For questions about estimation and how and when to estimate correctly

learn more… | top users | synonyms

1
vote
1answer
21 views

Given f(x) and two correlated random variables x & y, how do I estimate the correlation of f(x) & f(y)

I have a smooth continuous well-behaved function f(x), where f(x) is positive and mononically increasing with x, and x is positive real continuous variable. Given the mean, variance, and correlation ...
2
votes
1answer
45 views

Help show a statistic converges in probability given another statistic that converges in probability

Let $Y = (Y_1,\dots,Y_n)$ be a random sample from $N(\mu,1)$ and $\bar{Y}=\sum\limits_{i=1}^nY_i/n$ I am given that $\bar{Y}^2$ converges in probability to $\mu^2$ and now need to show that ...
0
votes
1answer
24 views

Determine an appropriate size of sample

Let's say I have a pool of $N$ balls, which can be of $n$ colors $A_1, \cdots, A_n$. $N$ is much bigger than $n$. What number of balls must I draw if I want to have a good estimate of $R_1, \cdots, ...
0
votes
0answers
22 views

Density - Excess Bunching - Bunching Estimator

Saez defines excess bunching at the kink as the area under the density in the dominated region: $$ B = \int^{z^*+d z}_{z^*} h(z)dz \approx h(z^*)dz^* $$ where income $z$ is distributed according to a ...
1
vote
1answer
39 views

How to figure out the respective sufficient statistic for a given vector of parameters?

Let $Y$ be a random sample from $N(\mu,\sigma^2)$ where both $\mu$ and $\sigma^2$ are unknown. Let $\theta$ be the vector of parameters of interest $\theta=(\mu,\sigma^2)$. I need to find the ...
0
votes
0answers
24 views

Show the linear combiantion is not sufficient for $p$

Let $X_1, X_2, X_3$ be a set of three independent Bernoulli random variables with unknown parameter $p = P(X_i = 1)$. Where it is given that $ \hat p = X_1 + X_2 + X_3$ is sufficient for $p$. Show ...
0
votes
1answer
26 views

Mean Square Error & Bias

The random variable $Y$ is related to the angle at which muon particles decay. Y has density function: $$f(y)=\frac{1+\alpha y}{2} \; \; -1 \leq y \leq 1$$ where $\alpha$ is a parameter satisfying ...
1
vote
0answers
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 ...
0
votes
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 ...
0
votes
1answer
134 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) ...
0
votes
1answer
106 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
37 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 ...
0
votes
1answer
97 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 $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
0
votes
1answer
22 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 ...
0
votes
1answer
34 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
0
votes
1answer
38 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, ...
0
votes
0answers
8 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 ...
2
votes
1answer
40 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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
41 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, ...
1
vote
1answer
30 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, ...
1
vote
3answers
65 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 ...
0
votes
0answers
37 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, ...
0
votes
0answers
37 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 ...
0
votes
0answers
12 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} ) ...
1
vote
0answers
8 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 ...
1
vote
1answer
78 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 ...
0
votes
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 ...
3
votes
4answers
136 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
votes
0answers
35 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 ...
1
vote
0answers
29 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
votes
0answers
39 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
votes
2answers
91 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 ...
0
votes
1answer
56 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 ...
2
votes
7answers
185 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
votes
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. ...
3
votes
1answer
39 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 ...
1
vote
1answer
109 views

Integrate by parts to prove this inequality

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 ...
0
votes
2answers
49 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 ...
0
votes
0answers
21 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 $y_1,\dots,y_n$. ...
1
vote
1answer
66 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 $L^p(0,1)$. ...
0
votes
2answers
75 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 $\theta > 0$. ...
0
votes
1answer
28 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 ...
0
votes
0answers
39 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 ...
2
votes
1answer
47 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 ...
1
vote
0answers
44 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) ...
1
vote
1answer
27 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 ...
1
vote
1answer
24 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 ...
0
votes
0answers
38 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 ...