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Questions tagged [mean-square-error]

This tag is for questions about mean-square-error. In statistics, the mean squared error (MSE) of an estimator measures the average of the squares of the errors or deviations, that is, the difference between the estimator and what is estimated.

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Calculating the Mean Square Error (MSE) in Wavelet Denoising

I´m currently reading the paper (to be more precise: it´s a chapter from the book "Shearlets, Multiscale Analysis of Multivariate Data" by Kutyniok and Labate) "Image Processing Using Shearlets" by G....
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1answer
46 views

Why MSE formula is looking so different?

Here's what I randomly found on course online (Google Course, Andrew NG, etc.) about Mean Squared Error (MSE) prediction - actual or ...
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1answer
12 views

Adding the variances of 2 dependent variables and covariance

$ E(\hat \theta_1 ) = E(\hat \theta_2) = \theta_1 $ , $ Var (\hat \theta_1) = \sigma_1 , Var(\hat \theta_2) = \sigma_2, Cov(\hat \theta_1, \hat \theta_2) = \sigma_{12}$ $\hat \theta_3 = a \hat \...
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0answers
14 views

MSE Minimized by Mean

In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $X \in L^2(\mathcal{F};\mathbb{P})$, is a random-variable in $\mathbb{R}$, then is the ...
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1answer
21 views

Determine the value of $α$ for which the $MSE(T)$ is minimal.

Let $X_1$ be an estimator for the probability $θ$ of unauthorized access. Let $X_2$ be another estimator for $θ$. Assume that $X_1$ and $X_2$ are independent, unbiased estimators for $θ$. ...
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1answer
56 views

How to quantify the confidence on a Bayesian posterior probability?

Consider a physical system that depends on a parameter $0\leq \phi <\infty$. I want to (i) find the probability that this parameter is smaller than a critical value: $\phi\leq \phi_c$, and (ii) ...
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2answers
53 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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Relationship between covariance eigenvalues and MSE

I have a matrix $A$, and $\tilde{A}$ is the approximation matrix calculated as the projection of $A$. Now I calculated the MSE between $A$ and $\tilde{A}$, and the $C$ as the covariance matrix of $A$. ...
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1answer
30 views

What does this notation in minimal square error mean?

My book defines minimal square error as: $$MSE(\theta, T) = E_\theta\Vert T - \theta \Vert^2$$ What does the $E_\theta$ mean? Is it an expectation? If yes, what does the theta supscript do there?
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What is best way to represent this false +ve error to reflect original scenario?

My aim is to represent false +ve error in a better way. I have 500 samples. Each samples contains some distinct values (there exists total 16 distinct values). Ideally, a sample should contain any 4 ...
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1answer
22 views

Finding the sample size given bound of error and standard deviation.

The personnel manager wants to determine if there is a difference in the average time lost due to absenteeism between two plants. From historical data, the estimated standard deviations of lost time ...
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1answer
62 views

Introduction Mathematical Statistics, finding the Mean Square Error of estimators

I'm working on a problem from my introduction to mathematical statistics course. So far, I've done the following work: Let $X_{1},...,X_{m}$ and $Y_{1},...,Y_{n}$ be independent samples form the ...
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1answer
98 views

MSE of the MME of $\theta$

A random sample of size $n$ is taken from a gamma distribution with parameters $α = 8 $and $λ = 1/θ$. The sample mean is $\bar{x}$ and $θ$ is to be estimated. Determine the mean square error of the ...
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1answer
52 views

Determine the variance of estimator T

I'm having trouble figuring out how to find the variance of the following estimator. Let $X_1,X_2,...,X_n$ denote random sample from a population which has a normal distribution with unknown mean $\...
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1answer
39 views

Is mean squared error equivalent to mean squared absolute error

It is a simple question. But, I want to make sure I am not missing any crucial assumptions. Is this $A = (\hat{Y} - Y)^2$ same as $B = (|\hat{Y} - Y|)^2$ Main concern: If $A$ returns negative ...
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1answer
247 views

Calculating variance of an estimator

Given that $ \operatorname{Var}(x)=\frac{3}{4}\theta^2$, I want t find the variance of estimator $\hat{\theta_1} = \frac{2n}{3}\sum_{i=1}^nX_i$. EDIT: $X_1,...,X_n$ are independent and identically ...
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1answer
60 views

moments estimation using Rayleigh distribution

consider Rayleigh distribution: $f(x;\theta) = \frac1\theta e^{\frac{-x^2}{(2\theta)}}$, x > $0$ and $\theta$ > $0$ Show that $E(X^2) = 2\theta$ On the basis of the proceeding item, construct an ...
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1answer
527 views

Minimum mean squared error of an estimator of the variance of the normal distribution

I am trying to find the estimator of the variance $\sigma^2$ of a normal distribution with the minimum mean square error. From reading up, I know the unbiased estimator of the variance of a Guassian ...
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0answers
12 views

How to find $min \mathbb{E} \Big[ || (X_1,X_2) - u(X_3) ||^2\Big]$ when $X \sim Dirichlet (\alpha_1, \ldots, \alpha_4)$

I'm trying to find the minimum of the following formula, where $(X_1, \cdots, X_4) \sim Dirich(\alpha_1, \cdots, \alpha_4)$. $\min_{u(\cdot)} \mathbb{E} \Big[ || (X_1,X_2) - u(X_3) ||^2\Big]$ Since ...
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1answer
138 views

MSE of estimator for normal distribution [closed]

Assume $X \sim N(\mu, \sigma)$. Find $c$ for which $ T = c \sum_i (X_i - \bar{X})$ has minimum MSE. We know that $ \operatorname{MSE}(T)= \operatorname{Var}(T)+ [E(T) - \theta]^2 $ Then: $$ E(T) = ...
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1answer
54 views

Mean squared error calculation [closed]

If $ X_1,...,X_n$ ~ $N(\mu, \sigma^2)$ where $\mu$ is known and $\sigma^2$ is unknown, calculate the MSE of $V^2$ $V^2 = \frac1n \sum_{X_i}^n Var(X_i) =\sigma^2$ Therefore: $MSE(V^2) = Var(V^2) = \...
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Comparison of errors of sample mean and sample median [closed]

I have a question considering the sample median of a mathematical sample. We have $n$ independent and identically $N(\mu,\sigma^2)$-distributed random variables $X_i$, $i=1,\dots,n$. Suppose that $\...
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1answer
61 views

Comparison of two estimators using the mean-square error

Let $ X $ be a single observation from Bernoulli density $ f(x;\theta) = \theta^{x}(1-\theta)^{1-x}$, $ x\in \lbrace 0,1\rbrace $, $ 0<\theta<1 $. Let $ t_{1}(X)=X $ and $ t_{2}(X)=\dfrac{1}{2} $...
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38 views

Limit Sup of mean square error of 2 Gaussian process

I have been struggling to proceed to how to proof this theorem. I saw this theorem in statistics specially in spatial statistics. I have seen a proof to which I understand but I want to use a ...
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1answer
22 views

Why the writer of this article divided the mean square error formula by 2 instead of MxN?

The link of the article, the intuition he made in his article is very convincing the understand the relation between mean square error and Gaussian distribution. but he changed the formula of the MSE ...
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0answers
86 views

What value of a minimises the MSE of this estimator

Consider estimators of the form a∑$(X_i − X)$$^2$ for $σ^2$ , the variance of a normal distribution with unknown mean µ and $σ^2$ , given a random sample of size n. Find the value for a that minimizes ...
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1answer
30 views

Comparing mean squared errors

Let $ X_{1}, X_{2}, \dots ,X_{n} $ be linearly independent random variables with the same distribution and probability density function given by $$ f_{\theta}(x)=\frac{1}{\theta+1}e^{-\frac{x}{\theta+...
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42 views

Can we here derive general expressions for Mean Squared Error and $R^2$?

Suppose we have the linear relationship $$ y= \beta _1 X_1+\beta _2 X_2+\beta _3 X_3 + \cdots + \beta _n X_n $$ and we have an estimate of $y$ given by $$ \hat{y}= \hat{\beta }_1 X_1+\hat{\beta }...
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114 views

Incremental (“online”) computation of the root-mean-squared-error (RMSE) for a linear regression

I've been able to calculate a least squares linear regression's slope incrementally. (https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Covariance shows one-pass "online" methods for ...
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2answers
453 views

Minimum mean squared error of uniform distribution

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\operatorname{Uniform}(-\theta,0)$. Now consider all of the estimates of the form $S_\rho=\rho \hat{\theta}_\text{MLE}$. I have to find which of these estimates has ...
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1answer
155 views

Mean Squared Error

How do we use Jensen's inequality to prove that $\operatorname E(T_2-\theta)^2 < \operatorname E(T_1-\theta)^2$, where $\theta$ is an unknown constant, $T_1$ is an estimator for $\theta$, and $T_2 =...
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0answers
70 views

Time Series Mean Square Convergence with given acvf

Let $(X_t)_{t\in \mathbb{Z}}$ be a stationary time series with $E[X_t]=0$ and acvf $\gamma(h)={1}_{0}(h)+\rho1_{h \ne0}(h)$ with $\rho \in (0,1)$. Show that $\frac{1}{n}\sum_{j=1}^nX_{-j}$ ...
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1answer
120 views

Derivation of complex number for Linear MSE estimator

Given a system $$y = h x + n$$ where $x$ is unknown to estimate, $y$ is observed data, $h$ is known, $n \sim \mathcal{N}(0,\sigma^2)$. They are complex numbers. I am trying to prove the linear ...
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43 views

For $X_i\sim N(\mu,\sigma^2)$, how to find the MSE of estimator: $\hat \sigma^2 = \frac 1{n+2}\sum_{i=1}^nX_i^2$.

This is part of a larger problem. I have already found the MSE of two other estimators: $\hat\sigma_1^2 = \frac1{n-1}\sum_i^n (X_i-\bar X)^2$ and $\hat\sigma_2^2 = \frac1{n+1}\sum_i^n (X_i-\bar X)^2$. ...
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0answers
56 views

Means Square Percentage Error vs OLS

I am trying to run a simple regression $$Y = a+bX +e$$ however I want to optimize it on Mean Square Percentage Error and not Mean Square Error as in OLS. Like this: $$argmin: e/Y = (Y-a-bX)/Y$$ ...
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3answers
628 views

Minimum Mean Square Error Estimate Example

We have data from 2D normal (gaussian) distribution. $$\begin{bmatrix}y\\x\end{bmatrix}\,\text{~}\,\mathcal{N}\left(\begin{bmatrix}2\\4\end{bmatrix},\begin{bmatrix}10&2\\2&20\end{bmatrix}\...
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1answer
99 views

Bias and variance of two alternative estimators

Apologies in advance as this seems like quite a simple problem but I have hit a wall and could use some guidance on next steps. Here is the problem and then where I am at: Suppose that the random ...
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1answer
212 views

A statistic minizing *Mean Absolute Relative Error*

Mean of values (x1, ... xi,...xn) will minimize Mean Square Error, Median of values will minimize Mean Absolute Error. I would intuite Median would also minimize Mean Absolute Relative Error (MARE) ...
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1answer
217 views

Unbiased Estimator Questions x2

Hi guys needed some help with these two questions. I've tried to go over my lecture notes, but I'm still struggling to get my head around the topic. I hope I've edited the questions properly using ...
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1answer
64 views

What is error covariance of a BLUE estimator equal to $E(\hat{x}-x)(\hat{x}-x)^T$?

I am working through some problems about BLUE estimators (Best Linear Unbiased Estimators). What I have discovered recently is that the error covariance of an unbiased estimator is not the same thing ...
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1answer
549 views

MSEs of Estimators of Variance in Normal Distribution

$\newcommand{\MSE}{\operatorname{MSE}}$Consider the mean squared error (MSE) of the following estimators of variance, where $X_i$ is given by the normal distribution: $$\MSE(S^2)=\MSE(\frac{1}{n-1}\...
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256 views

How to calculate RMSE for a non linear model?

I am doing a project in which I need characteristics for signal strength to distance. I have done measurements and made a curve fit of mean values. For a log-plot the model becomes linear and an RMSE ...
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1answer
693 views

Minimizing the expectation of the loss function

So i was reading Elements of Statistical Learning and found this in the Statistical Decision Theory part.I Did not understand it. The expected (squared) prediction error . By conditioning on $X$, ...
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0answers
32 views

Approximate function to reduce number of defining points

I have a function defined with a series of arbitrary points $(P_1,P_2,...,P_n); P_k(x_k, y_k)$ such that $x_k<x_{k+1}$, and $$f(x) = y_k\frac{x_{k+1}-x}{x_{k+1}-x_k}+y_{k+1}\frac{x-x_k}{x_{k+1}-x_k}...
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1answer
81 views

MATLAB need help in writing mean absolute error code

I am using this code but it is throwing error- Undefined function 'symsum' for input arguments of type 'double'. ...
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1answer
62 views

Basic Standard Deviation Question - Curtain rails

Hopefully you guys can help me. I'm completely stumped by normal distribution and the like. No matter what I do, I can't get it right. Hopefully someone here can save me! This is my question: A ...
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1answer
14 views

How to differentiate imaginary vector?

I am finding a vector $\mathbf{c}$ that minimizes $J(\mathbf{c})$, that is $J(\mathbf{c}) = \text{E}\left\{\left\vert\mathbf{c}^H\mathbf{r} - a_{t}\right\vert^2\right\}$. $\mathbf{c}$ and $\mathbf{r}...
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1answer
548 views

Expected test error in regression

I am unsure regarding the definition of the expected test error here. As far as I understand the definition it is the following. In a linear model the relationship between the random response ...
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1answer
72 views

Help with mean squared error

I am doing this question on mean squared error, but I don't know how to do any of the parts. This is the question: Any help? Thanks!
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0answers
73 views

MSE for Dependent, Unbiased Estimators

$\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ are unbiased, dependent estimators of $\theta$ with some $\rho$. I found that when $$\lambda = \frac{{\sigma_2}^{2} - \rho {\sigma_1} {\sigma_2}}{{\sigma_1}^{...