For questions about estimation and how and when to estimate correctly

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59 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} + ...
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1answer
22 views

Generate function from discrete data (time-series)

How to transform discrete data into continous function ? I am working extensively with time series data and I would like to reduce amount of data in our frontend application. It would be cool to ...
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1answer
602 views

Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...
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1answer
30 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
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76 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; ...
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0answers
83 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 ...
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1answer
48 views

How to estimate the upper bound of y in this situation? [closed]

How to estimate the upper bound of y in this situation? Given 1. a function $y=f(x_1,x_2,x_3,x_4,x_5)$ with 5 parameters ($y=f(...)$ can be any function). 2. for each $x_i$ there are $k_i$ possible ...
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4answers
378 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
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0answers
68 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 ...
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0answers
62 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 ...
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1answer
76 views

Rolling a die 100 times and adding results

Simple problem. We role a die 100 times and we add the results. What is the probability of getting sum between 330 and 380 ? I got this: $P(330 \le X \le 380) = P\left( \frac{330 - n * ...
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1answer
81 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 ...
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1answer
50 views

Some mean value limiting result

Let $\phi$ be continuous in a neighborhood of $0\in\mathbf{R}^3$ (you may assume it to be uniformly continuous, if you like). Do we have that $$\lim_{\epsilon\rightarrow ...
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1answer
69 views

Help show that a second derivative is always negative

How do I show that the second derivative is always negative? I've computed the second derivative to be: $\displaystyle\frac{n}{2\sigma^4}-\frac{1}{\sigma^6}\sum\limits_{i=1}^n(x_i-\mu)^2$ Then I ...
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3answers
94 views

Why does finding the $x$ that maximizes $\ln(f(x))$ is the same as finding the $x$ that maximizes $f(x)$?

I'm reading about maximum likelihood here. In the last paragraph of the first page, it says: Why does the value of $p$ that maximizes $\log L(p;3)$ is the same $p$ that maximizes $L(p;3)$. The ...
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1answer
692 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
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2answers
240 views

How do we prove the error estimation of the rectangle method

Let $f\in C^2[a,b]$. An approximation of the integral over $[a,b]$ is given by $$I[f]:=\int_a^bf(x)\text{ dx}\approx \frac{b-a}{n}\sum_{i=1}^nf\left(a+\frac{2i-1}{n}(b-a)\right)=:M_n[f]$$ I've spent ...
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2answers
1k views

Moment Estimate of theta

Consider a random variable $X$ whose pdf is $f(x;θ)=θx^{θ−1}$ for $0<x<1$ and zero otherwise. i) Show this is a density function ii) determine the moment estimate of theta on the basis of a ...
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0answers
79 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 $|y_1|+|y_2|\to \infty$ ...
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1answer
61 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
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0answers
22 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 ...
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0answers
936 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 ...
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0answers
50 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 ...
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0answers
41 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, ...
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1answer
36 views

How to find $X_i$ from this equation

Suppose $X_i=\nu_i+\frac{m-i}{m}X_{i+1}+\frac{i}{m}X_{i-1},\quad 1\le i\le m$ where $X_0=X_{m+1}=0$. I need to find an expression for $X_i$ in terms of $v_i$, $i$, and $m$. I know how to find it ...
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2answers
140 views

Finding the variance of a statistic.

$X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define $$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I have shown that it is an unbiased estimator of the ...
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0answers
22 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 ...
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1answer
74 views

sine function description using three points

Is there any way to find the parameters of a sine wave ($A$, $w$, and $ \phi $ for $A \sin(wt+\phi)$ ) using just three points (samples)? Thank you in advance for your help.
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4answers
206 views

Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
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1answer
49 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...
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1answer
168 views

Variance of maximum likelihood estimator for discrete distribution

Lets say we have a discrete distribution with following probabilities: $P(X=0)=\frac{1}{3}\theta, P(X=1)=\frac{2}{3}\theta, P(X=2)=\frac{2}{3}(1-\theta), P(X=3)=\frac{1}{3}(1-\theta)$ Estimating ...
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2answers
82 views

Estimating arctan to below

How can I estimating $$ \arctan(\lVert x-y\rVert), $$ to below (where $(x,y)\in\Omega\times\Omega, x\neq y$, $\Omega\subset\mathbb{R}^n, n>1$ bounded domain)? Can you give me a hint please? ...
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0answers
92 views

Find a weakly singular kernel function for an estimation of a kernel

Let $\Omega\subset\mathbb{R}^n (n>1)$ be a bounded domain and $0<\alpha<n$. Show, that the kernel function $$ k(x,y):=(\arctan(\lVert x-y\rVert))^{-\alpha}\text{ for }x\neq y $$ ...
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0answers
101 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 ...
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2answers
33 views

Calculating a sample's representativeness to confirm/refute a given hypothesis?

Why hello! I'm fairly new to statistics, which is why I'm somewhat confused as to how I can approach this problem in a scientific way. The problem: Experiments are conducted to find the probabilities ...
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1answer
28 views

Boundedness, Estimate $L_2$-Norm by $H^2$

I wanted to prove the estimate: $$ \| \Delta u \|_{L_2(\Omega)} \leq C \| u \|_{H^2(\Omega)} \ \ \forall \ u \in H^2_0(\Omega) $$ Where $ C > 0$ is a constant not depending on $ u $. On the ...
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1answer
62 views

Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
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0answers
44 views

Variance of a difference in estimated proportions with trivariate discrete distributions

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...
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0answers
34 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 ...
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67 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 ...
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1answer
58 views

Why is it called the score of the log likelihood function?

Since the score of the log likelihood function is just the gradient of the log likelihood function, why give it a special name? Why not just call it the gradient?
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0answers
56 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?
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1answer
85 views

How to estimate mean from sanples of multiple correlated random variables?

Suppose we have $n$ normal random variables with variance $1$ and unknown mean. Suppose we have $n$ samples of size 1 from those random variables. Suppose also we know the correlations between the ...
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1answer
26 views

formulate a power series

Could anyone advise me if there is any way to estimate/formulate the following series $$ \sum_{i=m..n} \left(\frac{c}{i}\right)^i $$ where m,n and c are positive integers.
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0answers
67 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: ...
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1answer
41 views

How rapidly can a polynomial grow in a proximity of the real segment comparing to the values on the segment?

Let $P_n$ be a polynomial of degree $n$ with complex coefficients. Does for any $l>0$ and small $\varepsilon>0$ there exist $C=C(l,\varepsilon)>0$ and $q=q(l,\varepsilon)>1$ s.t. in the ...
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1answer
20 views

Hyperbola's estimator

I have a set of data. This set represents an approximation of a hyperbola. Is there a good way to estimate a rectangular hyperbola's function from this set?
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1answer
24 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
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0answers
27 views

Is that estimation right on $B_1(0)$?

Is the estimation $$ \frac{n}{2}-1 < \sum_{i=1}^n\frac{1}{1+x_i^2} $$ right, if $x\in B_1(0)\subset\mathbb{R}^n$? I guess $B_1(0)$ is the open unit ball. Anyhow it is: $$ ...
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1answer
100 views

Is my determination of this maximum correct?

Consider $\Omega:=B_1(0)\subset\mathbb{R}^n$ (it is the open unit ball), $\mathbb{R}^n$ is provided with the euclidean norm $\lVert\cdot\rVert_2$. Now I want to determine the following ...