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3
votes
0answers
48 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
17 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
57 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
105 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates.

I've stuck in unpleasant problem with noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
2
votes
0answers
90 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
131 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
35 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
33 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
24 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
30 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
7 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
33 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
62 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
40 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
35 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
16 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
33 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
16 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
67 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
39 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
vote
0answers
52 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
41 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
65 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
57 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
48 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
304 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
50 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_(⌊qn⌋) and ...
0
votes
0answers
14 views

Find the covariance of estimation error

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
0
votes
0answers
4 views

WLS: Finding minimum variance linear estimator of 3 observations

I am given three observations from a distribution with different variances (ie heteroskedastic errors). \begin{align} Y_i = \mu + u_i \end{align} The summary statistics of the three observations, ...
0
votes
0answers
35 views

Can I estimate Variance of Gamma from Negative Binomial distributed data, given NB is Poisson-Gamma mixture

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a mixture of Poisson and Gamma. The variance of this Gamma distribution is ...
0
votes
0answers
19 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
0
votes
0answers
42 views

Probabilities and Estimation of average and standard deviation

I've done a good bit of this number but I have trouble with part 2. I'll show you my work and the questions I can't figure in bold. A guy has a machine that scans his apples. The machine rules are : ...
0
votes
0answers
54 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
0
votes
0answers
5 views

EKF to fuse gyroscope and accelerometer readings

I found it interesting to implement EKF for fusing gyroscope and accelerometer data. Trying reach my goal i discovered a lab with some theory explaned, also it has nice app for the phone to stream ...
0
votes
0answers
8 views

Maximum Likelihood parameters

I have a generative model with class conditional probability distribution $\Bbb P(x | C_k)$ and class priors $\Bbb P(C_k)$. I am having trouble with deriving the Likelihood function and hence the ...
0
votes
0answers
32 views

Unbiased estimators in an exponential distribution

We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 ...
0
votes
0answers
21 views

Problem on population estmation

Can anyone help with this question.I can show my progress provided you need it
0
votes
0answers
22 views

Parabolic regression with restricted shape

How can I calculate the parabolic regression with vertex at minimum. Is it possible? I have a set of points from which I estimate the parabola using the (I believe) standard equation (from ...
0
votes
0answers
39 views

MME of $\theta^2x^2e^{-{\theta}x}$

I need to find the Method Moment Estimator of parameter $\theta$ based on a random sample $X_1…X_n$ with the following pdf: $f(x;\theta)=\theta^2x^2e^{-{\theta}x}$; $0<x$, zero otherwise; ...
0
votes
0answers
27 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
0
votes
0answers
52 views

Covariance of $(\bar{X}, \bar{Y})$ under Simple Random Sampling

$\newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Var}{\operatorname{Var}}$ Under Simple Random Sampling without replacement, and two variables of interest $X$ and $Y$, want to estimate "$r=Y/X$". ...
0
votes
0answers
32 views

Commulative degree distribution of nodes in a scale-free network

In a Barabasi-Albert model, which is a special kind of scale-free graphs, the degree distribution of each node is $$P(k) \sim k^{-3}$$ Given $\| V \|$ (number of nodes), how can I compute "number of ...
0
votes
0answers
35 views

This function is continuous and has the following estimation?

Let $\varphi$ be a positive linear function $\varphi:C(\mathbb{R})\rightarrow\mathbb{C}$ such that for all $n\in\mathbb{N}$ there exists $f_n\in C(\mathbb{R})^+$ with $f_n(x)\leq1$ if $|x|\leq n$ and ...
0
votes
0answers
187 views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
0
votes
0answers
36 views

What is $E[T^2]$ of $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$ ??

What is second moment i-e $E[T^2]$ of random-variable: $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$, Where $X[n]$ and $W[n]$ are both 'independent' of each other and 'stationary'. Moreover, ...
0
votes
0answers
12 views

Estimate for Leray Operator acting on Gaussian Kernel

I just wonder if someone could give me an estimate for: $$ |{(\Pi \Phi)(x)}| \leq c(1+|x|)^{-n} $$ where $\Phi(x)= \pi^{ -n/2}e^{-|{x}^ {2}}|$ and $\Pi$ is the Leray operator $\Pi:L^{2}\to L^{2}$ ...