For questions about estimation and how and when to estimate correctly

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1answer
27 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
117 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 ...
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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
68 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 $a_{1}>0$, $a_{2}>0$, ...
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0answers
22 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
55 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: ...
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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 ...
0
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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
25 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
29 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 ...
0
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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 ...
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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
29 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
136 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}, ...
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1answer
46 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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0answers
27 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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0answers
100 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 ...
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2answers
40 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
73 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
42 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 ...
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51 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 ...
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0answers
9 views

How to estimate a time step to obtain required accuracy when simulating linear dynamic system?

Assume the the linear dynamic system is $$\dot{x}=Ax$$ , and the initial state is $x_0$, where $|x_0|<K$, $K$ and the matrix $A$ is known. The state after time $t_0$ will be $e^{At_0}x_0$. The ...
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0answers
33 views

Point estimation of expected value - disease spread

Firstly I want to put big disclaimer here. This particular problem is a smaller part of my homework. Since even after discussion with my fellow classmates we are not sure how to handle it we decided ...
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0answers
24 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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1answer
33 views

Understanding the point estimation of the expected value

I am trying to understand this problem, however I can't get past some of the definitions used when estimating the expected value. What I would need is to confirm or disprove my conclusions - I read ...
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1answer
64 views

Estimating $\sum\limits _{n=1}^k \sin \frac x n$ in the form $f(k,x) \sin(g(k,x))$

When you plot the function for a reasonably large $k$ ($300$ in this case) you get something like this... This seemed like it could be estimated the way I stated previously. The accuracy of that ...
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0answers
223 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
2
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1answer
35 views

Estimating the number of books in the world from randomly chosen overlapping lists

Suppose I have lists $L_1 , \dots , L_n$ of, say, books. Assume further that these are uniformly chosen from the set of all books (probably unrealistic for obvious reasons, and if this assumption can ...
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1answer
63 views

Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Is it possible to evaluate or at least to estimate the following integrals? $$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$ and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$ I ...
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0answers
47 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, ...
2
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0answers
53 views

For what $r,s$ exist unbiased estimation of $f(p) = p^{r}(1 - p)^{s}$ for binomial distribution?

We have sample $x_1, ..., x_n$ generated by independent binomial random variables $\xi_1, ..., \xi_n$. We know parameter $k$ but don't know probability $p$. k is number of tests: $\xi_i \sim ...
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1answer
30 views

Estimation, Upper limits, Lower limits

Two rods of length 2.6 cm and 3.5 cm are measured correct to the nearest 0.2 cm. The two rod are joined together, find the lower and upper limit of the new rod. I get stuck. HOw to do?
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57 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
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1answer
13 views

derivation of $\theta(x)=\int_{a}^{x}\varphi (y)dy - (\int_{a}^{b}\varphi(y)dy)\psi(x)$ and °L^2$-norm estimation

Let $I=(a,b)$, $u\in L^2(I)$ and $\psi\in C^{\infty}(I)$ such that $\psi=0$ on $(a,a+\epsilon)$ and $\psi=1$ on $(b-\epsilon , b)$ for sufficient small $\epsilon$. Let $\varphi\in C_C^\infty (I)$ and ...
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1answer
40 views

Estimate parameters of a quadratic function

Suppose that we have two data points which tell us about the output of some function $f(x)$: $(0, 50)$ $(10, 150)$ We know that the function is quadratic (so it's something like $ax^2 + bx + c$). ...
2
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1answer
13 views

Saturating space so that at least two lines are close enough

All lines in what follows pass through the origin. The only reason for the angle $2\pi/3$ below is that this is how I began wondering about these questions. Picture the unit disc $S^2$, by which I ...