For questions about estimation and how and when to estimate correctly

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15 views

What is the distribution for this function?

Suppose we let X1, X2,...,Xn be an IID random sample of observations on the random variable X. So what my question is: Assuming that X ~ (μ,σ^2), find the distribution of √n(μ̂-μ)/σ. This is a ...
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1answer
15 views

On the estimation of the average of a deteriministic, scalar, real-valued function of two variables.

Suppose we have a scalar, real-valued function of two variables $F(x,y)$ where ($x$,$y$) belongs to a discrete domain $D$ of $F$, which has finite number of elements. Let $\mid D \mid$ be the number ...
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1answer
170 views

Exponential decay estimate for $(x,t) \in U_T$

Suppose $u$ is a smooth solution of $$\begin{cases}u_t - \Delta u + cu = 0 & \text{in }U \times (0,\infty) \\ \qquad \qquad \quad \, \,u=0 & \text{on } \partial U \times [0,\infty) ...
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1answer
137 views

Exponential decay estimate

Assume $u$ is a smooth solution of $$\begin{cases} u_t - \Delta u = 0 & \text{in }U \times (0,\infty) \\ \qquad \quad u=0 & \text{on }\partial U \times [0,\infty) \\ \qquad \quad u = g ...
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1answer
26 views

Sample Size Estimation with unknown variance

After spending a lot of time I'm still nowhere. What I have: Let d denote the desired margin of error so that the interval for µ is of width 2d. Let (1 − α) denote the probability associated with ...
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1answer
40 views

probability, unbiased median?

I am trying to work a problem from the book. Problem: As an alternative to imposing unbiasedness, an estimator's distribution can be "centered" by requiring that its medium be equal to the unknown ...
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1answer
99 views

The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$

We have $n$ sensors $X_i$ which estimate the scalar value $\alpha$ with different relative accuracies $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
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1answer
24 views

Two chips drawn from urn, calculate $P(|\hat\theta - 3| > 1.0)$.

Two chips are drawn without replacements from an urn containing 5 chips, numbered 1-5. The average of the two drawn is to be used as an estimator, $\hat\theta$, for the true average of all the chips ...
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1answer
37 views

Likelihood and maximum likelihood

what is the likelihood, log-likelihood and MLE of; $$θ(θ+1)x^{θ−1}(1−x)$$ any help greatly appreciated
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1answer
39 views

Is it possible to estimate $e$ based on $N$?

Consider a sequence of random numbers $u_1,\dots,u_n$ obtained from a continuous distribution $F$. Let $N$ be the first one that is greater than its immediate predecessor. In othe words, ...
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0answers
9 views

Estimating mortality rates when $n_t=0$

I am using a series of observations to calculate mortality rates: $m=1-(n_t/n_0)^{1/t}$ $n_0$ number of individuals in the first observation $n_t$ number of individuals at time $t$ I am comparing ...
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1answer
42 views

Find the MLE estimator for θ

Let $Y_1 ,Y_2 ,...,Y_n$ be a random sample from a distribution with pdf $f(y) = e^{-(y -θ) }$ for $y \geq 0 $ and $0$ else a) Find the Method of Moments estimator for θ b) Find the MLE estimator ...
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0answers
27 views

Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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0answers
42 views

Estimate uncertainty of a function

I have a question about estimate the uncertainty of a function. I have a variable $x$. I assume that I can predict the variable $x$ is followed by a function $f(x)$ such as $$f(x)=\exp(-x^2)$$ Now, ...
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1answer
36 views

Compound distribution with unknown distribution of its hyperparameter

Suppose $X\sim \mathcal{N}(0,\sigma)$, and $\sigma$ is another random variable in a sense that we only know that it is some constant random variable with finite support, i-e $\sigma \in [\sigma_\max, ...
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3answers
78 views

Estimate the number of typos there are in a book, based on two editors' finds

This is one question from an interview I have just taken: Suppose there is a book full of typos. Tom and Jerry found $x$ and $y$ typos throughout the book, respectively. There are $z$ typos that ...
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0answers
39 views

Why is my approximation of this alternating series incorrect?

I've been working on some calc problems and I'm stuck on the second part of a problem consisting of estimating the value of a series with a given error. I tried calculating it by hand.. wolfram, ...
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0answers
38 views

Estimating $\int_0^1\sin(x^2)dx$ using the Taylor expansion of $\sin(x)$

Problem: a) Find the Taylor polynomial $T_6(x)$ for $f(x) = \sin(x)$ about $x=0$. I found this to be $x-\frac{x^3}{6} + \frac{x^5}{120} + O(x^6)$. b) Use this to estimate $\int_0^1\sin(x^2)dx$ with ...
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0answers
13 views

Optimum delay of complex signals for LS minimization

Say I have a complex periodic signal $s(t)$ with period $T$. Note that $s(t)$ is not necessarily a complex sinusoide. Now I define $h(t)$ as \begin{equation} h(t) = \sum_{i}^{N} s(t - \tau_{i} ) ...
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0answers
9 views

How to do density estimation restricted to a linear model (multidimensional plane)?

I would like to fit a plane to sampling data. So I would like to do a density estimation but restricted to linear models on the main domain. Is there any standard method? This is also a problem in ...
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1answer
94 views

Deduce the following estimate

Let $u$ be a smooth solution of the uniformly elliptic equation $Lu=-\sum_{i,j=1}^n a^{ij}(x)u_{x_i x_j}$ in $U$. Assume that the coefficients have bounded derivatives. Set $v:=|Du|^2+\lambda ...
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0answers
17 views

Decomposing change in estimates

In this scenario, an estimate of a variable is given by the observed data multiplied by a representative weight. Let the observed data at the current time be $d_t$ and the weight $w_t$. The difference ...
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4answers
158 views

What is the smallest positive integer that has never been mentioned?

The set of positive integers is infinite. The set of explicitly mentioned positive integers is finite. Therefore, there is a non-empty set of positive integers that have never been mentioned, and ...
2
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1answer
45 views

Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
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0answers
34 views

Covariance matrix estimation

I will talk about the estimation of an unknown covariance matrix from a sample (of N points) when: The mean vector (MV) is known. The mean vector is unknown. In the case of when the mean vector is ...
2
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0answers
47 views

How to estimate parameters of exponential functions?

I am a new bird to math! I have an expression as follows, $$e^{-ux}+e^{-uy}=z$$ I have at least ten pairs of $(x,y,z)$, so how can I estimate the value of $u$? And how to evaluate my results? Hand ...
2
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2answers
171 views

Approximate the second largest eigenvalue (and corresponding eigenvector) given the largest

Given a real-valued matrix $A$, one can obtain its largest eigenvalue $\lambda_1$ plus the corresponding eigenvector $v_1$ by choosing a random vector $r$ and repeatedly multiplying it by $A$ (and ...
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1answer
57 views

estimation of the difference between two step functions by 1/sqrt(n)

Given are $ g_n ~and~ k_n$ two step functions, such that for $f\geq0$, which is Riemann-integrable the following holds: $0\leq{g_n}\leq{f}\leq{k_n}$ We diefined Riemann's integrability of $f$ in our ...
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7answers
202 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
2
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0answers
43 views

Verify that the unbounded function belongs to $W^{1,n}$ [duplicate]

Verify that if $n > 1$, the unbounded function $u = \log \log \left(1+\frac 1{|x|}\right)$ belongs to $W^{1,n}(U)$, for $U=B^0(0,1)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 14. ...
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1answer
40 views

Show that the variance is biased

I am trying to understand the proof that the uncorrected sample variance is biased (given here) $$ \begin{eqnarray} E[S^2] &=& E \left [ \frac{1}{n} \sum \limits_{i=1}^{n} (X_i - \bar ...
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1answer
143 views

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise ...
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2answers
59 views

Robust estimator

What does it mean that an estimator is robust? How can you tell whether an estimator is robust or not in statistics? I need to discuss whether the maximum likelihood estimators of the normal ...
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0answers
24 views

Double mass analysis and statistical errors

Double mass analysis The normal field of application for double mass analysis is to find out inconsistencies between sets of data. Say you have two sets of data $x_1,\dots, x_n$ and $y_1,\dots,y_n$. ...
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1answer
75 views

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

Assume $n=1$ and $u \in W^{1,p}(0,1)$ for some $1 \le p < \infty$. (a) Show that $u$ is equal a.e. to an absolutely continuous function and $u'$ (which exists a.e.) belongs to $L^p(0,1)$. ...
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2answers
80 views

Asymptotic Maxwell MLE distribution

Consider i.i.d. random samples $X_1,...,X_n$ from the Maxwell Density: $$ f_\theta(x)=\sqrt{\frac{2}{\pi}}\dfrac{x^2}{\theta^3}e^{-\frac{x^2}{2\theta^2}}I_{(0,\infty)}(x) $$ with $\theta > 0$. ...
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1answer
40 views

Getting Distribution Information from given random Sample - using Histograms

as the title mentions it I'm having some trouble understanding the following. Assume we have $X_1 \dots X_n$ i.i.d. random variables with a known distribution and plotted histogram. For Example, use ...
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0answers
45 views

L2 error of a convection-diffusion equation

I am given a pde of the following type: \begin{align} -\Delta f + \vec{a}\cdot \nabla f + f &= g\quad \text{in }\Omega:=B_1(0)\subset\mathbb{R}^2\\ f &= 0\quad \text{on }\partial\Omega ...
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1answer
53 views

Obtaining this estimate

How do I obtain this following estimate: $$\max_{0\le t \le T} \| \mathbf{u}(t) \|_{L^2(U)} \le C(\|\mathbf{u}\|_{L^2(0,T;H_0^1(U))}+\|\mathbf{u'}\|_{L^2(0,T;H^{-1}(U))}), \tag{10}$$ the constant ...
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0answers
45 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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1answer
29 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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1answer
30 views

Checking the consistency and Bias of $\frac{\sum X_i +\sqrt{n}/2}{n+\sqrt{n}}$

Let $X_1,\ldots,X_n$ be i.i.d. $B(1,\theta)$ random variables, $0<\theta<1$. Then, as an estimator $\theta$, check if $T(X_1,\ldots,X_n)= \dfrac{\sum_{i=1}^n X_i +\sqrt{n}/2}{n+\sqrt{n}}$ is ...
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0answers
46 views

Cramer-Rao bound for a parameter that can take only a finite set of values

My question is related to the bound on the variance of an estimation of a parameter that can take only a finite set of values. I copy and paste what is written on wikipedia to have a common starting ...
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2answers
112 views

Is this estimator biased?

I am struggeling to understand the how an estimator is arrived and whether it can be determined it is biased or not. I have this example Let $X_1 , X_2 ,\ldots, X_7$ denote a random sample from a ...
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4answers
298 views

A calculation that goes awfully wrong if we let $\pi=22/7$

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation ...
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1answer
78 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
2
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1answer
48 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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0answers
50 views

equivalent inner-product vector for one

I have a map that projects a $k$ dimensional vector $x$ to an $m$ dimensional vector $\phi(x)$. The vector function (map) $\phi$ can be any linear or non-linear function of $x$, which is not ...
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1answer
29 views

how to use the maximum likelihood function to acquire the maximum likelihood (ML) estimate

I'm reading this book Tracking and Data Association. The author gives an example for estimating an unknown non-random variable $x$ given some observations $z_{j}$ corrupted by Gaussian noise $ w \sim ...
1
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2answers
61 views

How to find smallest integer which is greater than N positive primes

I know this can't be computed exactly, but I just need a rough estimate. I know one can compute a rough estimate of the number of primes less than N using the famous formula: ...