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Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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13 views

Best error term in $\sum_{(n,q)=1}\frac{1}{n}$ (harmonic series with coprimality condition)

It is very well known and not difficult to prove that $\displaystyle\sum_{\substack{0<n\leq X\\ \\(n,q)=1}}\frac{1}{n}=\left(\log(X)+\gamma+\sum_{p|q}\frac{\log(p)}{p-1}\right)\frac{\phi(q)}{q}+O\...
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17 views

How to estimate the following sum?

I am looking at the following sum, $$ \sum_{i = 1}^{k-1} \left( \frac{i}{f^2(k-i)+i} \right)^2, $$ for any $f \geq 1$, in particular $f$ can be considered as a function of $k$. For instance, I ...
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1answer
25 views

Deriving the UMVUE for Rayleigh scale parameter

Let $X_1,...,X_n$ be iid with the pdf given by $f(x|\theta)=2\theta^{-1}xe^{-x^2/\theta}$ for $x>0$. My task is to find the UMVUE for $\theta$, and I’m given the following hint: “$U(X)=\sum_{i=1}^...
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1answer
21 views

Deriving Rao-Blackwellized version of unbiased estimator

Let $X_1,...,X_n$ be iid Poisson($\lambda$) with $n\geq 4$. We are given the unbiased estimator $T(X)=I(X_1=0 \cap X_2=0 \cap X_3=0)$ for $f(\lambda)=e^{-3\lambda}$, and my task is to derive the Rao-...
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14 views

Most powerful test for discrete uniform Neyman Pearson Lemma

This is with regard to the question whose link is given below- Most powerful test for discrete uniform I obtained the most powerful test function as- $\phi(x)$ = 1 if X < 3 ; ...
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0answers
9 views

Determine initial state using measurement error cost function

Given dynamic system $$x(k+1) = Fx(k)$$ $$y(k+1) = Hx(k+1) + v(k+1)$$ where v(k) is zero mean white noise. I need to derive an estimator for the initial state $\hat{x}(0)$ using the cost function $$J(...
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7 views

What are the connections (if any) between “kernel” in “kernel density estimation,” “kernel of a matrix,” and “kernel method”?

I'm getting my understanding of kernel density estimation from pages 6-7 of this PDF. If there are conceptual relationships between the "kernels" in each of these topics, I'd like to understand them.
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13 views

Reasons for Unbiasedness and Consistency of the following estimator

I am slightly stuck with these two problems, has anyone an idea on how to solve them? a) Assume the simple regression model satisfying all Gauss-Markov assumptions yi = β0 + β1xi + ui Somebody ...
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10 views

calculating confidence intervals for a weighted binomial distribution

We have n binomial distributions {$b_i$} - each with m trials, and a probability of success $p_i$ somewhere in the range [0,1]. Also, each binomial distribution $b_i$ is assigned some weight $w_i$, ...
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26 views
+50

Lower bound for sum of Hecke eigenvalues

Let $\lambda$ be weakly multiplicative, $\lambda(n)\geq0$, $p$ prime and $S(x)=\sum_{n\leq x}\lambda(n)\log(\frac{x}{n})$ for real $x$. How can I show $S(x)\gg \left(\sum_{p\leq \sqrt{x/3}}\lambda(p)\...
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12 views

Which loss function does the maximum likelihood estimator minimize?

I'm trying to understand Maximum Likelihood estimators in the context of general estimation theory. I know Bayesian estimator minimizes mean squared loss, MAP estimator minimizes all-or-nothing loss (...
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10 views

Division of two population variances

Why do we divide variances of two samples / population while estimating while for mean and proportion we take difference of two population .What is the reason behind division of variance?
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1answer
57 views

How to approximate exponential function or under approximate this function

I am trying to find the minimum of this function analytically: $$ G(x)=\frac{c(a.x^3+b)}{x}+2(a.x^3+b)\sum_{m=1}^{\infty}\frac{e^{-\beta^2m^2(T-\frac{c}{x})}-e^{-\beta^2m^2T}}{\beta^2m^2} $$ where $0&...
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2answers
76 views

Proving continuity in the origin $\frac{xy}{\sqrt{x^2 + y^2}}$

Let $g: \mathbb{R^2} \to \mathbb{R}$. How can I prove that $g$ is continuous in its origin, but not totally differentiable? If I take $$g(\frac{1}{n},\frac{1}{n}) = \frac{1}{n\sqrt{2}} \to 0 \...
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53 views

“Straightforward” estimate on intersection-probabilities of Brownian Motion

I am currently working on a paper and there appears this so called "straightforward estimate": $$\mathbf{P}\{B[0,1] \cap B[3,n] = \emptyset\} \leq \frac{c}{\ln(n)} \quad \text{where}\ c<\infty\ \...
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1answer
70 views

Estimating $\int_0^1 x^{2n}(1-2x)^{2n}e^xdx$

To find a bound for $\int_0^1 x^{2n}(1-2x)^{2n}e^xdx$ I did the following: \begin{align} &x(1-2x)\leq\text{max}[x(1-2x)]\\ &x(1-2x)\leq\ 1/8 \:\:\: \text { the maximum occurs at $x=1/4$ }\\ ...
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0answers
14 views

Estimation of 2 parameters with Maximum likelihood and a function depending on 2 random variables

I have the following PSF (Point Spread Function) (Moffat PSF) : I want to estimate the parameters $\alpha$ and $\beta$ ($\theta=[\alpha,\beta]$ represents the vector of parameters to estimate) with ...
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0answers
27 views

Finding BLUE of $\theta$ where $X_1,\ldots,X_n$ have common pdf $f(x)=\frac{1}{2\theta}e^{-|x|/\theta}$

Let $X_1,...,X_n$ have the common pdf $$f(x)=\frac{1}{2\theta}\exp\left(-\frac{|x|}{\theta}\right)$$, where $x$ can be any real number and $\theta$ is positive. I’m trying to construct the best ...
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1answer
59 views

Estimation for $a_n$ where $\int_0^1 x^ne^xdx = a_ne+b_n$

\begin{align} I_n=\int_0^1 x^ne^xdx = a_ne+b_n \end{align} where $a_n,b_n \in \mathbb{Z}$ and $n\geq0$ is a integer. How to find a bound for $a_n$ in the form $a_n<f(n) \: ?$ I'm not sure what ...
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17 views

Definition of BLUE

I am tasked with finding the best linear unbiased estimator (BLUE) for the population mean based on $X_1,...,X_n$ iid $Poisson(\lambda)$. My question is, am I supposed to find the best linear unbiased ...
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19 views

Minimal sufficiency and completeness

Let $X_1,X_2... X_n$ be i.i.d $N(\theta,\theta^2)$.Then why is $(X_{bar}, S^2)$ not complete despite the fact that $f(x,\theta)$ belongs to $k$-parameter exponential family and jointly minimal ...
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1answer
26 views

Proving the uniqueness of an unbiased estimator

Let $X$ be a random variable having pmf: \begin{array}{ll} p(x)=2 \theta \ \ \ \text{if} \ x=-1 \\ p(x)=\theta^2 \ \ \text{if} \ x=0 \\ p(x)=1-2\theta-\theta^2 \ \ \text{if} \ x=1\\ ...
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1answer
45 views

Attempt at Finding the Best Linear Unbiased Estimator (BLUE)

Let $X_1,...X_n$ be iid $N(0,\sigma^2)$ where $\sigma$ is unknown. My task is to find the BLUE for $\sigma$ within the set of linear functions of $|X_i|$ for $i=1,...,n$. Here is my work thus far: ...
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0answers
18 views

Estimate the probability of missing links based on partially observed graphs

Suppose the underlying true graph $G^T$ is generated from some known random graph model. We are able to obverse a partial graph $G^O$ (Assume the difference between $G^T$ and $G^O$ is that some links ...
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0answers
12 views

Numerical evaluation of a complex integral, when to neglect complex exponential

I have to evaluate the order of magnitude of a certain function which is given by the formula \begin{equation} \psi(\vec{r}) = K\int_{\mathbb{R}^3} f(\vec{r}')U(\vec{r}')\frac{e^{i k|\vec{r} - \vec{r}'...
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8 views

Can using Gauss-Hermite quadrature for estimation, introduce a bias?

I am using Gauss-Hermite quadrature to estimate E[h(x)] where x is a log normal random variable. I happen to see some bias in my final results. I was wondering if there is a formula specifying an ...
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0answers
12 views

Lipschitz-type estimate for a function of two variables

Let $f:[0,1]\times\mathbb R\to\mathbb R$. For $r>0$ consider the estimate \begin{align*} |f(s,u)-f(s,v)-f(t,u)+f(t,v)|\leq L_r|s-t||u-v|,\tag{$\star$} \end{align*} for all $s,t\in[0,1]$ and $u,v\in[...
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2answers
19 views

Definition of accuracy to three decimal places.

Let us say that we have estimated the value of some constant $S$ as a number $T$ with $|S-T|<0.001$. Does this mean that the number $T$ is correct to $3$ decimal places? My textbook seems to ...
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22 views

Understanding of sufficient statistics

I have came across a question regarding sufficient statistics, but I cannot understand it under this context. Let $P$ be a finite family with densities $p_i, i = 0, \dots , k$ and for any x, let $S(x)...
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0answers
35 views

Strongly consistent estimator for uniform distribution on $[-\theta, \theta]$

Let $X$ be a random variable having uniform distribution on the segment $[-\theta, \theta]$. I construct the following estimator for unknown parameter $\theta$. $$ \hat{\theta}(x_1,\ldots,x_n) = \frac{...
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1answer
35 views

Asymptotically normal estimator for distribution

Let $X$ be a random variable with the following distribution. $P[X = 1] = 1-a-a^2$, $P[X = 2] = a^2$, $P[X = 3] = a$. I want to construct an asymptotically normal estimator for parameter $$\frac{1}{...
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A question about the consistency proof of Lasso.

I'm reading Knight and Fu (2000), on the asymptotics for Lasso-type estimator and got the following question about part of the proof: In the proof of THEOREM 1 , $\hat{\beta}_{n}=\arg\min Z_{n}(\Phi)$ ...
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3answers
144 views

Prove $\sum_{1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2 $

Prove that $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2$$ I have found that $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} < \pi / 2 $ with integrating from $1$ to ...
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0answers
39 views

Estimate on Limit of Recursive Sequence

How can I estimate (via a lower bound) the limit of the recursive sequence $$P_{n+1}=P_n-\frac{C(P_n-1)^2}{(2^n+C)(P_n+C2^{-n})}$$ where $0<C<1$ and $1<P_0<2$. Let $P_{\infty}=\lim_{n\to\...
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1answer
23 views

Bayesian probability estimation

Consider a sequence of independent bernoulli random variables $X_1 X_2 ... X_n$ with parameter $\theta$ and $0<\theta<1$,where $P(X_i=1)=1-P(X_i=0)=\theta$. Assume the prior of $\theta$ follows ...
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1answer
38 views

Proving a binomial coefficient inequality per induction

$\left(\begin{array}{c}2n\\ n+k\end{array}\right) < \left(\begin{array}{c}2n\\ n\end{array}\right), n,k \in \mathbb{N}, 1\leq k \leq n$ Proving this by induction on $k$ is no problem. However, I ...
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32 views

Cramer-Rao Lower Bound for a Conditional Likelihood Function

I'm here looking for assurance that my interpretation is correct. Let the likelihood function under consideration be a conditional likelihood given by $$p(r|x;\theta)$$ where $r$ is some random ...
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1answer
38 views

Calculation cube root by hand using division method, not working for $\sqrt[3]{4} $

I have been studying the division method to calculate cube roots by hand (in preparation for an exam in which one is not allowed to use a calculator). This division method for calculation of cube ...
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0answers
30 views

Estimation in presence of signal dependent noise

Given a model as below: $$y_1 = x + \eta_1$$ $$y_2 = x + \eta_2$$ where $n_1 \sim N(0,\sigma_1^2)$ and $n_1 \sim N(0,\sigma_2^2)$, $N$ denotes a Gaussian distribution and $\sigma_1^2$ and $\sigma_2^2$...
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0answers
19 views

How to Solve Argmin Optimization Problem in Closed Form?

I am reading this article which is related to the 3D facial point reconstruction from the given images (a.k.a triangulation) Under the section 5.2 the author represent the algorithm to reconstruct ...
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0answers
30 views

Estimating a double series

I am having problems to analyse the behavior (for big $m$) of a (kind of) double series (a little difficult one). The problem is the following: Fix $N\in\Bbb N$ and take $p\in \Bbb N$ as big as you ...
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0answers
14 views

Estimating time to solve based on skill and difficulty

My question may be a bit too straight forward for this section but I would like to have some incentives here, as I feel like I am missing something. Consider the following simple problem: you give ...
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0answers
24 views

What to choose for the distribution of $x$ to estimate $f(x)$ as linear

I have to estimate $y = f(x)$ as a linear function of $x$ on a given range. I can measure $y$ for as many $x$ values as I want. Each measure has its own measurement error. Assuming this error is ...
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0answers
21 views

MSE of Gaussian random vector

first time to use this website, hope I made it correctly. Suppose $\underline{X}\sim n(\underline{\mu},\Sigma)$ where $\Sigma= (\begin{matrix}&\Sigma_{11} &\Sigma_{12}\\&\Sigma_{21} & ...
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1answer
63 views

Unbiased estimator for Gamma distribution

Let $Y_1, Y_2,...,Y_n$ be a random sample from a Gamma(2, $\beta$) distribution with pdf $$f(y) = \frac{y e^{-y/\beta}}{\beta^2} \;, \quad y>0$$ If the maximum likelihood estimator for $\...
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1answer
50 views

Method of Moments Pareto Distribution

Find a formula for the method of moments estimate for the parameter $\theta$ in the Pareto pdf, $$f_Y(y;\theta) = \theta k^\theta\bigg(\frac{1}{y}\bigg)^{\theta+1}$$ Assume that $k$ is known and ...
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1answer
37 views

Given the mean and the range of a normal distribution to find the variance [closed]

I am wondering is it possible to estimate the variance based on the mean and the range (max, min) of a normal distribution. I only need an approximated result.
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1answer
90 views

MLE for Uniform (0,theta)

I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$. I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelyhood ...
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1answer
28 views

Is $\limsup_{a\rightarrow0^+}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{1-|-a+(1-a)e^{it}|}\operatorname{d}t<\infty$?

Is $\limsup_{a\rightarrow0^+}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{1-|-a+(1-a)e^{it}|}\operatorname{d}t<\infty$? I've found this integral doing estimatation on the integral over circles ...
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3answers
57 views

Proof verification: $\lim_{n\to\infty}(\sqrt{n^2+1}-n)=0$

I'm having issues forming the discussion part of the proof because I am not sure if I am coming up with the right estimation. Is this an appropriate way of coming up with an estimation? I wrote: We ...