# 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. ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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{cases} \...
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1 vote
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### 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 ...
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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?