For questions about estimation and how and when to estimate correctly

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1answer
43 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
116 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
31 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
8 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
43 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 ...
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1answer
18 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 ...
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1answer
69 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.
2
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1answer
40 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
42 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 ...
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1answer
34 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 ...
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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
24 views

Statistics application problem - estimate number of items by weight

An experiment was conducted by weighting 20 sets of items with known quantity and the weight of items in each trail were obtained. We also know the weight of each item is supposed to be $W$ kg ...
2
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1answer
43 views

$\frac{1}{x^{\alpha}(\log x)^{\beta}}$ in $L^p((1,\infty))$?

Consider $$f(x)=\frac{1}{x^{\alpha}(\log x)^{\beta}},$$ $\alpha,\beta\in\mathbb{R}$. For which $\alpha,\beta \in \mathbb{R}$ is $u \in L^p((1,\infty))$, $1\le p\le \infty$? For $p=1$ I know how to do ...
1
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1answer
42 views

How to Find the Linear Approximation of $\ln(8-4x)$ at $x = 7/4$, and Use it to Estimate $ln(0.99)$

I am trying to determine how to find the linear approximation of $\ln(8-4x)$ at $x = 7/4$, and use it to estimate $\ln(0.99)$. So far, I have made the following steps: 1) Find the derivative of ...
0
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1answer
24 views

Estimation method starting with too big and too low values

I not sure to what field exactly this question belongs, but math/statistics seemed closest to me. So here we go: It is a method of estimating a value that informally goes like this (bear with me). ...
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0answers
14 views

Series Reversion for $n$ power series

I have $n$ functions with power series representation as $F_i(X)=\sum_{k_1,\dots k_m}a^{i}_{k_1,k_2,\dots k_m}x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$, where $X=[x_1,x_2,\dots,x_n]$ and ...
0
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1answer
26 views

MLE of β in the gamma distribution?

So I have the pdf for the gamma distribution, $$f(x) = \frac{1}{\Gamma(\alpha)} \beta^\alpha x^{\alpha - 1} e^{-\beta x} $$ and I'm having trouble getting to the MLE of $\beta$, which should be ...
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1answer
51 views

Generalizing Algebraic Problem

Molly went to the store to purchase ink pens. She found three kinds of pens. The first cost 4 dollars each; the price of the second kind was 4 for 1 dollar, and the cost for the third kind was 2 for 1 ...
0
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1answer
19 views

MLE for the mean of a distribution?

Let $X_1, \ldots, X_n$ be a random sample of size $n$ from a distribution having the pdf $$f(x)=\frac{2x}{\theta^n}.$$ I need to find the MLE for the mean of the distribution but am not sure how.
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4answers
66 views

Prove that $x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\geq\sqrt{2}$

Let $(x_n)_{n\in\mathbb N}$ be a recursively defined sequence with $x_1=9$ and $$x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\text{ for }n\geq 1.$$ Show that $x_n\geq\sqrt{2}$ for all $n$. Because ...
1
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1answer
19 views

estimates on an improper integral associated with normal distributuion

Show that $\int_{x}^{\infty}e^{-\frac{t^2}{2}}dt\geq e^{-\frac{t^2}{2}}(\frac{1}{x}-\frac{1}{x^3}) $ for all positive $x$ Does it require the mean value theorem, or the Taylor series expansion? It is ...
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3answers
38 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,\ ...
1
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1answer
31 views

MLE of uniform distibution again

I've struggled for hours with a seemingly simple problem, I'm supposed to compute the MLE for $\theta$. We have $(y_1, y_2...y_n)$ obervations with a uniform distribution. The density function is as ...
1
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1answer
43 views

Deriviative estimates in finite $L^p$ space

I'm stuck as to how to solve the following exercise: If $U$ is open and bounded with smooth boundary, $1<p<\infty$, $\epsilon>0$, and $u\in C^{\infty}(\bar U)$, show $\exists C$ s.t. ...
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0answers
29 views

Exponential Family, derivative of Marginal Likelihood zero at MLE

My question refers to the proof of theorem 6.3 in Lehmann/Casella (1998): Theory of point estimation. We have a Bayes setting: $ X_i \mid \eta \stackrel{ind.}{\sim} f_i(x_i \mid \eta),\quad i = 1, ...
2
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0answers
43 views

Ratio estimator in sampling

Let the population $U=(1,2,3)$. We want to estimate $R=\frac{\mu_y}{\mu_x}$.Consider the estimators $$\hat{R_1}=\frac{\overline{y}}{\overline{x}},\hat{R_2}=\frac{\overline{y}}{\mu_x}$$ where ...
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0answers
38 views

How to propagate uncertainty into the prediction of a neural network?

I have inputs $x_1\ldots x_n$ that have known $1\sigma$ uncertainties $\epsilon_1 \ldots \epsilon_n$. I am using them to predict outputs $y_1 \ldots y_m$ on a trained neural network. How can I obtain ...
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0answers
6 views

How to find the partial likelihood estimates of $\rho_1$ and $\rho_2$?

I am having problems with the following exercise: I am given a model $$ X_t = \rho_1 X_{t-1} + \rho_2 V_t + \epsilon_t~~~~for~t\in \mathbb{N}$$ We have the intial value $X_0=0$, $\vert \rho_1 ...
0
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2answers
41 views

Numerical approximation of a dx/dy derivative

I have to find numerical approximation of the derivative of dx/dy where y(x)=exp(sin^2(x)+cos(x)exp(x^2)) at the point Xo=0.5. As far as I understand, I have to pick a close point to X0 for example ...
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4answers
50 views

How do you determine how many digits of pi are necessary?

It is said that you only need to calculate pi to 62 decimal places, in order to calculate the circumference of the observable universe, from its diameter, to within one Planck length. Most of us are ...
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1answer
68 views

When to we accept a hypothesis when using Wald test statistic? [closed]

Hello I had to test two hypothesis, one hypothesis gave a wald test statistic with value 0.00015 and the other a value of approximately 40. Is it true when I then say that we accept the hypothesis ...
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0answers
4 views

How to computing Wald test in R based on what I have so far?

I have to simulate an AR(1) process with $\rho = 0.5$ and then estimate $\rho$ based on the first 100 $X_T$ values I did this the following way: ...
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2answers
23 views

When does it make sense to build a confidence interval for the mean with known standard deviation?

While estimating with confidence interval the mean value for a population, there are two options: If the standard deviation is known, and If the standard deviation is unknown. But in the first ...
0
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1answer
120 views

Upper bound for the error on the Fourier series for $e^{x}$

I have been given the following problem: Find the Fourier series for $e^{x}$ over the interval $-\pi \le t \le \pi$. Hence find the upper bound of its error. To spare me typing a huge expression and ...
1
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1answer
59 views

An asymptotic estimate for density of eigenvalues

this is the screenshot of the useful part of the cited book Let $\{\lambda_n\}$ be constants such that ${\lambda_n}=n^2\pi^2+\int_{0}^{1} q(t)\,dt +c_n \qquad \text{for} \quad n\rightarrow \infty$ ...
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1answer
32 views

problem using MLE on gamma distributed variable

I am making some kind of systematic error(s), while working with maximum likelihood estimations. Could someone please point these out to me? In my last assignment, I tried to find the MLE of $\beta$ ...
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1answer
32 views

Finding number pattern

I have a program which I originally thought was linear, meaning that if I had twice the data, the program would take twice as long to process. I was wrong, and it seems to be somewhat exponential. ...
0
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1answer
16 views

on a step of a proof of the Levinson density theorem

Let $n(r): \mathbb{R}\rightarrow\mathbb{N}$ be a monotone (increasing) function such that $\int_{1}^{r} \frac{n(u)}{u}du \leq \frac{1}{2}\log (r)+ A$ where $A$ is a certain constant. I should deduce ...
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0answers
18 views

Finding the closest vector to an observation

I have a collection of vectors (a codebook in hand) which are presented within a matrix $A$ $$ A = [ a (\theta_1), \, a (\theta_2), \, a (\theta_3), \, a (\theta_4), \,\cdots]$$ We have obtained ...
1
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1answer
42 views

Estimate of a (integral) function

I should show that function $H(w)=\int_{-\pi}^{\pi}f(x) e^{iwx}dx$, where $f(x)\in L^2(-\pi,\pi)$, is such that $H(re^{i\theta})=O(e^{\pi r |sin(\theta)|})$. Any suggestion?
1
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1answer
17 views

Required number of simulation runs

I have the following problem: One wants to estimate the expectation of a random variable X. A set of 16 data values (i.e. simulation outputs) is given, and one should determine roughly how many ...
1
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2answers
21 views

Significant figures addition/subtraction rounding?

I thought you round to the same place as the number with the addend with the least precision. For example, if you had $25.63+ 42.3$ the answer would be rounded to the tens place ($67.9$). However, my ...
1
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1answer
49 views

Estimate the Green function for the Laplace equation in 2D

The Green function for the Dirichlet problem for the Laplace equation in the unit disk in $\mathbb R^2$ has the following form: $$ G(x,y) = \frac{1}{2\pi}\ln \frac{|x-y|}{|x| \bigl|y - ...
3
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2answers
41 views

Is it possible to determine which yields a better approximation for $\pi$?

If I use the trapezoidal rule, using two equal partitions, to estimate $$\int_{0.5}^{1} \sqrt{1-x^2}dx$$ I can obtain an approximation of $\pi$, as the integral's exact value can be found without much ...
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0answers
20 views

Calculating a 95% confidence interval for µ

A test was conducted to determine the length of time required for joggers to run a particular path. The joggers were instructed to run the course at the maximum speed at which they could without ...
2
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1answer
88 views

Expectation of largest and smallest order statistic from uniform distribution

Given is a random sample of size n from a uniform distribution with parameters $-\theta$ and $\theta$, $\theta>0$. I'm asked to find a constant $c$ such that $c(X_{n:n}-X_{1:n})$ is an unbiased ...
0
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1answer
32 views

Deriving a bound on an integral

I'm trying to follow a result in a paper and I can't get it to work out. The result requires deriving the bound \begin{equation} \int_{\mathbb{R}^N\times\mathbb{R}_+} ...
2
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3answers
112 views

An interview question: $2.1^{3.1}$ vs $3.1^{2.1}$,$ 2.1^{4.1}$ vs $4.1^{2.1}$, which is larger?

While Mathematica told me that $2.1^{3.1} - 3.1^{2.1} = -0.786932$ and $2.1^{4.1} - 4.1^{2.1} = 1.58855$, I wonder how to compare them quickly, by hand. I see $2^3 < 3^2$, so perhaps we have ...
0
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0answers
21 views

When regularity conditions for the CRLB to hold are not met, can a lower bound be computed numerically with the MLE?

I have a probability density $p(\hat{d}|d)$ such that $\int_d^\infty p(\hat{d}|d) = 1$ where $d$ is the parameter to be estimated based on observations $\hat{d}$ so the regularity conditions for a ...