For questions about estimation and how and when to estimate correctly

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13 views

Fitting a model to a collection of binomial proportions, based on varying (large) sample sizes.

I have a multi-parameter bivariate function, say $f(i,j)$ that I want to use to predict the entries of a matrix $M(i,j)$, the entries of which are binomial probabilities based on varying sample sizes, ...
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13 views

Maximum Likelihood Estimation with 2 parameters for a Poisson distribution

I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same ...
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0answers
14 views

Lower bound of KL-divergence between unknown and known distribution

Suppose I have two multivariate continuous random variables $X$ and $\hat{X}$ with underlying probability distributions $P_{X}$ and $P_{\hat{X}}$, where $P_{\hat{X}}$ is a Gaussian approximation of ...
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0answers
15 views

About PDE solution: H^1 norm bounded = bounded in L^2?

Let $\Omega \subset \mathbb{R}^d~(d=2,3)$ be an open bounded set with Lipschitz continuous boundary $\Gamma$. We assume that $\Gamma$ consists of two disjointed parts, i.e, $\Gamma = \Gamma_{c} \cup ...
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0answers
10 views

Estimation effectiveness of two normal-distributed variables

You have two processes of measuring the air pollution, $X$ and $Y$. Both processes deliver values which are normal distributed around $\mu$: $X ~ N(\mu, \sigma_x^2)$ and $Y ~ N(\mu, \sigma_y^2)$. I ...
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0answers
14 views

FIR estimator for IIR system

Suppose that we have a dynamical system of which the impulse responses are infinite (IIR). Now I found methods on papers (http://dx.doi.org/10.1109/9.839942) estimating states or outputs of such a ...
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0answers
7 views

lower bound for integral of curve derivative

Let $\gamma_k:[0,1]\rightarrow\mathbb{R}^3$ be some arc-length parametrized $C^1$-curve, more specifically a knot. $\gamma: [0,1]\rightarrow\mathbb{R}^3$ is the limit of a sequence of such knots in ...
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1answer
25 views

$|f(z)|\le\frac{M}{|z|^{\alpha}}$ for all $z\in U_r(0)\setminus \{0\}.$ Why is $0$ a removable singularity of $f$?

Let $0<r<1$, $f:U_r\setminus\{0\}\to\mathbb{C}$ holomorphic. Let $\alpha <1,\; M\ge 0$ such that $$|f(z)|\le\frac{M}{|z|^{\alpha}}$$for all $z\in U_r(0)\setminus \{0\}$. Prove that $0$ is a ...
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12 views

Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
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119 views
+50

Regularity of heat kernel

I'm trying to find some references dealing with regularity and properties of the heat/Gaussian kernel $$ G_t\left(x,y\right) = \frac{1}{\sqrt{2\pi t}}\, e^{-\left.\left(x-y\right)^2\right/2t}, ...
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1answer
32 views

“Guesstimation” problems within pure mathematics

Wikipedia defines a “guesstimate” as “an estimate made without using adequate or complete information, or, more strongly, as an estimate arrived at by guesswork or conjecture.” Guesstimation problems ...
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16 views

Peak requests per second, The Social Network Movie's Face Mash

In the movie The Social Network, the Harward network crashes after facemash.com gets too much traffic. They mention 22000 page views during the 4 hours it was online. Viral websites usually follow an ...
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0answers
82 views

Why is there no unbiased estimator for $\frac{1}{\theta}$ for Poisson Distribution?

Suppose that $X_1,\dots,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$. To do ...
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2answers
23 views

Is “non-random parameter estimation” the same thing as maximum likelihood estimation?

In one book and a few papers, mostly on navigational tracking, I have found reference to the method of "non-random parameter estimation" but this term is not on the Wikipedia and not in a lot of ...
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1answer
42 views

Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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30 views

Kalman filter on Timeseries

After a lot of research on Kalman filter I can't find anywhere how exactly the filter works on timeseries.Specifically, I want to know about fοrecasting with Kalman filter on Timeseries, point ...
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0answers
24 views

Norm Estimation

I'm studying partial differential equations, and I have a question on the estimation of solution norms. Let $\Omega \subset \mathbb{R}^d~(d=2,3)$ be an open bounded set with Lipschitz continuous ...
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1answer
13 views

unbiased estimator for sample covariance?

I'm new to statistics and and I need some help: Let $X_1,...X_n$~$N(\mu_x,\sigma^2)$, $Y_1,...Y_m$~$N(\mu_y,\sigma^2)$. All r.vs. are i.i.d and $\mu_x,\mu_y,\sigma$ are unknown I was told that ...
5
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2answers
114 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
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0answers
22 views

Compute the limit of the error term of Taylor polynomial of $f(x)=e^{3x}-1$

I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$). First, I found the Taylor polynomial of $f(x)$, which is $$T_n ...
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0answers
45 views

Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,…$ defines a probability distribution for the random variable $X$.

Let $p \in (0,1)$ and $q=1-p$. (a) Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,...$ defines a probability distribution for the random variable $X$. (proved) (b) Given the single observation ...
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8 views

How to estimate a time step to obtain required accuracy when simulating linear dynamic system?

Assume the the linear dynamic system is $$\dot{x}=Ax$$ , and the initial state is $x_0$, where $|x_0|<K$, $K$ and the matrix $A$ is known. The state after time $t_0$ will be $e^{At_0}x_0$. The ...
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0answers
30 views

Point estimation of expected value - disease spread

Firstly I want to put big disclaimer here. This particular problem is a smaller part of my homework. Since even after discussion with my fellow classmates we are not sure how to handle it we decided ...
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0answers
17 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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1answer
22 views

Understanding the point estimation of the expected value

I am trying to understand this problem, however I can't get past some of the definitions used when estimating the expected value. What I would need is to confirm or disprove my conclusions - I read ...
0
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1answer
61 views

Estimating $\sum\limits _{n=1}^k \sin \frac x n$ in the form $f(k,x) \sin(g(k,x))$

When you plot the function for a reasonably large $k$ ($300$ in this case) you get something like this... This seemed like it could be estimated the way I stated previously. The accuracy of that ...
2
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0answers
111 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
2
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1answer
31 views

Estimating the number of books in the world from randomly chosen overlapping lists

Suppose I have lists $L_1 , \dots , L_n$ of, say, books. Assume further that these are uniformly chosen from the set of all books (probably unrealistic for obvious reasons, and if this assumption can ...
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1answer
56 views

Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Is it possible to evaluate or at least to estimate the following integrals? $$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$ and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$ I ...
2
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0answers
46 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, ...
2
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0answers
46 views

For what $r,s$ exist unbiased estimation of $f(p) = p^{r}(1 - p)^{s}$ for binomial distribution?

We have sample $x_1, ..., x_n$ generated by independent binomial random variables $\xi_1, ..., \xi_n$. We know parameter $k$ but don't know probability $p$. k is number of tests: $\xi_i \sim ...
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1answer
18 views

Estimation, Upper limits, Lower limits

Two rods of length 2.6 cm and 3.5 cm are measured correct to the nearest 0.2 cm. The two rod are joined together, find the lower and upper limit of the new rod. I get stuck. HOw to do?
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0answers
28 views

Inverse sum headache

I'm now extreamly tierd of not pulling off this equation. $$\sum_{i=1}^n (y_i-\alpha)^2= \frac{2n\sum_{i=1}^n (y_i - \alpha)}{\sum_{i=1}^n (\frac{1}{y_i - \alpha})}$$ Solve for $\alpha$, y is a ...
2
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0answers
56 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
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1answer
13 views

derivation of $\theta(x)=\int_{a}^{x}\varphi (y)dy - (\int_{a}^{b}\varphi(y)dy)\psi(x)$ and °L^2$-norm estimation

Let $I=(a,b)$, $u\in L^2(I)$ and $\psi\in C^{\infty}(I)$ such that $\psi=0$ on $(a,a+\epsilon)$ and $\psi=1$ on $(b-\epsilon , b)$ for sufficient small $\epsilon$. Let $\varphi\in C_C^\infty (I)$ and ...
0
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1answer
26 views

Estimate parameters of a quadratic function

Suppose that we have two data points which tell us about the output of some function $f(x)$: $(0, 50)$ $(10, 150)$ We know that the function is quadratic (so it's something like $ax^2 + bx + c$). ...
2
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1answer
12 views

Saturating space so that at least two lines are close enough

All lines in what follows pass through the origin. The only reason for the angle $2\pi/3$ below is that this is how I began wondering about these questions. Picture the unit disc $S^2$, by which I ...
0
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1answer
41 views

Why integral is equal to zero

I wonder why under assumption that w>>$\frac{1}{T}$ then $\int_{0}^{T} sin(wt)dt$ is approximately zero? Since the integral should be like- $\frac{cos(wt)}{w}$ from $0$ to $T$ and after plugging the ...
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4answers
106 views

How to estimate numbers like $(19/20)^{30}$

Is it possible to estimate by hand what is the value of expresion like $(19/20)^{30}$? $$19/20 = 0.95$$ but $$(19/20)^{30} \approx 0.2146$$ So it is totally different number.
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1answer
21 views

Unbiased estimator problem

Let $X_1, X_2,\dots, X_n$ be a sample of size $n$ from a distribution with unknown mean $−\infty<\mu<\infty$, and unknown variance $\sigma^2 > 0$. Show that the $Y = (X_1 + 2X_2 + 3X_3 ...
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0answers
7 views

To find sharp infimum (lower bound) of function with indicator function

Let $(x_\varepsilon,y_\varepsilon)\in[0,1]\times[0,x_\varepsilon)$ be a sequence such that $(x_\varepsilon,y_\varepsilon)\to(x,y)\in[0,1]\times[0,x)$ as $\varepsilon\to0$. Is there an integrable ...
0
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1answer
35 views

what is the covariance between $\hat Y$ and$\hat \beta_1$?

I'm having a crisis of faith here, I'm trying to prove that $\beta_0$is unbiased. The formula for $\beta_0$(the parameter) is: $$\beta_0=\mu_Y-\beta_1\mu_X$$ The formula for $\hat \beta_0$(the ...
0
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1answer
16 views

How do i calculate the number of subintervals n in Midpoint method?

I want to calculate the least error (o) in order to obtain the exact answer for integration using the midpoint method. However I am having trouble doing so since i was given a functions whose second ...
0
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1answer
41 views

Estimating Cable Length on a Reel

I have been searching all areas of the internet to try and find a reliable formula for estimating cable length on a reel, I'm trying to create a faster and more reliable way to estimate cable to ...
2
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1answer
47 views

estimation of a series $3^n/( 4^n -1 )$

I am trying to show that the series $$ \sum_{i=0}^\infty \frac{3^i}{4^i-1}$$ is convergent, but do not see how to get rid of the one in order to get a bigger series. Thanks for helping.
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1answer
36 views

The maximum-likelihood estimators of $\sigma^2$

A sample of size $n$ is drawn from each of four normal populations, all of which have the same variance $\sigma^2$. The means of the four populations are $a+b+c$, $a+b-c$, $a-b+c$ and $a-b-c$. What ...
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2answers
35 views

Does the contour integral of a rational fraction function in the complex plane vanish in large radius limit?

Let $f(z)=\frac{z^m+az^{m-1}+\cdots+b}{z^n+cz^{n-1}+\cdots+d}$ be a rational fraction function of complex variable $z$, where the integers $n-m\geqslant 2$. Is the following integration limit ...
0
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1answer
28 views

Maximum Likelihood Estimator of $\theta$

I have the following question I tried to answer I got answer that same like this answer Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer ...
1
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1answer
15 views

Does the correlation between the measurement noise influence the result of the LS?

Consider a linear measurement process with some noise: $$y=Hx+v$$ with $$v \sim \mathcal{N}(0,\Sigma)$$ the covariance matrix $\Sigma$ is not a diagonal matrix. As we know, using LS, the $\hat{x}$ is ...
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0answers
26 views

Derivative of conjugate transpose of matrix 2

Following the development of this post Derivative of conjugate transpose of matrix I really don't get why take the derivative w.r.t. $z∗$ and not take the derivative with respect to $z$ (look the ...