For questions about estimation and how and when to estimate correctly

learn more… | top users | synonyms

1
vote
3answers
65 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
2
votes
1answer
36 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
2
votes
0answers
10 views

How to handle Finite-state-machine with correlated inputs?

My system can be represented by the following state-diagram. The inputs to this FSM are correlated. This implies that I can no longer make "independent input" assumption. My question is: How ...
1
vote
1answer
44 views

Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then $$\lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ ...
0
votes
1answer
14 views

Estimate line in [theta, rho]-space given 2 points

Given 2 points (x1,y1), (x2,y2) I wish to estimate a line defined by [cos(θ) sin(θ) -r], where r is the distance from origin to the line along a vector perpendicular to the line, and the angle theta ...
3
votes
1answer
192 views

Estimate of the derivative

Show that if $f(x)=x^2+O(x)$, and $f$ is differentiable with non-decreasing derivative $f'(x)$, then $f'(x)=2x+O(\sqrt{x})$. I know that if $f'$ is not non-decreasing, then the statement is not ...
-1
votes
1answer
10 views

Analogue Tape how long do I have to record?

If I have 1200ft (feet) of tape. How long will I be able to record for at 7.5ips (inches per second) Thank you
1
vote
1answer
35 views

Find zeros of a function or at least say things about their location?

Let $a>0$ be a fixed parameter. I would like to find the (I think there are only two) $x\in \mathbb{R}$ such that $$(x-a)e^{-\frac{1}{2}(x-a)^2} = (x+a)e^{-\frac{1}{2}(x+a)^2}.$$ I know this might ...
1
vote
2answers
84 views
+100

Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: ...
1
vote
0answers
26 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
0
votes
0answers
14 views

Check inequality

Let $x\in \mathbb{R}^d$ and $e_j$ be a basis vector with 1 at the $j$-position (otherwise $0$). Is it true that $\frac{1}{\mid x+e_j\mid^{d-2}}-\frac{1}{\mid x\mid^{d-2}}=O(\mid x\mid^{-d+1})$? Does ...
2
votes
3answers
451 views

Simpson's rule to estimate distance traveled given velocity at certain points

Problem: A boat drives a steady course with a variable speed for 4 hours. The speed is registered at regular intervals in meters per second. The registration shows $2.4, 4.4, 7.6, 8.4, 8.6, 7.9, ...
0
votes
1answer
15 views

variance of a Maximum Liklihood estimator

I am reading a book on Bayesian Estimation and Sensor Fusion and I want to know where the formula below come from. In fact, what is the relation between the variance and the second derivative of the ...
3
votes
3answers
53 views

Question for the estimation of $\sum_{i=1}^x \frac{1}{w+i}$ as $x \to \infty$.

I have a question of the estimation of this summation: $$ \frac{1}{w+1}+\frac{1}{w+2}+\cdots+\frac{1}{w+x}$$ Which is: $$\sum_{i=1}^x \frac{1}{w+i}$$ What I have tried: applying limit to the ...
0
votes
0answers
8 views

If a family of densities is not complete then is it necessary that there isn't any MVUE?

The question is about the truth of this statement: "If the family $\{f(x;\theta):\theta\in\Omega\}$ is not complete, then there doesn't exist any MVUE" MVUE is an abbreviation for "Minimum Variance ...
0
votes
0answers
13 views

Computing an expression with limes and limit superior and floor-function

Let $2\leq e\leq r$. I am trying to compute or estimate (from above) $$ \lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log\left[(e+r+1)^{2(n+k)-1}-((e+r)\cdot 2r+e+r+(e+r+1)\cdot r^2)^{c_{n,k}}\cdot ...
-1
votes
0answers
21 views

Interpolation of random data

I have points $(t_1,x_1(t)),(t_2,x_2(t)), \cdots , (t_n,x_n(t))$ and I would like to estimate values of $x_k(t)$ where $1 < 2 < \cdots < k <\cdots <n$. How can I do this. I have read ...
0
votes
2answers
31 views

What's the best guess for the parameter of an exponentially distributed sample?

I have a sample of size $N$ values. I know the values are exponentially distributed, i.e. they are distributed according to this probability density function: $$ f(x;\lambda) = \begin{cases} \lambda ...
0
votes
0answers
33 views

Estimating distribution from two distributions

I have been doing a survey on Family Incomes in India. The income of male and females are denoted by x and y. x and y are strictly positive. Per chance, individual values of y were deleted. I only ...
0
votes
0answers
12 views

How to improve a poisson based estimator using variance reduction techniques

Given a random number $X \sim Pois(\mu)$ for some random, i.i.d $\mu$, I'm trying to estimate $P(X \ge x)$ by simulation. The approach is to use a raw/classic estimator, where I generate a bunch of ...
0
votes
0answers
11 views

Estimate average change over time with matlab

I'm trying to use matlab to estimate the average difference in location and height for tectonic plates for a university assignment. Not alot of guidelines or pushes in any direction as how to solve ...
4
votes
1answer
105 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
0
votes
2answers
120 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
1answer
17 views

An issue related to the expectation maximization algorithm for a coin toss experiment

I just read a very nicely written introduction paper for the expectation maximisation algorithm published in Nature biotechnology by Do and Batzoglou ...
0
votes
1answer
22 views

Stability (wrt parameters) of elliptic partial differential equation

consider the equation $$\mathcal Lu=f \quad \text{in } \Omega $$ With some appropriate boundary condition, $\Omega$ regoular as you like, $ \mathcal L$ to be defined by $$\mathcal ...
0
votes
2answers
27 views

Optimal estimation of the fusion of two measurements

Suppose I have a sensor measuring a quantity $\text R$. For example the sensor could be a radar estimating the range of a target. We can write: $$R(t)=r(t)+\nu_0(t)$$ where $r(t)$ is the real range ...
0
votes
0answers
9 views

Estimate Results From a Database

I have some data in an Excel document, from differents projects i've made. Now I would like to use this data in order to estimate the results for new projects with different input values. Each ...
0
votes
1answer
29 views

Hyperbolic energy estimate in Evans PDE book

Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how ...
0
votes
0answers
13 views

Error propagation in complex formula

I'm currently trying to use error propagation formulae to calculate an estimate for the error in the following molecular dynamics formula: $C_v^* = \frac{3}{2}\bigg[1-\frac{2}{3NT^{*2}}\big\langle ...
0
votes
1answer
17 views

Predicting values based on given data

num value 8 ? 10 ? 12 24.33 16 22.5 20 22.29 24 22.41 28 22.55 32 ? 36 ? let's say i have the following ...
0
votes
1answer
24 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
0
votes
1answer
16 views

Extimate error, payment rate, continous compounded

Give an estimate of the error, when the payment rate $x_m = r P_0 \frac{(1+r/m)^{mT}}{(1+r/m)^{mT}-1}$ (compounding and repayment m times per year) ist approximated bei $x_{\infty}=\frac{r P_0* ...
2
votes
1answer
24 views

Can a prediction interval be interpreted as a probability?

Suppose I find a 90% prediction interval for some data distribution. This implies that if I sample large enough data from this distribution, then 90% of such data will lie inside the prediction ...
1
vote
0answers
12 views

Representation of the optimal filter measure as the measure of a diffusion process

In "Mitter SK, Newton NJ. A Variational Approach to Nonlinear Estimation. SIAM J Control Optim. 2003 Jan;42(5):1813–33", it is shown that the path estimation measure $P_{X|Y}(\cdot,y)$ for the ...
0
votes
1answer
33 views

Bias and variance of estimator

I have the following estimator, $E = 1/\bar{X}$ of $E = 1/\lambda$ where X is exponentially distributed with parameter $\lambda$. I'm trying to find the bias and variance of this estimator. For the ...
0
votes
1answer
18 views

Given the estimator find wheter

X has an uniform distribution on interval $(0,\theta]$ where $\theta$ is a positive parameter Given the estimator: $$T(X_1,X2, \ldots, X_n)=\frac{2}{n} \sum_{i=1}^n X_i$$ Find whether this estimator ...
1
vote
2answers
25 views

Estimation of a defined integral

I need to show that $$\left|\int\limits_{0}^{\frac{1}{2}}\frac{(2t-1)^3}{(\sqrt{1+t})^7}dt\right|<\frac{16}{125}$$ Evaluating it would be my last hope, but it wouldn't be easy wither. Is there a ...
0
votes
0answers
4 views

Computing the log-likelihood ratio for a large dataset

I have a sparse binary matrix $M$ of size $500k \times N$ where rows are objects and columns are features. $M_{ij} == 1$ if there is an observation of feature $j$ in item $i$. For each object I would ...
0
votes
0answers
16 views

Numerical Estimation and Integration by Parts

I have hit a roadblock in a project and I wonder if anyone with better quantitative skills (shouldn't be hard to find :p) can lend me a hand. I need to compute the following: $\int^a_bxf(x)dx$ ...
0
votes
1answer
37 views

Unbiased Estimator of $\sigma^2$

Let $X_1, X_2, ..., X_n$ be a random distribution such that the mean $\mu = 0$ and the variance $\sigma^2$ is unknown. I'm finding a constant $c$ such that $$U(X) = c \sum^{n-1}_{i = 1}(X_{i+1} - ...
0
votes
0answers
28 views

How to estimate probability of a binomial passing a threshold?

Let $X$ be a binomial random variable $X \sim Bi(p, t)$. ($t$ is the number of tosses) Is there a way to estimate $$P(X \ge \alpha t + \beta)$$? I know that I can write the probability exactly but ...
0
votes
1answer
42 views

Asymptotic variance of MLE of normal distribution.

I am trying to explicitly calculate (without using the theorem that the asymptotic variance of the MLE is equal to CRLB) the asymptotic variance of the MLE of variance of normal distribution, i.e.: ...
0
votes
0answers
9 views

Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all ...
4
votes
4answers
74 views

How to estimate $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}$ with error less than $0.01$?

How to estimate $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}$ with error less than $0.01$? In order to solve the question, I think we need to write out the terms. So ...
2
votes
0answers
23 views

Parameter estimation - Holt's Two parameter Linear Exponential Smoothing

The reference for the below equations can be found in the Link . Note that $k$ is the timestamp and $i$ is the $i^{th}$ entry of a vector or $(i,i)^{th}$ entry of a matrix, $F$ in this case Equation 1 ...
2
votes
1answer
443 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
0
votes
1answer
41 views

Simple expected value manipulations (in estimation problem)

Let $\hat{\theta}_{1}$ $\hat{\theta}_{2}$ $\hat{\theta}_{3}$ be three estimators of the parameter $\theta$. E($\hat{\theta}_{1}$) = E($\hat{\theta}_{2}$) = $\theta$, E($\hat{\theta}_{3}$) $\ne$ ...
1
vote
2answers
79 views

Earth population growth rate is exponential or logarithmic?

How many points on a monotonically increasing curve is needed to determine if it is exponential or logarithmic? For example can we tell that in the most recent history population is increasing ...
0
votes
1answer
21 views

Simplifying linear algebra division

So yesterday, I posted a question but it seems I wrote it too confusingly. Now, I will simplified the question so that probably you have some opinion or suggestion. Suppose, I have an function like ...
0
votes
0answers
15 views

Addition of two estimated means to become new estimated mean

In order to estimate population mean there were conducted two independent questionnaire survey. They have mean estimates $\hat \mu_1$ and $\hat \mu_2$ respective. And their standart deviations are ...