For questions about estimation and how and when to estimate correctly

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3
votes
1answer
61 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
1
vote
0answers
22 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
1
vote
0answers
866 views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
1
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0answers
50 views

what is the bias of an estimator

The point estimator $\hat\theta$ of a parameter $\theta$ is some function of the sample $D=\{x_1,...,x_n\}$, $$\hat\theta=g(D)$$, since $\hat\theta$ depends on the sample $D$ we're using, so ...
0
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0answers
41 views

What is $E[T^2]$ of $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$ ??

What is second moment i-e $E[T^2]$ of random-variable: $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$, Where $X[n]$ and $W[n]$ are both 'independent' of each other and 'stationary'. Moreover, ...
0
votes
1answer
36 views

How to find $X_i$ from this equation

Suppose $X_i=\nu_i+\frac{m-i}{m}X_{i+1}+\frac{i}{m}X_{i-1},\quad 1\le i\le m$ where $X_0=X_{m+1}=0$. I need to find an expression for $X_i$ in terms of $v_i$, $i$, and $m$. I know how to find it ...
2
votes
2answers
130 views

Finding the variance of a statistic.

$X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define $$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I have shown that it is an unbiased estimator of the ...
1
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0answers
22 views

Estimator of Absolute Error?

Given $X \sim B(n, p)$, we know that $\hat{p} = X / n$ is the obvious estimator for unknown parameter $p$, and the following quantity $$\frac{\hat{p}(1-\hat{p})}{n-1}$$ has the property that its ...
1
vote
1answer
74 views

sine function description using three points

Is there any way to find the parameters of a sine wave ($A$, $w$, and $ \phi $ for $A \sin(wt+\phi)$ ) using just three points (samples)? Thank you in advance for your help.
3
votes
4answers
205 views

Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
0
votes
1answer
46 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...
1
vote
1answer
164 views

Variance of maximum likelihood estimator for discrete distribution

Lets say we have a discrete distribution with following probabilities: $P(X=0)=\frac{1}{3}\theta, P(X=1)=\frac{2}{3}\theta, P(X=2)=\frac{2}{3}(1-\theta), P(X=3)=\frac{1}{3}(1-\theta)$ Estimating ...
0
votes
2answers
81 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? ...
0
votes
0answers
92 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 $$ k(x,y):=(\arctan(\lVert x-y\rVert))^{-\alpha}\text{ for }x\neq y $$ ...
1
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0answers
98 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 ...
0
votes
2answers
32 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 ...
1
vote
1answer
28 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
votes
1answer
60 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 ...
0
votes
0answers
44 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 ...
1
vote
0answers
34 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 ...
2
votes
0answers
67 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 ...
1
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1answer
55 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?
1
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0answers
55 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?
0
votes
1answer
83 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 ...
0
votes
1answer
26 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
votes
0answers
67 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
vote
1answer
39 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
vote
1answer
20 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
votes
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
votes
0answers
27 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
100 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
votes
1answer
25 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
votes
1answer
16 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
vote
1answer
29 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
votes
1answer
114 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
vote
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
votes
0answers
137 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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vote
0answers
23 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
votes
4answers
55 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
vote
1answer
77 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
vote
1answer
536 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
293 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
votes
1answer
35 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
518 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
167 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
vote
2answers
84 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
votes
1answer
58 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
168 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
votes
1answer
56 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
vote
1answer
84 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 ...