The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
2answers
12 views

upper bound on generalized eigenvalues

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computation, i was trying ...
1
vote
0answers
25 views

Statistics question: Estimating mean when standard deviation is known

I am reading a textbook to learn more about statistics. This section is about estimating the mean of a population when standard deviation of the population is known. My simple question is this: How ...
-2
votes
0answers
12 views

Estimator of random process [closed]

$ \begin{cases} Y_1(t)=S(t)+N_1(t) \\ Y_2(t)=\frac{dS(t)}{dt}+N_2(t) \end{cases} $ It is impossible to send $S(t)$ through a low pass filter. $N_1(t)$ and $N_2(t)$ are both white noise and ...
1
vote
0answers
22 views

When does l1 regularisation give a sparse solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...
1
vote
0answers
14 views

Bias of the MLEs for the two-parameter Weibull distribution

Is it possible to obtain a formula for or an equation on the exact bias of the MLE-vector for the two-parameter Weibull distribution (both parameters unknown). I've read papers offering Monte Carlo ...
2
votes
1answer
60 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
0
votes
0answers
16 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
0
votes
1answer
44 views

How to estimate $\sum_{x=1:n}{xf(x)}$ having $\tilde{f}$

I have an estimator $\tilde{f}(x)$ whose error is at most $\epsilon$, i.e., $\frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \leq \epsilon$. I want to estimate $\sum_{i=1:n}i.f(i)$ with a small error. But if I ...
3
votes
1answer
33 views

Oscillations about equilibrium for coupled differentail equations

I have the following system of equations: $$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$ I would like to study the property of solutions to this function about ...
0
votes
0answers
33 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable?

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
1
vote
1answer
40 views

Stratified random sampling without replacement

I came across this statement and can't decide if it's true or false. Statement: In a stratified random sampling without replacement, with proportional allocation to the population size, the sample ...
2
votes
2answers
57 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
4
votes
1answer
66 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
27 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
0
votes
0answers
21 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
0
votes
1answer
35 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
0
votes
0answers
20 views

How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
0
votes
1answer
33 views

Kalman filter innovation residual inversion

I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume ...
0
votes
0answers
23 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
0
votes
1answer
31 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
1
vote
0answers
22 views

Improving Schauder estimate for a linear elliptic PDE with oblique boundary

Let $\Omega \subset \mathbb R^n$ a $C^{2,\alpha}$ domain, $f \in C^{0,\alpha}(\overline{\Omega})$, $g \in C^{1,\alpha}(\overline{\Omega})$, $h \in C^{1,\alpha}(\overline{\Omega};\mathbb{R}^n)$ such ...
0
votes
1answer
116 views

why is the answer 21,845 and not 218,450?

How can you tell whether $$\frac{(250,000)(5.47)}{6.26}$$ is closer to 21,845 or 218,450 without calculating it exactly? Thank you.
2
votes
1answer
57 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
0
votes
0answers
15 views

regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error term. We know covariance matrix ∑ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would ...
0
votes
1answer
47 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
0
votes
0answers
11 views

Lp estimates from Elliptic Equation

Using the theorem: Let $f \in L^{p}(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$, $w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy$. Then $w\in W^{2,p}(\Omega), \Delta w=f$ a.e and ...
0
votes
0answers
14 views

Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
1
vote
1answer
18 views

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't ...
0
votes
0answers
30 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
2
votes
0answers
94 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
0
votes
1answer
105 views

Test if estimator is unbiased

I'm having problems with the following question for my econometrics homework. Is $\ \ \hat \beta_2 = (y_n - y_1)/(n - 1)\ $ an unbiased estimator of $\beta_2$ for $\ \ y_t\ =\ \beta_1\ +\ \beta_2 \ ...
1
vote
1answer
97 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
0
votes
0answers
27 views

Estimating “task duration” variance through questions in a questionnaire

I am working to estimate the average task duration for complex processes - a total of 19 tasks / processes (each task made of several subtasks of varying duration). I will be organising a ...
0
votes
1answer
22 views

Show that $\hat{\delta}_1=\hat{\beta}_1+(X_1^T X_1)^{-1} X_1^TX_2\hat{\beta}_2$

Let $\hat{\beta}=(\hat{\beta}_1,\hat{\beta}_2)^T$ be the least squares estimator in the regression model $Y=X_1\beta_1+X_2\beta_2+u$. Let $\hat{\delta}_1$ be the least squares estimator of the ...
0
votes
0answers
33 views

Linear model: Show that $\hat{\theta}$ and $\hat{e}$ are independent

Show that under the assumptions $Y\sim N(X\theta,\sigma^2I_n)$and $\text{rang}(X)=\text{rang}(\theta)$ the residual vector $\hat{e}$ and the least squares estimator $\hat{\theta}$ are ...
0
votes
1answer
36 views

Estimating the Growth Rate of Worms

I would like to know how much worms I'd have on hand given an initial amount and period of time. Here are some metrics regarding the growth rate of these worms. ...
5
votes
1answer
97 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
2
votes
1answer
30 views

Estimating sum of n elements by throwing away half of elements

I've got a task where i need to proove the asymptotic big-Theta equation: $$ \log n! = \Theta(n \, \log n) $$ $ \ $ Since $f(\mathit{n}) \in \Theta(g(\mathit{n}))$ means that $g(n)\cdot k{_1} \leq ...
0
votes
0answers
13 views

Estimating the accuracy of precipitation odds.

Modern weather forecasts often include a percentage chance of rain. If this estimate was a constant it would be a simple matter of calculating the ratio. But the reality is quite another. This is the ...
2
votes
1answer
48 views

Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$ I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x, $$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
1
vote
3answers
54 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
0
votes
1answer
28 views

Maximum-Likelihood Estimator: What problems occur if data is not i.i.d.?

This is a question from an exam: You want to estimate the parameters for a gaussian distribution using the Maximum-Likelihood Method for an i.i.d. set of data. What role does the property ...
1
vote
1answer
23 views

show that the sample median of normal distribution is median unbiased

$X_i \sim N(\mu,\sigma^2)$, $i = 1,2,\ldots,n$. Show that the sample median is unbiased in median for $\mu$. I have obtained the pdf of sample median for $n=2m+1$ as: $$f(x_{(m+1)}) = \frac ...
0
votes
0answers
15 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers
1
vote
0answers
60 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
0
votes
1answer
20 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
1
vote
1answer
21 views

How to estimate the lower bound of a given Toeplitz matrix's eignvalue?

A given Toeplitz matrix is $$\left( \begin{matrix} 1 & a & a^2 & \cdots & a^n \\ a &1 &a & \cdots & a^{n-1} \\ a^2&a & 1 & \cdots& \cdots ...
1
vote
1answer
69 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
2
votes
1answer
193 views

What initial guess is used for finding n-th root using Newton-Raphson method?

I would like to know what is an optimal initial guess for use with Newton-Raphson method when finding n-th root. I develop some program which uses GMP C++ library. GMP manual says: The initial ...
1
vote
2answers
39 views

Estimate of L2 norm of a product of functions

Assume that $f,g, f\cdot g\in L^2(\mathbb{R}^n)$. Is the estimate $\|fg\|_{L^2(\mathbb{R}^n)}\leq \|f\|_{L^2(\mathbb{R}^n)}\|g\|_{L^2(\mathbb{R}^n)} $ correct? Thank you!