combination with a L1-Norm penalized LLH to get exact zeros?

I am working along a paper Predicting the Long term Stock Market Volatility:A GARCH MIDAS Model with Variable Selection It uses a PLLH = LLH - L1-Norm (equation 7) And always speaks of "non-zero&...
mexx's user avatar
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2 votes
1 answer
70 views

Finding $MSE$ of optimal estimator

Background of the question: We know that $X$ is a continuous random variable that has $P\left(-1\le X\le 1\right)=1$ and $f_X\left(x\right)<\infty $ We define $X(n)=cos(\pi n X)$ $E[X]=\mu$ and $...
Analysis_Complex_Study's user avatar
0 votes
0 answers
21 views

Convergence of coefficients in multivariate regression

In this thread, the convergence of coefficient for univariate dependent variable is proven. I wonder, assuming the same setup, how can the convergence be extended to multivariate as: $$Y=XW+\epsilon$$ ...
statwoman's user avatar
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0 answers
100 views

Corollary 1.17 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: The proof is very much ambiguous to me. For example: 1- The claim $\int_1^...
Ali's user avatar
  • 281
0 votes
0 answers
19 views

Evaluating $|E_K(x)| = s(x) - \sum_{k=1}^K \dfrac{\sin 2 \pi k x}{\pi k}$ on $0<x \le 1/2$

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I could understand the full proof but the last paragraph, i.e. to show ...
Ali's user avatar
  • 281
0 votes
1 answer
18 views

Non-decreasing expectated value of non-decreasing function with family of probability density satisfying monotone likelihood ratio property.

I am trying to prove the Lemma(3.4.2)(1) of book "Testing statistical hypotheses" by Lehmann and Romano, its statement is " If $\{p_{\theta}(x)\}$ be a family of densities on the real ...
Naveen Kumar's user avatar
0 votes
0 answers
25 views

Need Help Understanding the Proof of Lower Bound on Expectation of Maximum Gaussians

I am trying to follow the proof given here: http://www.gautamkamath.com/writings/gaussian_max.pdf. In particular, I would like to understand the following crude bound mentioned in the paper: My ...
Partial T's user avatar
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0 votes
0 answers
46 views

Understanding an estimate

I am trying to understand an estimate from the post: Lower bound for expectation of maximum absolute value of standard normal random variables. In particular, in the answer posted there, we used the ...
Partial T's user avatar
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1 vote
1 answer
106 views

How to show that $O(\sum_1^{\infty} n^{1-{2\sigma}} e^{-\delta n} \sum_1^{n/2} 1/r )=O({\delta}^{2 \sigma -2} \log \dfrac{1}{\delta})$?

The following lemma is from Titchmarsh's The Theory of the Riemann Zeta-Function: I have difficulties in getting both the estimates: 1- $O((\sum_1^{\infty} n^{-{\sigma}} e^{-\delta n})^2) = O((\int_1^...
Ali's user avatar
  • 281
2 votes
1 answer
82 views

Basic analysis question to estimate the growth of a function

Let $f \in C(\mathbb R,\mathbb R)$, $\beta_0 >0$, and $\alpha\in (0,2)$ be given such that \begin{equation}\tag{1}\label{1} \lim_{|s|\to \infty}\frac{|f(s)|}{e^{\beta s^2}} = \begin{cases} 0 & \...
toothlessninjafrog's user avatar
1 vote
0 answers
59 views

Understanding Theorem 5.5. of Titchmarsh's Book

I have two questions regarding the proof of Theorem 5.5. of Titchmarsh's book The Theory of the Riemann Zeta-Function: Green-underlined: How from the $O$-term $$\mathcal{O} {\Bigg(\mu + \sum_{r=1}^{\...
Ali's user avatar
  • 281
0 votes
0 answers
37 views

Calculating the average of a parameter estimate

Suppose $X_1, ..., X_n$ are independent random variables on $[0,1]$ ~ $\beta(\alpha,\alpha)$, where $\alpha$ is a positive parameter. A program generates $10000$ random samples of size $10$ and ...
lcthaha's user avatar
0 votes
1 answer
56 views

How to calculate $\int_{\frac12 \eta}^{\infty} e^{-\frac{{\lambda}^2}{4 \pi}} {\lambda}^n d{\lambda}$?

In the following text how the first blue-underlined O-term has changed to the second one? I don't know how writing the integrand as $$e^{-\frac{{\lambda}^2}{8 \pi}} \times e^{-\frac{{\lambda}^2}{8 \...
Ali's user avatar
  • 281
0 votes
1 answer
25 views

Find a matrix that maps a several broad region in $R^n$ to a small regions in $C^n$

I have several vectors $y_1, y_2, y_3 \cdots y_n \in R^n$ that I need to linearly map to a single vector $x_0 \in C^n$. The same matrix should also map $z_1, z_2, z_3, \cdots z_n \in R^n$ to a single ...
user3284182's user avatar
2 votes
1 answer
76 views

What makes the Mean Squared Error so special compared to other upper bounds?

Let $X$ be some measure space (esp probability space), $Y$ some metric space and we consider $g:X\to Y$ as an estimator for $f:X\to Y$. It seems convenient to consider an estimation as 'good' in case ...
kuemmel's user avatar
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