For questions about estimation and how and when to estimate correctly

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11 views

Estimate integration

Suppose $n<<k$ and $n<i<k$. How can I estimate this integral: $\int_{1}^{k-i} x^{-0.4}*(x+i-1)^{-0.4}dx$. I would like to get the result in the form of O(f(k,i)). Since the integrand is ...
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8 views

Estimation of binomial probabilities $f(r)$ over $r \in [0,\frac{1}{2}]$

I want to fit a (decreasing) univariate function, \begin{equation} f(r), \end{equation} over $r \in [0,\frac{1}{2}]$ to a series ($r =\frac{1}{100}, \frac{2}{100}, \frac{3}{100} ,\ldots,\frac{1}{2}$) ...
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17 views

Maximum likelihood bivariate normal

I have the following model: $$x_{t+1} = x_t + a + \epsilon_{t+1}$$ $$y_{t+1} = b y_t + \delta_{t+1} $$ Where all the $x_t$ and $y_t$ are known and where the $(\epsilon_t,\delta_t)$ are independent ...
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11 views

How to calculate a time estimate and plot it?

I recently made a website for a game's community. The game basically has an online dragon which people can fight. Each fight chips off a little bit health. On the next encounter with the dragon, ...
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1answer
9 views
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1answer
46 views

How can I calculate missing values from a table listing of areas and prices? [on hold]

I have a set of objects of different sizes (measured in square metres). I know the price of some of them. I want to use the known prices to find the missing prices. Here is the data I have: ...
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24 views

Cramer-Blackwell estimator for uniform distribution.

I've got two estimators of parameter $\alpha$ in the distribution $X=X_1,...,X_n$, where $X_i$s are i.i.d. uniform random variables on the interval of $(0,\alpha)$. These two estimators are: ...
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1answer
16 views

Estimating confidence Interval for unknown Variance, Normal distribution

I've been stuck with this question for a while: I've learnt how to use t-distribution to estimate CIs for an unknown variance, but I'm unsure how that applies to this situation. Any help would be ...
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1answer
36 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
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2answers
45 views

Probability of island having 8 people born with disease, estimate?

The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that ...
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1answer
132 views

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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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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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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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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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
27 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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1answer
33 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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17 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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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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2answers
26 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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84 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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1answer
43 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
25 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
14 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 ...
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1answer
392 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
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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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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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1answer
219 views

$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$

$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$ $\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$ $\Rightarrow \log ...
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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 ...
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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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1answer
23 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 ...
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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
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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120 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 ...
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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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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 ...
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 $n$, ...
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3answers
100 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
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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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3answers
37 views

Initial values of a exponential decay

How can I estimate the initials values ($A$, $B$, $C$) of a exponential decay? I got the function and a set of experimental points. $p(t) = Ae^{-1.5t} + Be^{-0.3t} + Ce^{0.05t}$ $p(0.5)=6,\ ...
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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?
2
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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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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 ...
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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 ...