For questions about estimation and how and when to estimate correctly

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0
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2answers
23 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
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0answers
7 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
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1answer
13 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
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0answers
7 views

initial value issue in 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 ...
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1answer
12 views

finding the expectation of the MLE for $\mathbf{\Sigma}$ in a multivariate Gaussian [on hold]

I am trying to find the expectation of the MLE for $\mathbf{\Sigma}$ for the multivariate gaussian. $E(\mathbf{\Sigma}_{ML}) = E\left (\dfrac{1}{N} \sum (\mathbf{x}_n - \mathbf{\mu})(\mathbf{x}_n - ...
0
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0answers
8 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 ...
-2
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0answers
14 views

Mle of phone calls [closed]

Assume the number of emails received per minute by a service centre is a Poisson random variable with parameter lambda . If in 10 minutes 20 emails are received, find the MLE of lambda .
0
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1answer
16 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
23 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
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1answer
14 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
29 views

Calculation of estimators [closed]

Let $X_1$ and $X_2$ be i.i.d. from some distribution $f(x;\theta)$ with mean $\mu$ and variance $\sigma^2$. Note that we do not know the precise form of the density function $f(x)$. Consider the ...
0
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0answers
15 views

Out-Of-Sample N-periods ahead forecast with MATLAB [closed]

I getting blind in front of my laptop trying to find out a way to code the following regression in a proper way. My data are a matrix $X=(ones(315,1), a \hspace{3mm}b \hspace{3mm} c \hspace{3mm} d)$ ...
2
votes
1answer
23 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 ...
0
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0answers
11 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
32 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 ...
3
votes
1answer
103 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 true. ...
1
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2answers
24 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 ...
-1
votes
0answers
44 views

Nonlinear constraints replaced by parameters and estimated iteratively

I have an optimization problem with nonlinear constraints in the following form: $x + y + 0.5(x+y)^2-z = 0$ $s+(x+y)*t\ge M$ I linearize these constraint by replacing the nonlinear terms by ...
0
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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
36 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} - ...
-1
votes
0answers
4 views

Hierarchical model without fixed intercept

I understood it is common sense to almost always use the intercept in a model. Let's assume I have a model withouth intercept, that measure heights. The predictor are also physical measurements ...
0
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0answers
25 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
38 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
8 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
73 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
21 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
428 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
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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
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2answers
72 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
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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
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0answers
14 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 ...
1
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1answer
19 views

Estimation Question - London Eye

Good afternoon, I was recently at an assessment center and was asked an estimation question. This was the first one I've ever done so was wondering how everybody else would go about solving the ...
0
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0answers
16 views

Underestimation of absolute value of integrals

If we have a function for which it is not possible to calculate the definite integral explicitly, we may want to approximate it somehow, for example by giving bounds for its absolute value. There are ...
0
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0answers
9 views

General approach of MLE for function of parameters.

everyone. Imagine that you have two random variables $X(\theta)$ and $Y(\alpha)$. How to estimate using MLE function of this two parameters $f(\theta,\alpha)$? For example, sum or difference, or ...
2
votes
1answer
61 views

How to show that $E[E[X\mid Y]\mid Y] = E[X\mid Y]$

I'm reading a little proof about how the estimation error (based on the conditional expectation estimator, $E[X\mid Y]$) has zero expectation, and at one point the author used the equality that the ...
1
vote
1answer
25 views

Showing $X_{(n)}$ is an unbiased and consistent estimator for $\theta$.

Let $X_1,X_2,\ldots, X_n$ is distributed iid $\mathrm{Uniform}(0, \theta)$ with $\theta$ in being real positive. Show $\frac{n+1}{n}X_{(n)}$ is an unbiased and consistent estimator for $\theta$. I ...
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2answers
3k 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
1answer
113 views

Find a 95% confidence interval of the mean serum cholesterol of patients on the special diet.

A physician who has a group of thirty-eight female patients aged 18 to 24 on a special diet wishes to estimate the effect of the diet on total serum cholesterol. For this group, their average serum ...
1
vote
2answers
49 views

Obtaining exact decimals in bisection method

While studying the bisection for the approximation of roots of non-linear equations I was given the following bound for the error: $|x_n-s| \leq \frac{(b-a)}{2^{n+1}}$ where $x_n$ is the n-th ...
0
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0answers
28 views

Using Log Likelihood to Find Sufficient Statistic

So, I've been given the following problem from Wackerly's Mathematical Statistics with Applications (specifically 9.60). I'm aware of one way to find the solution, but I'd like to know if this works ...
1
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1answer
21 views

Calculate average time of person in system when ins and outs are roughly equal

I have a system that has people going through a sensor for in, a sensor for out. I'm measuring ins and outs through the day, ...
-1
votes
1answer
61 views

Show that Cov(X,A)=X [closed]

please help, I don't understand what the problem is asking
0
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0answers
28 views

Weird behaviour of CLT's application to binomial.

I am carrying out the simulations of the following experiment for all $n$ in the set $\{1,2,3,...,100\}$. (0) Set $k=0$. (1) Generate $n$ $Bernoulli(0.9)$ trials. (2) Construct estimate ...
47
votes
11answers
3k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
1
vote
0answers
60 views

ML estimation for bivariate data.

Consider the random variables $X$ and $Y$ with joint distribution $$F(x,y)=[1-e^{-a_{1}x}]^{\theta}+[1-e^{-a_{2}y}]^{\theta}-[1-e^{-a_{1}x-a_{2}y}]^{\theta},$$ where $a_{1}>0$, $a_{2}>0$, ...
0
votes
1answer
14 views

Estimation effectiveness of two normal-distributed variables

You have two processes of measuring the air pollution, $X$ and $Y$. Both processes deliver values which are normal distributed around $\mu$: $X ~ N(\mu, \sigma_x^2)$ and $Y ~ N(\mu, \sigma_y^2)$. I ...
0
votes
0answers
21 views

Parabolic interpolation in $k$ dimensions

I know the values of a smooth function of $k$ variables at $3^k$ points on a cube in $k$ dimensions (where $k=2,3$ or $4$). The central value is known to be the largest. I want to estimate the ...
0
votes
1answer
28 views

Why is $m(n)\approx\log_2(n)$?

Why is $m(n)\approx\log_2(n)$ ? If $m(n)=\inf\{m:2^{-m}m^{-3/2}\le\frac1n\}$, taking log of $m(n)$ I get $-m(n)-\frac32\log_2(m(n))\le-\log_2(n)$ (This appears in the solution of an exercise in ...