For questions about estimation and how and when to estimate correctly

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0
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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.
3
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0answers
65 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: ...
1
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1answer
38 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 ...
1
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1answer
19 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
24 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 ...
0
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0answers
26 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
votes
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 ...
0
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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 ...
0
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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 ...
1
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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 ...
0
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1answer
113 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 ...
1
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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 | <= ...
0
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0answers
134 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 , ...
1
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0answers
21 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 ...
2
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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 ...
1
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1answer
76 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 ...
1
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1answer
506 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 ...
1
vote
1answer
276 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 ...
0
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1answer
34 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
votes
1answer
455 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
votes
1answer
161 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
1
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2answers
78 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: $p(\mu) = N(\mu | \mu_0, \sigma_0^2)$ ...
0
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1answer
56 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
votes
1answer
165 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
55 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 $S$ ...
1
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1answer
79 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 ...
0
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1answer
50 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?
1
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0answers
45 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
votes
1answer
76 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 ...
1
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1answer
102 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
218 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, ...
0
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2answers
278 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 ...
1
vote
2answers
177 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} ...
0
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1answer
94 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
1
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2answers
569 views

Biased/Unbiased estimator

I'm trying to solve a statistic exam and i got lost with this exercise. 1) Consider a sample from a continuos probability distribution with density: $$ f(x) = \begin{cases} (1+\theta x)/8 & ...
0
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1answer
127 views

square root estimator

Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots. This form also seems ...
1
vote
1answer
213 views

Using the estimation lemma

I have the question: Prove using the estimation lemma, for a function $f$ which is continuous in some region $D$ that: $\lim_{r \mapsto 0}\displaystyle\int_{\Sigma}\dfrac{f(z)}{(z-z_0)}\ dz = 2\pi ...
2
votes
1answer
176 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...
21
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8answers
933 views

Mental estimate for tangent of an angle (from $0$ to $90$ degrees)

Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable. I can ...
1
vote
1answer
324 views

Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the ...
0
votes
1answer
274 views

MLE of Poisson Variable

Consider a random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Find the MLE of $\theta$ and show that it is a consistent estimator. --We have ...
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2answers
76 views

Poisson Estimators

Consider a simple random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Let $T=\sum X_{i}$. Show that $\tilde{\theta}=[(n-1)/n]^{T}$ is an unbiased estimator of ...
0
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0answers
33 views

Confidence Interval for the Survival Probabiltiy

Let $X_{1},\dots,X_{n}$ be random samples from an exponential distribution with pdf $f(x)=\mu^{-1}\exp(-x/\mu)$ over $x\geq0$ ($0$ otherwise) with $\mu>0$. In this case, $\mu$ is the mean. Find ...
1
vote
1answer
311 views

Gaussian Curve Fitting - Parameter Estimation

I was redirected here because someone in SO pointed out this is more of a math question than a programming question: I have to fit a Gaussian curve to a noisy set of data and then take it's FWHM for ...
1
vote
2answers
2k views

Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. ...
5
votes
2answers
208 views

Naively estimating the factorial

A naive way to estimate the factorial is $n! \geq (a+1) (a+2) \dots n \geq a^{n-a}$ for any $a$. For example, it gives $n! \geq (n/2)^{n/2}$ and slightly better $n! \geq (n/3)^{2n/3}$. I am interested ...
1
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0answers
71 views

Estimation (Newton potential)

Let $f$ be a $C^2$-function on $\mathbb{R}^n$ with compact support. Let $N$ be the Newtonpotential on $\mathbb{R}^n$ and $u:=N\star f$ (i.e. a solution of the potential equation $\Delta u=f$). ...
0
votes
1answer
126 views

What is $\sum\ln{(x_i!)}$?

I started learning statistics and in my homework i should find the Maximum Likelihood Estimate. The function is $f_x(x)=e^{-\lambda n}\prod_{i=1}^n \frac{\lambda^{x_i}}{x_i!}$ Now i take the ...
1
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1answer
156 views

Evaluation/Estimation of a Gaussian integral

Is there a closed form expression for the following definite integral: $$ F(u) = \frac{1}{2}\int_{-u}^u e^{-\frac{\alpha^2}{x^2}-\beta^2 x^2}\,dx = e^{-2\alpha\beta} \int_0^u ...
2
votes
1answer
326 views

Method of Moments and Maximum Likelihood estimators?

The random variables $X_1,...X_n$ are independent draws from continuous unifirm distribution with support $[0,\theta]$. Derive a method of moments and maximum likelihood estimators of $\theta$. Your ...