For questions about estimation and how and when to estimate correctly

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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 ...
0
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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
65 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
vote
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 ...
0
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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
42 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
28 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 ...
0
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0answers
35 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 ...
0
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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
votes
2answers
35 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 ...
0
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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 ...
-2
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1answer
60 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 ...
0
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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: ...
1
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2answers
22 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
119 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 ${\lambda_n}=n^2\pi^2+\int_{0}^{1} q(t)\,dt +c_n \qquad \text{for} \quad n\rightarrow \infty$ ...
1
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1answer
30 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 $\beta$ ...
0
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1answer
30 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
16 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
vote
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
47 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
votes
2answers
40 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 ...
0
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0answers
19 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
85 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
31 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
votes
3answers
110 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
20 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 ...
3
votes
2answers
168 views

Finding an error estimation for the De Moivre–Laplace theorem

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...
1
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1answer
37 views

$\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx$

Let $\lambda,\mu$ be positive constants. Let $I_{\epsilon,\eta}=\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx\:\:\epsilon,\eta >0$. Show that, by integration by part, ...
1
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0answers
49 views

Estimate on the exponential integral of a complex argument (a reference needed)

Consider the exponential integral of the complex argument defined by $$ Ei( z ) = \gamma + \ln(-z) +\sum\limits_{ n = 1 }^{ \infty } \frac{ z^n }{n n!}, $$ where $ z \in \mathbb{C} \backslash ( ...
2
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1answer
65 views

Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that ...
1
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0answers
16 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
1
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1answer
33 views

Improving estimatied parameters of a known distribution

Assume there is a Set of data which follows a known distribution (e.g. normal distribution). $$S = \left\{ a_0,a_1 ... a_n \right\}$$ When taking a subset from S $$S_k = \left\{ a_0,a_1 ... a_k ...
0
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0answers
20 views

Histogram estimator, why dividing by smoothing parameter?

In histogram estimator: $$\widehat{f}(x)=\frac{1}{nh} \sum_i^n I\left(-\frac{1}{2} \le \frac{x_i-x}{h} \le \frac{1}{2}\right)$$ why do we have to divide by smoothing parameter $h$ in first term ...
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0answers
22 views

Is this estimation of an double integral right?

I'm in the setting, where $\Omega$ is a bounded domain, $t_0 > 0$ is fix and I have $g, \ h \in L^\infty((0,t_0);L^1(\mathbb{R}^n \setminus \Omega))$. So this should say, that $g$ and $h$ are ...
0
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0answers
33 views

Big Oh notation problem

Retaining the three terms in the series, estimate the remaining series using "Big Oh" notation with the best integer value possible, as $x\to 0$. The series is $$\ln (\tan (x)) =\ln(x)+ ...
0
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0answers
25 views

Variable value estimation for given product/fracture values

I have a data set (time series) with given values for certain fractions xy = x/y (where x,y are not constant over time) Thus, there are following fractions: AB = A/B CB = C/B AD = A/D CD = C/D AE = ...
2
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0answers
38 views

Method of moment estimator. Deconvolution

The distributions $Y, X, Z$ and $W$ are related as follows: $$Y_1 = X + Z$$ $$Y_2 = X + W,$$ that is $X$ (random variable) is a common factor to the random variables $Y_1$ and $Y_2$, which ...
0
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0answers
10 views

How to interpret the Quantified properties of estimator?

https://en.wikipedia.org/wiki/Estimator The link provides a very good explanation of the estimator. I am beginner to statistics and inference , so i have some confusion about the quantified properties ...
1
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1answer
27 views

Sufficient statistics problem

$X_1, X_2, \ldots, X_n$ are iid $N(0,\theta), 0 < \theta < \infty$ Show $$\sum_{i=1}^{n} X_i^2$$ is a sufficient statistic for $\theta$. My attempt at this is $S = (X_1^2 + ...
4
votes
1answer
43 views

Cramer-Rao lower bound for any unbiased estimator

The first part of a question I am trying to solve asked to find the maximum likelihood estimator for $\theta$ for a pdf $f_X(x)=\frac{2x}{\theta^2}$, $0 < x \le \theta$ , $0$ otherwise. ($X_1, ...
0
votes
0answers
22 views

Minimum mean square estimator for closing value of the NYSE

I am attempting to develop an estimator for the closing value of the NYSE $x(n)$ based on previous $N$ closing values, $x(n-1), x(n-2), ... x(n-N)$. I want to find the Minimum Mean-Square estimator ...
1
vote
1answer
140 views

Cost Function of Neural Network (Forward Propagation)

This question is related to Andrew Ng's machine learning course on Coursera. Basically, when I calculate the cost function of a neural network, I use the following formula that was described by Ng: $$ ...
1
vote
1answer
118 views

Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T ...