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3
votes
0answers
62 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths ...
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} = ...
2
votes
0answers
39 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
2
votes
0answers
21 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
2
votes
0answers
62 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
2
votes
0answers
92 views

Approximability of continuous sampling by discrete sampling - definitions?

Are there standard definitions that express the notion that sampling from a given continuous random variable can be approximated "to any desired degree of accuracy" by sampling from an appropriately ...
2
votes
0answers
167 views

How can you calculate actual values when all you have is rolling averages?

Let's say you have a set of data that is rolling 6 month averages of the actual monthly data. Good data collection would mean you saved the actual values and then calculated the rolling averages, but ...
1
vote
0answers
27 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 ...
1
vote
0answers
23 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
15 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 ...
1
vote
0answers
23 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 ...
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} + ...
1
vote
0answers
23 views

Derive Maximum Likelihood Estimator of a Generalised Linear Regression Model

I understand how to find the MLE estimator for $b$ if it is a simple linear regression model. However, when $u\sim N(0,\sigma^2\Omega)$ where $\Omega\ne I$. I am getting confused. The model is: ...
1
vote
0answers
47 views

How to find the MLE of the mean of Gamma distribution

If I parameterize Gamma distribution in the way as $\Gamma(\alpha,\frac{\mu}{\alpha})$, am I able to find the maximum likelihood estimator of $\mu$. Here, $\alpha$ is the shape parameter, ...
1
vote
0answers
39 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
1
vote
0answers
39 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
1
vote
0answers
26 views

ML estimate of sum of guassian variables?

consider the sum $z=x_{1}+...+x_{k}$, where the scalar variables $x_{i}$ are statistically independent and Gaussian, each having the same mean $0$ and variance $\sigma^2_{x}.$ how can I construct the ...
1
vote
0answers
38 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
1
vote
0answers
11 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
1
vote
0answers
48 views

Convergence in Probability of an estimator

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
1
vote
0answers
68 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
1
vote
0answers
50 views

Calculating a metric to compare multiple posterior probability distributions

I am beginner in mathematics/statistics and apologise in advance for my faulty use of language. Especially because I assume this to be a simple problem. I am working on a problem in statistical ...
1
vote
0answers
45 views

Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
1
vote
0answers
18 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
1
vote
0answers
43 views

what is the bias of an estimator

The point estimator $\hat\theta$ of a parameter $\theta$ is some function of the sample $D=\{x_1,...,x_n\}$, $$\hat\theta=g(D)$$, since $\hat\theta$ depends on the sample $D$ we're using, so ...
1
vote
0answers
20 views

Estimator of Absolute Error?

Given $X \sim B(n, p)$, we know that $\hat{p} = X / n$ is the obvious estimator for unknown parameter $p$, and the following quantity $$\frac{\hat{p}(1-\hat{p})}{n-1}$$ has the property that its ...
1
vote
0answers
81 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
1
vote
0answers
31 views

Estimate starting with variational formula

I'm working on an a priori estimate, using equality's like Young, Cauchy,... But I'm stuck with my testfunction. I've got the following problem: $\frac{\partial u}{\partial t} - \Delta u + \int_\Omega ...
1
vote
0answers
47 views

Finding an unbiased estimator for function of Poisson

Let $X_1,...,X_n \sim Poi(\lambda)$ then unbiased estimator for $\lambda$ is obviously $\bar{X}$. What about $\tau(\lambda)=\sqrt{\lambda}$? Also how would one derive UMVUE for this lambda?
1
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0answers
56 views

Asymptotic stability

we know from the theory of ODE that $\left\|\exp(tA)\right\|\leq Ke^{-\delta t}$ for $K,\delta >0$ and $t\in\mathbb{R}^+$ if the real part of all eigenvalues are strict non-positiv. My question is: ...
1
vote
0answers
19 views

How do I compute the variance (or confidence interval) of a Maximum Spacing estimator?

I am trying to solve a problem using a Maximum Possible Spacing estimator (see Maximum spacing estimation on wikipedia for links). Details on what I am trying to do can be found in the following ...
1
vote
0answers
42 views

Approximating arccos(a/(a+x)) for the sake of simplfying an integral

I recently tried to evaluate $$\int e^{\beta\arccos(a/(a+x))}dx$$ (everything constant except $x$) and got a complicated answer involving a hypergeometric series with complex arguments. Can anyone ...
1
vote
0answers
68 views

Estimation (Newton potential)

Let $f$ be a $C^2$-function on $\mathbb{R}^n$ with compact support. Let $N$ be the Newtonpotential on $\mathbb{R}^n$ and $u:=N\star f$ (i.e. a solution of the potential equation $\Delta u=f$). ...
1
vote
0answers
62 views

Fast way to estimate cardinal number of subset

I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets, $ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
1
vote
0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
1
vote
0answers
49 views

Estimate the size of a set given random sub sets.

Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible. I'm not going to quibble ...
1
vote
0answers
39 views

Sample estimated normal distribution - what will be the expected effect of another sample?

Assume I already have n samples of a 2D variable. I can compute the sample mean and variance. If I assume that the samples are taken from a normal distribution, then using the mean and variance I get ...
1
vote
0answers
320 views

Calculating a fisher information

$\ (X_1,X_2,X_3,X_4)$ has a multinomial distribution with parameters 3$\theta$/5, $\theta$/5,$\theta$/5,(1-$\theta$) Calculate the fisher information where $\theta$ is in [0,1]. So I wrote out the ...
1
vote
0answers
54 views

Consistent estimator of percentile or Value-at-Risk

Given a sequence of i.i.d. random variables X_1,...,X_n I would like to estimate VaR_q(X) (see http://en.wikipedia.org/wiki/Value_at_risk#Mathematical_definition) for some 0 1) VaR_q(X)=X_(āŒŠqnāŒ‹) and ...
0
votes
0answers
29 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
0
votes
0answers
7 views

Using EM versus estimating all parameters directly

My question is really a request for a clarification, and pertains to whether or not there are questions for which it is necessary to use EM or if it's more of a convenience. Let's presume we have n ...
0
votes
0answers
8 views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
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
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 ...
0
votes
0answers
28 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
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
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
0answers
15 views

regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error term. We know covariance matrix āˆ‘ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would ...
0
votes
0answers
13 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 ...