For questions about estimation and how and when to estimate correctly

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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: $\int^a_bxf(x)dx$ ...
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0answers
31 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 ...
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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} - ...
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1answer
45 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.: $$\...
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1answer
16 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 $r&...
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4answers
78 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 $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}...
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0answers
24 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 ...
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1answer
42 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($\hat{\theta}_{3}$) $\ne$ $\...
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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 ...
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2answers
87 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 ...
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0answers
16 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 $\...
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1answer
22 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 ...
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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 ...
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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 ...
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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 ...
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1answer
31 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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1answer
121 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 ...
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0answers
30 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 ...
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2answers
50 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 ...
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1answer
22 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, ...
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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 $\hat\theta=...
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1answer
61 views

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

please help, I don't understand what the problem is asking
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0answers
81 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<\...
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0answers
25 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 ...
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1answer
30 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 ...
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0answers
13 views

Estimation of binomial probabilities $f(r)$ over $r \in [0,\frac{1}{2}]$

I want to fit a (decreasing) univariate function, \begin{equation} f(r), \end{equation} over $r \in [0,\frac{1}{2}]$ to a series ($r =\frac{1}{100}, \frac{2}{100}, \frac{3}{100} ,\ldots,\frac{1}{2}$) ...
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1answer
10 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
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1answer
24 views

Estimating confidence Interval for unknown Variance, Normal distribution

I've been stuck with this question for a while: I've learnt how to use t-distribution to estimate CIs for an unknown variance, but I'm unsure how that applies to this situation. Any help would be ...
0
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1answer
56 views

How can I calculate missing values from a table listing of areas and prices? [closed]

I have a set of objects of different sizes (measured in square metres). I know the price of some of them. I want to use the known prices to find the missing prices. Here is the data I have: \begin{...
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2answers
45 views

Probability of island having 8 people born with disease, estimate?

The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that ...
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1answer
45 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
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0answers
14 views

Fitting a model to a collection of binomial proportions, based on varying (large) sample sizes.

I have a multi-parameter bivariate function, say $f(i,j)$ that I want to use to predict the entries of a matrix $M(i,j)$, the entries of which are binomial probabilities based on varying sample sizes, ...
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0answers
30 views

Maximum Likelihood Estimation with 2 parameters for a Poisson distribution

I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same ...
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0answers
30 views

Lower bound of KL-divergence between unknown and known distribution

Suppose I have two multivariate continuous random variables $X$ and $\hat{X}$ with underlying probability distributions $P_{X}$ and $P_{\hat{X}}$, where $P_{\hat{X}}$ is a Gaussian approximation of $...
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1answer
16 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 ...
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0answers
17 views

FIR estimator for IIR system

Suppose that we have a dynamical system of which the impulse responses are infinite (IIR). Now I found methods on papers (http://dx.doi.org/10.1109/9.839942) estimating states or outputs of such a ...
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0answers
9 views

lower bound for integral of curve derivative

Let $\gamma_k:[0,1]\rightarrow\mathbb{R}^3$ be some arc-length parametrized $C^1$-curve, more specifically a knot. $\gamma: [0,1]\rightarrow\mathbb{R}^3$ is the limit of a sequence of such knots in ...
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1answer
30 views

$|f(z)|\le\frac{M}{|z|^{\alpha}}$ for all $z\in U_r(0)\setminus \{0\}.$ Why is $0$ a removable singularity of $f$?

Let $0<r<1$, $f:U_r\setminus\{0\}\to\mathbb{C}$ holomorphic. Let $\alpha <1,\; M\ge 0$ such that $$|f(z)|\le\frac{M}{|z|^{\alpha}}$$for all $z\in U_r(0)\setminus \{0\}$. Prove that $0$ is a ...
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0answers
18 views

Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
2
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1answer
137 views

Regularity of heat kernel

I'm trying to find some references dealing with regularity and properties of the heat/Gaussian kernel $$ G_t\left(x,y\right) = \frac{1}{\sqrt{2\pi t}}\, e^{-\left.\left(x-y\right)^2\right/2t}, \hspace{...
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1answer
47 views

“Guesstimation” problems within pure mathematics

Wikipedia defines a “guesstimate” as “an estimate made without using adequate or complete information, or, more strongly, as an estimate arrived at by guesswork or conjecture.” Guesstimation problems ...
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28 views

Peak requests per second, The Social Network Movie's Face Mash

In the movie The Social Network, the Harward network crashes after facemash.com gets too much traffic. They mention 22000 page views during the 4 hours it was online. Viral websites usually follow an ...
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106 views

Why is there no unbiased estimator for $\frac{1}{\theta}$ for Poisson Distribution?

Suppose that $X_1,\dots,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$. To do so, I ...
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2answers
42 views

Is “non-random parameter estimation” the same thing as maximum likelihood estimation?

In one book and a few papers, mostly on navigational tracking, I have found reference to the method of "non-random parameter estimation" but this term is not on the Wikipedia and not in a lot of ...
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1answer
80 views

Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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0answers
44 views

Kalman filter on Timeseries

After a lot of research on Kalman filter I can't find anywhere how exactly the filter works on timeseries.Specifically, I want to know about fοrecasting with Kalman filter on Timeseries, point ...
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1answer
28 views

unbiased estimator for sample covariance?

I'm new to statistics and and I need some help: Let $X_1,...X_n$~$N(\mu_x,\sigma^2)$, $Y_1,...Y_m$~$N(\mu_y,\sigma^2)$. All r.vs. are i.i.d and $\mu_x,\mu_y,\sigma$ are unknown I was told that $...
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2answers
116 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
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0answers
24 views

Compute the limit of the error term of Taylor polynomial of $f(x)=e^{3x}-1$

I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$). First, I found the Taylor polynomial of $f(x)$, which is $$T_n f(x)=\frac{3^1}{1!}...
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0answers
52 views

Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,…$ defines a probability distribution for the random variable $X$.

Let $p \in (0,1)$ and $q=1-p$. (a) Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,...$ defines a probability distribution for the random variable $X$. (proved) (b) Given the single observation ...