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Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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1answer
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Concluding consistency of estimators

Say we have a set of $n$ iid rvs with variance $\sigma^{2}$ and an estimator T of some parameter $\theta$. If we know that $Var(T) = {\sigma^{2}\over n}$, is that enough for us to conclude that our ...
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On a nonlinear regression problem

Consider the function $f\colon \mathbb{R}^2\to \mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data $\{(x_1^{(i)},...
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Show $\frac{X_{(i)}-X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ if $X_i$'s are i.i.d $U(\theta_1,\theta_2)$

Suppose $X_i$'s are i.i.d $U(\theta_1,\theta_2)$. Show that $\frac{X_{(i)} - X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ for all $2\le i\le n-1$. Specifically I'm trying to ...
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Is absolute value of the empirical mean unbiased/biased?

Given a random variable $x$ defined on $\mathcal{X}$ and corresponding i.i.d. observations $X_i$, $i=1,...,n$, could we estimate a following probability? $$P\left(\left| \left|\mathbb{E}(x)\right|-\...
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3answers
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Solve for Y in classic bond formula

I'm trying to set up an excel spreadsheet that solves for $Y$ in the classic bond formula: $P = \frac{C}{(1+Y)}+\frac{C}{(1+Y)^2}+\frac{C}{(1+Y)^3}+_{......}+\frac{(C + Q)}{(1+Y)^n}$ Where "C" is a ...
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32 views

Deriving the least squares estimate of $\beta_{k-1}$

Let $y_i=\Sigma^k_{j=0} x_{ij} \beta_j+\epsilon_i$ $\epsilon_i$ is $NID(0,\sigma^2)$ and $x_{ij}, i=1,...,n, j=0,...,k$ is the $(i,j)^{th}$ elelement of the $n \times (k+1)$ matrix $X$, which is of ...
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1answer
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How to show that MSE of ML estimator is greater than that of Bayesian posterior mean?

This question is based on problem 9 from chapter 4 of Gelman et al.'s Bayesian Data Analysis. Suppose we observe $y\sim N(\theta,\sigma^2)$ and wish to estimate $\theta$, with $\sigma^2$ known. We ...
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31 views

Mean estimator of a Gaussian variable with positive mean for quadratic loss

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian, and $X\sim\mathcal{N}(\theta,1)$, in which $\theta>0$ is positive but othrewise unknown. We want to estimate $\theta$ ...
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centroid algorithm robust to missing poins

I need to find a center point of a person given the coordinates of all the joints. The joints of a person can be represented as a nodes of a graph with a fixed structure. The catch is some of the ...
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Identifying all the 1s in a sea of 0s

I have an $N$-length binary string, the total number of 1s in the string is D. Assume N is much larger than D. Also, D is fairly larger than 1. (For example, N=1000, D=40). Now, my aim is to find ...
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Modelling conditional distribution based on multiple variables of various types?

I have a looking basic statistics problem: basing on a large sample of multivariate data, model conditional probability distribution (continuous) of one variable based on the remaining ones: a few ...
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1answer
39 views

Finding UMVUE for uniform distribution $U(\alpha, \beta)$

Let $X = (X_1, X_2, \ldots, X_n)$ be a sample from uniform distribution $U(\alpha, \beta): \alpha, \beta \in \mathbb{R}, \alpha < \beta$. I am to find UMVUE for the parameters $\alpha, \beta$. ...
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1answer
49 views

How do you estimate the remaining degree distribution?

Let $q_{j,k}$ be defined as the joint probability distribution of the remaining degrees of the two nodes at either end of a randomly chosen edge. Let $G=(V,E)$ be an undirected graph with nodes $V=(...
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Is there any relationship between efficiency and correlation coefficient?

Let $t_1$ be the most efficient estimator and $t_2$ be the less efficient estimator with efficiency $e$ and let $r$ be correlation coefficient between the two estimator $t_1$ and $t_2$.Define ...
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1answer
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Demonstrate that solution of differential equation is bounded

Let $b(t) \in C^1([0,+\infty))$. I have to find a formula for the solution of this Cauchy's problem: $$ \left\{\begin{aligned} x''(t)+x(t)&=b(t) \\ x(0)&=x_0\\ x'(0)&=x_1 \end{aligned}\...
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Fitting a sinusoid vs. DTFT

I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized: $$\text{...
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1answer
52 views

Finding a upper bound (estimation) for the infinite series: $\frac{n^5!}{(n+1)^5!}+\frac{n^5!}{(n+2)^5!}+\frac{n^5!}{(n+3)^5!}+\cdots$

I have the infinite sum: \begin{align} S_n = \frac{n^5!}{(n+1)^5!}+\frac{n^5!}{(n+2)^5!}+\frac{n^5!}{(n+3)^5!}+\cdots \end{align} I must find a upper bound for $S_n$, that is, $0<S_n<f(n)$, ...
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Numerical estimation

Show that the numerical value of the expression $\frac{2\times4\times6\times\dots\times2020}{1\times3\times5\times\dots\times2019}$ is between 44 and 64.
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What's the error in this birthday-problem estimation?

Theorem. Choose $Q$ random natural numbers in the set $\{1,2, ..., M\}.$ The probability of getting at least one collision is $$P_C(Q) = 1 - \frac{M - (Q - 1)}{M} P_{\neg C}(Q-1).$$ Notation. By $...
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1answer
67 views

Existence of MLE

I have a problem with MLE's definition: Casella Berger in Statistical Inference and Nitis Mukhopadhyay in Probability and Statistics said that MLE for a parameter $\theta\in\Theta$ is respectively $...
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Evaluating variance of scale parameter estimators

Let $\{Y_{i}\}_{i=1}^{n}$ be a random sample of the random variable $Y_{i}\sim \mathcal{N}(0,\sigma^{2})$, we define the following estimators for $\sigma^{2}$ $U=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-...
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What is the unbiased estimator of covariance matrix of N-dimensional random variable?

Suppose $x$ is a random vector in $\mathbb{R}^n$ which is distributed according to $D$. What is the unbiased estimator of covariance matrix of an N-dimensional random variable? When $y$ is a i.i.d. ...
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1answer
40 views

Infer the variance of a distribution

Assume that the distribution of firms' income is log-normal, and that we know the sum of the all the revenues and the number of firms in the market (so the mean of the revenue distribution is known). ...
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1answer
43 views

Example of a maximum likelihood estimator that is not a sufficient statistic

I am currently researching on providing some bounds on estimation using some information theoretic tools (I won't expend on that here for now, I may make a post about it later) and turns out that ...
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0answers
8 views

Complexity of histogram scheme

Due to work by Lugosi and Nobel available here I know that for consistent density estimation complexity of histogram scheme should increase sub exponentially. By complexity we mean: Given a number of ...
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0answers
29 views

Estimating the value of the eigenvalue

I was trying to see whether the eigenvalue $\lambda_{\pm}$ lies on the unit circle or inside the unit circle? but I am trying to estimate it but unable to do so as they are variables, but there is ...
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10 views

Relation between parzen window estimate and histogram

Can histogram density estimation be seen as approximation to parzen density estimation with the usual hyper cube function $1/h^d \Sigma_{1 \le i \le n} \varphi(\frac{x-x_i}{h})$ where $\varphi(y)=1$ ...
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Convergence of density estimates with parzen window

I am trying to understand why $lim_{||u|| \rightarrow+\infty}{\varphi(u)}\prod_{i=1}^{d}u_{i} = 0$ is necessary for convergence of Parzen density estimates. Similar question has been asked here ...
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1answer
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Unbiased estimate for a parameter

They ask me to estimate any parameter and I do not if the solution is correct: The life time X of a battery is considered to be a random variable with density function $f (x; Θ) =\frac{ 2 }{ Θ²} (Θ -...
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1answer
34 views

Unbiased estimate for a parameter.

The time T that it takes to execute an optimization algorithm is assumed to be a random variable with the parameter distribution Θ > 0 $f (t; Θ) = \frac{t} {Θ^2} e ^\frac{-t}{Θ} $ if t > 0 Let T1, ...
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$X_1$,$X_2$…$X_n$ are iid N($\mu,\sigma^2$). Derive a confidence interval for the parametric function $\mu+\sigma$ and $\mu/\sigma$.

I know how to find confidence interval for each of the parameters $\mu$ and $\sigma^2$ but don't know how to find in the case of parametric function.
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1answer
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Sufficient statistics for $p$ for a random sample from $\text{Ber}(p)$ distribution

Let $X_1,X_2, X_3$ be a random sample from Bernoulli distribution $B(p)$. Which of the following is sufficient statistic for $p$ ? $(A)\ \ X_{1}^{2}+X_{2}^{2}+X_{3}^{2}$ $(B) \ \ X_1+2X_{2}+X_{3}$ $...
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44 views

An estimate for a 1d hyperbolic PDE

Let $L, T, \lambda> 0$ be fixed, and let $f \in C^1([0,T];H^1(0,L))$, $g \in C^1([0,T];H^1(0,L)) \cap C^2([0,T];L^2(0,L))$ and $v^0 \in H^1(0,L)$. Consider the problem $$ \begin{cases} \partial_t ...
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4answers
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Limit $\lim_{x\to\infty}\left(1-\frac{a^2}{x^2}\right)^{x^2}$

I found this example in a textbook, and I understand the author's reasoning and I also reached the same answer using L’Hôpital’s rule. However, I have two issues: Firstly: For any finite $a$, then ...
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0answers
28 views

Reducing the confidence interval with T distribition

My question is the following: The population: weight of apples (in grams). We do not know anything about the population distribution except that it is a normal distribution. We want to find a 90% ...
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Finding Best Unbiased Estimator of Uniform Distribution

Let $X_i$, $i=1,...,n$ be iid with $f(x,\theta) = \frac{1}{2\theta}$ for $-\theta<x<\theta$. Find the best unbiased estimator of $\theta$ if one exists. So I first tried $T(X)=X_{(n)}$, which ...
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Why residuals are a good estimator of random disturbance?

Let a linear OLS model: $$Y= X \beta + u$$ Where $u$ is a random disturbance. If we define the residual of the regression as $$e = Y - X \widehat{\beta}$$ where $\widehat{\beta}$ is the OLS vector of ...
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0answers
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Sample moments Intuitive explanation

Can anyone intuitively explain to me the definition I have posted above? Also, what is the purpose of this definition i.e. how do you use it?
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0answers
33 views

How to show that the following integral is infinite

For an application I want to show the following claim: Let $x\in(0,1)$ and $y\in[0,1]$ be two parameters, then $$\int_{\mathbb{R}}\frac{1+\cos(\pi x (y-0.5))|z|^{x}}{1+2\cos(\pi x (y-0.5))|z|^x+|z|^{...
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1answer
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Error when trying to derive variance of sample mean

Assume that $X$ is a random variable with mean $E[X]$ variance $\sigma^2$. Let $\mu = \frac{1}{N}\sum_{n=1}^N X_n$, where $X_n$ is i.i.d with respect to $n$ having mean $E[X]$ and variance $\sigma^2$, ...
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estimate population sum given population size and a sample mean and variance

I have a question which seems easy but I'm not sure how to solve it. Assume we have a population of $N$ integer numbers, $x_1, x_2, ..., x_N$. So $N$ is population size. I'm interested in the sum of ...
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1answer
33 views

Find max likelihood estimator for a 2 variable function

I have been given a problem in which I am given a set of independent random variables, X1, X2, ... Xn with the distribution: $f_x(x ; b, m) = \frac{1}{2b}e^{-|x-m|/b}$ for $-∞ < x < ∞$ and I ...
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1answer
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Proving an estimate of $ \mathcal{L}^n (A_1 \cap A_2)$ from estimates of $\mathcal{L}^n (A_1)$ and $ \mathcal{L}^n (A_2)$.

Consider two measurable sets $A_1, A_2 \subset \mathbb{R}^n$ and let $B$ be an open ball in $\mathbb{R}^n$. Assume that $A_1 \subset B$, $ A_2 \subset B$ and that for a certain $0 < \epsilon < 1 ...
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Which information gives least minimum mean square error?

out of : probability density of Y Realization of another Random variable X = x Realizations of X i.e a number of realizations of X Each have their Mean square errors, which one has the least MSE ?
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1answer
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Estimating an integrand

Given an integral $$\int_1^T \frac{f(t)}{t} \, {\rm d}t$$ where $f(t)$ is oscillating and I want to make an estimate I can do the following $$\left|\int_1^T \frac{f(t)}{t} \, {\rm d}t\right| \leq \...
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0answers
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Showing that a estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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2answers
44 views

Check if the estimator is unbiased

For $X_i\sim U[0,a]$ where $i=1,2,\dots,n$ so, $E(X_i)=\dfrac a2$. Is $a'=\max\{X_1,X_2,\dots,X_n\}$ an unbiased estimator of $a$? This is what I thought. Since $a'=\max\{X_1,X_2,\dots,X_n\}=X_k$ ...
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1answer
44 views

$\lim_{r\to\infty}\int_{c-ir}^{c+ir}z^{-\lambda}(\log(z))^{-1}dz$

Let $c>1$ and $\lambda\in\mathbb{C}$ with $\Re(\lambda)\geq1 $. Show that $\lim_{r\to\infty}\int_{c-ir}^{c+ir}z^{-\lambda}(\log(z))^{-1}dz=0$, where the path of integration is the straight line and ...
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0answers
34 views

Deriving UMVUE for $\mu\sigma^k$ when both $\mu$ and $\sigma$ are unknown and are the parameters of a normal distribution

Let $X_1,...,X_n$ be iid $N(\mu,\sigma^2)$, and define $f$ as $f( \theta)=f(\mu,\sigma)=\mu\sigma^k$. I'm attempting to find the UMVUE for $f(\theta)$ via the Lehmann-Scheffe approach, i.e. I'm ...
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0answers
10 views

Bias of a PCA Estimator

For the model $\mathbf{y} = \gamma\mathbf{X}+\mathbf{\epsilon}$ we do a PCA regression and estimate the parameter $V\hat \gamma= \hat\beta = \sum_{i=1}^k \lambda^{-1}_i \mathbf{v_i} \mathbf{v_i}^...