For questions about estimation and how and when to estimate corectly

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1answer
26 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
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0answers
55 views

Probabilities and Estimation of average and standard deviation

I've done a good bit of this number but I have trouble with part 2. I'll show you my work and the questions I can't figure in bold. A guy has a machine that scans his apples. The machine rules are : ...
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1answer
23 views

How to call $(E\hat{x} - x)^2$?

Let $\hat{x}$ be an estimation of $x$. Quantity $E(\hat{x} - x)^2$ is called Mean Squared Error. How one would call $(E\hat{x} -x )^2$?
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0answers
41 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
2
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1answer
55 views

Expectation of the MLE $e^{-\frac{1}{\overline{X}}}$

I am having a bit of a problem with examining the properties of a maximum likelihood estimator. I feel like I am missing something simple, but I have been unable to find someone doing an example quite ...
3
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1answer
35 views

Why is the MLE a special case of the minimum contrast estimator?

In my statistics lecture, we had two definitions, namely Let $X_1,\ldots.X_n$ be iid random variables, each with density $p_{\Theta_0}(x)$. Furthermore, let $\varrho$ be a real function such that ...
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0answers
76 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
0
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1answer
45 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
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0answers
48 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
1
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0answers
26 views

ML estimate of sum of guassian variables?

consider the sum $z=x_{1}+...+x_{k}$, where the scalar variables $x_{i}$ are statistically independent and Gaussian, each having the same mean $0$ and variance $\sigma^2_{x}.$ how can I construct the ...
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0answers
10 views

Maximum Likelihood parameters

I have a generative model with class conditional probability distribution $\Bbb P(x | C_k)$ and class priors $\Bbb P(C_k)$. I am having trouble with deriving the Likelihood function and hence the ...
2
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0answers
22 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
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0answers
39 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
0
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1answer
386 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 ...
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0answers
14 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
1
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0answers
50 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} + ...
0
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1answer
19 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 ...
0
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1answer
236 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 ...
0
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1answer
26 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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0answers
62 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
39 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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0answers
69 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 ...
-1
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1answer
42 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 ...
3
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4answers
272 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 ...
3
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0answers
64 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 ...
1
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0answers
53 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 ...
1
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1answer
64 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 * ...
0
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1answer
70 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 ...
1
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1answer
46 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 ...
2
votes
1answer
67 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
85 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 ...
2
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1answer
481 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 ...
0
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0answers
36 views

This function is continuous and has the following estimation?

Let $\varphi$ be a positive linear function $\varphi:C(\mathbb{R})\rightarrow\mathbb{C}$ such that for all $n\in\mathbb{N}$ there exists $f_n\in C(\mathbb{R})^+$ with $f_n(x)\leq1$ if $|x|\leq n$ and ...
0
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2answers
135 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 ...
0
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2answers
749 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 ...
1
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0answers
55 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 $|y_1|+|y_2|\to \infty$ ...
3
votes
1answer
59 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}) \, ...
1
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0answers
20 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 ...
0
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0answers
592 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 ...
1
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0answers
46 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 ...
0
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0answers
39 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, ...
0
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1answer
34 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 ...
2
votes
2answers
127 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 ...
1
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0answers
20 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 ...
1
vote
1answer
67 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.
3
votes
4answers
192 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 ...
0
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1answer
42 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 ...
0
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0answers
15 views

Estimate for Leray Operator acting on Gaussian Kernel

I just wonder if someone could give me an estimate for: $$ |{(\Pi \Phi)(x)}| \leq c(1+|x|)^{-n} $$ where $\Phi(x)= \pi^{ -n/2}e^{-|{x}^ {2}}|$ and $\Pi$ is the Leray operator $\Pi:L^{2}\to L^{2}$ ...
1
vote
1answer
145 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 ...
0
votes
2answers
74 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? ...