For questions about estimation and how and when to estimate corectly

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2answers
75 views

Estimating arctan to below

How can I estimating $$ \arctan(\lVert x-y\rVert), $$ to below (where $(x,y)\in\Omega\times\Omega, x\neq y$, $\Omega\subset\mathbb{R}^n, n>1$ bounded domain)? Can you give me a hint please? ...
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0answers
60 views

Find a weakly singular kernel function for an estimation of a kernel

Let $\Omega\subset\mathbb{R}^n (n>1)$ be a bounded domain and $0<\alpha<n$. Show, that the kernel function $$ k(x,y):=(\arctan(\lVert x-y\rVert))^{-\alpha}\text{ for }x\neq y $$ ...
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0answers
86 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
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0answers
18 views

Estimation (weakly singular)

Let $\Omega\subset\mathbb{R}^n (n>1)$ be a dominated domain and $0<\alpha<n$. Show that the following kernel function $k$ can be estimated by a kernel function thyt is weakly singular. ...
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2answers
31 views

Calculating a sample's representativeness to confirm/refute a given hypothesis?

Why hello! I'm fairly new to statistics, which is why I'm somewhat confused as to how I can approach this problem in a scientific way. The problem: Experiments are conducted to find the probabilities ...
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1answer
27 views

Boundedness, Estimate $L_2$-Norm by $H^2$

I wanted to prove the estimate: $$ \| \Delta u \|_{L_2(\Omega)} \leq C \| u \|_{H^2(\Omega)} \ \ \forall \ u \in H^2_0(\Omega) $$ Where $ C > 0$ is a constant not depending on $ u $. On the ...
3
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1answer
52 views

Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
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0answers
42 views

Variance of a difference in estimated proportions with trivariate discrete distributions

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...
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0answers
22 views

Density estimation for two sets of samples

Imagine two sample sets $A$ of size $\#A$ and $B$ of size $\#B$. First density estimation for both is done separately which yields $P_A$ and $P_B$ based on those densities. In what way is ...
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0answers
31 views

Estimate starting with variational formula

I'm working on an a priori estimate, using equality's like Young, Cauchy,... But I'm stuck with my testfunction. I've got the following problem: $\frac{\partial u}{\partial t} - \Delta u + \int_\Omega ...
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0answers
63 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
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1answer
48 views

Why is it called the score of the log likelihood function?

Since the score of the log likelihood function is just the gradient of the log likelihood function, why give it a special name? Why not just call it the gradient?
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0answers
50 views

Finding an unbiased estimator for function of Poisson

Let $X_1,...,X_n \sim Poi(\lambda)$ then unbiased estimator for $\lambda$ is obviously $\bar{X}$. What about $\tau(\lambda)=\sqrt{\lambda}$? Also how would one derive UMVUE for this lambda?
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1answer
68 views

How to estimate mean from sanples of multiple correlated random variables?

Suppose we have $n$ normal random variables with variance $1$ and unknown mean. Suppose we have $n$ samples of size 1 from those random variables. Suppose also we know the correlations between the ...
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1answer
25 views

formulate a power series

Could anyone advise me if there is any way to estimate/formulate the following series $$ \sum_{i=m..n} \left(\frac{c}{i}\right)^i $$ where m,n and c are positive integers.
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0answers
61 views

Asymptotic stability

we know from the theory of ODE that $\left\|\exp(tA)\right\|\leq Ke^{-\delta t}$ for $K,\delta >0$ and $t\in\mathbb{R}^+$ if the real part of all eigenvalues are strict non-positiv. My question is: ...
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0answers
12 views

When would you use a triangular or box kernel instead of gaussian?

Just a conceptual question regarding density estimation. Empirically, the gaussian kernel gives me lower MISE values than triangular or box. Epanechnikov gives me the best MISE values if the ...
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1answer
37 views

How rapidly can a polynomial grow in a proximity of the real segment comparing to the values on the segment?

Let $P_n$ be a polynomial of degree $n$ with complex coefficients. Does for any $l>0$ and small $\varepsilon>0$ there exist $C=C(l,\varepsilon)>0$ and $q=q(l,\varepsilon)>1$ s.t. in the ...
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0answers
42 views

Maximization of The Likelihood Function of Vector Entries and Its Norm

I'd be happy for assistance with the maximization of the likelihood function of the following model. The Parameters Vector $ \mathbf{\Theta} = [{x}_{1}, {x}_{2}] $. The measurement vector is $ ...
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1answer
17 views

Hyperbola's estimator

I have a set of data. This set represents an approximation of a hyperbola. Is there a good way to estimate a rectangular hyperbola's function from this set?
2
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1answer
22 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
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0answers
24 views

Is that estimation right on $B_1(0)$?

Is the estimation $$ \frac{n}{2}-1 < \sum_{i=1}^n\frac{1}{1+x_i^2} $$ right, if $x\in B_1(0)\subset\mathbb{R}^n$? I guess $B_1(0)$ is the open unit ball. Anyhow it is: $$ ...
3
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1answer
99 views

Is my determination of this maximum correct?

Consider $\Omega:=B_1(0)\subset\mathbb{R}^n$ (it is the open unit ball), $\mathbb{R}^n$ is provided with the euclidean norm $\lVert\cdot\rVert_2$. Now I want to determine the following ...
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1answer
24 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
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1answer
15 views

Estimate of Subtraction when Observed Outcome May Result from Subtraction and Addition

The problem I am trying to answer is as follows: 1) I have a stack of parcels in my house. 2) Each day, the post may bring more parcels and/or I may send parcels away. 3) At the end of the day my ...
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1answer
28 views

Estimating error in calculation

I'd like somebody to verify my solution of the following problems: Let's assume, that float arithmetics $fl()$ has precision $\nu$ for standard operations $(+\ -\ \cdot \ \div)$. a.) Estimate ...
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0answers
13 views

What is the variance of the largest sample among a set of independent samples?

Say x1, x2, ..., xn are independently derived from a uniform distribution [0, d] (or any other kind of distributions). What will the variance of max(xi) be? Does it follow some sort of distribution ...
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1answer
111 views

Basics for estimation/prediction based of historical data

this is a very basic question. For my master thesis I need to estimate the power consumption for the current month. I have a lot of historical data of the power consumption. I have data for every ...
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1answer
41 views

Overestimate of $|\oint_{|z|=R} f(z) \mathrm{d}z|$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1\mathrm{with} \;a \in \mathbb R$

How can I overestimate, $|\oint_{|z|=R} f(z) \mathrm{d}z |$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1 \; \mathrm{with} \;a \in \mathbb R$ ? I tried this: $|\oint_{|z|=R} f(z) \mathrm{d}z | <= ...
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0answers
103 views

unbiased estimator of the area of the circle

the radius of a circle is measured with an error of measurement which is distributed normal with mean $0$ and variance $\sigma^2$,$\sigma^2$ unknown.Given $n$ independent measurements of the radius , ...
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0answers
20 views

How do I compute the variance (or confidence interval) of a Maximum Spacing estimator?

I am trying to solve a problem using a Maximum Possible Spacing estimator (see Maximum spacing estimation on wikipedia for links). Details on what I am trying to do can be found in the following ...
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4answers
54 views

Can a sampling based method estimate how many species exist?

I've got in to a bit of a debate online and I'm hoping some people here can help clear it up. The position I'm arguing against is "It's impossible even come up with a ballpark estimate for how many ...
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1answer
69 views

Advanced urn problem

Imagine there are two urns — urn A and urn B. Urn A contains 3 blue balls and 7 red balls. Urn B contains 7 blue balls and 3 red balls. Balls are now randomly drawn from one of these urns where the ...
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1answer
462 views

How can I show that sample mean has the smallest variance?

Let the population distribution is $N(\mu,1)$. Sample mean: $\bar{X_n}=\frac{\sum_{i=1}^{n} X_i}{n}$ Then $E(\bar{X_n})=\mu$ and $V(\bar{X_n})=\frac{1}{n}$ It is an unbiased estimator, and as $n ...
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1answer
232 views

How many iterations of Taylor series for n correct decimal digits

I'm using Taylor series to estimate trigonometric functions. So I need to know exactly how many iterations of Taylor series (say for sine) are needed for n decimal digits precision? (I'm writing a ...
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1answer
30 views

Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the ...
2
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1answer
369 views

Error Term for Fourier Series?

Suppose I have a piecewise smooth $2 \pi$-periodic function $f$ on $\mathbf{R}$ with a Fourier series $\sum_{n \in \mathbf{Z}}a_n e^{inx}$, a number $x_0 \in \mathbf R$, and $N>0$. I would like an ...
0
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1answer
146 views

Method of Moments, MLE, and Estimation Question

This is just a practice question. Not a take-home exam or a homework or an extra credit. It is not related with course work at all. Can anyone please give me detailed solution? Thank you
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2answers
73 views

Do prior hyperparameters update as you take successive measurements in the case of Gaussian unknown mean?

I am trying to use conjugate priors to estimate the mean $\mu$ of a Gaussian with known variance, $\sigma^2$. Derived was that the choice of prior should be: $p(\mu) = N(\mu | \mu_0, \sigma_0^2)$ ...
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1answer
55 views

How can I estimate the Euclidean distance?

I read in an article the Euclidean distance formula can be estimated with about 6% relative error with the following formula. Would you please why this is true and where can I find such estimations? ...
2
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1answer
146 views

Likelihood of a Uniform Distribution

I have been looking at this solution for two days and still can't understand the solution. The question is as follows: Given $w[i], i = 1, 2, \ldots, N$ are IID following a distribution of $U[0, ...
2
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1answer
54 views

Estimating the number of edges in a subset-poset in terms of the total number of elements

Let $S$ be a finite set of sets, with $N = \sum_{s \in S} |s|$ the 'total cardinality' of $S$; i.e., the sum of the cardinalities of all the sets in $S$. Now, consider the poset $(S, \subset)$ of $S$ ...
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1answer
75 views

Estimate divergence by gradient in H1

I am currently trying to fully understand the stationary Stokes equations of incompressible fluid. In the mixed form (homogeneous boundary data), for $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, a ...
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1answer
49 views

Order of magnitude of a variable.

What will be the order of magnitude of a variable whose value varies between 0 and 1? And why?
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0answers
43 views

Approximating arccos(a/(a+x)) for the sake of simplfying an integral

I recently tried to evaluate $$\int e^{\beta\arccos(a/(a+x))}dx$$ (everything constant except $x$) and got a complicated answer involving a hypergeometric series with complex arguments. Can anyone ...
2
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1answer
74 views

Inverse of the German Tank Problem?

I have a problem that maps to estimating the discrete distance to a goal. The sample space is n discrete positions on a circle labeled sequentially; n is known. A target position is randomly ...
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1answer
95 views

Find $\sin(1/10)$ to within error of $10^{-7}$

The maclaurin series of $\sin(x)$ is $x- x^3/3! + x^5/5! - \cdots + (-1)^n x^{2n+1}/(2n+1)!$. My teacher wants me to use Taylor's inequality theorem on page 607 to solve this problem. I know that ...
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2answers
183 views

How to calculate this ratio for use in pro-rata forecasting

This is really a very simple question, it's more the understanding I need rather than a simple answer. If I have two arrays, A with 10 elements (A1, A2, ...), and another, B, with 5 elements (B1, B2, ...
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2answers
242 views

Estimate a value knowing the values of: the function, the derivative and the second derivative in 0

Please suppose you have an unknown function r(x). This function r(x) is defined in the range: [-5; 5] You know that: r(0) = 1; r'(0) = -1; r"(0) = 1. Please estimate the value of r(x) in the ...
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2answers
161 views

Maximum likelihood function (MLE) for Levy distribution

I am a student who is writing a little thesis belonged in the applied mathematics category. I choose a "Levy distribution" defined as, \begin{equation} \lambda(t;u,c) = \begin{cases} ...