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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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17 views

Finding estimator that minimizes the weighted mean-square error

I have continuous multivariate random variable $x$ in $R^n$ with known prior $p(x)$ over the latent random variable. I observe $z$ and want to come up with a estimator for $x$ that minimizes the ...
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22 views

What's the intuitive purpose of RMSE (root mean square error) compared to MAE (mean average error)?

When I want to find out what the average difference / error there is between two datasets, such as a predicted output vs. observed output of any system (i.e.: I predict output to be 100V, how does the ...
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1answer
29 views

Given the Laplace transform of a function $f(t)$, can I find the “total squared error” $\int_0^\infty f(t)^2\ dt$?

I'm modeling the behavior of an aircraft using ordinary differential equations. I've written an equation for the angle of attack $\alpha(t)$, and then taken the Laplace transform of this equation to ...
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16 views

Normalise the root mean square error

I have $10$ people in a group and they undergone a surgery. I have the root mean square of each subject before and after the surgery. ...
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1answer
42 views

$Y_1,Y_2,Y_3$ are uncorrelated rvs such that $E(Y_1)=\beta_1+\beta_2$,$E(Y_2)=2\beta_1$ and $E(Y_3)=\beta_1-\beta_2$

Let $Y_1,Y_2,Y_3$ be uncorrelated random variables with common variance $\sigma^2 > 0 $ such that $E(Y_1)=\beta_1+\beta_2$,$E(Y_2)=2\beta_1$ and $E(Y_3)=\beta_1-\beta_2$ where $\beta_1$ and $\...
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12 views

Size of replicates needed to calculate MSE in R?

I'm calculating MSE for cauchy distribution estimators in R. My code is the following ...
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24 views

Finding MSE for cauchy distribution estimators in R. Is this method and findings correct?

Hi I'm required to find the MSE of a few estimators of the cauchy distribution for location $\theta$. I am using the following R code but my MSE values are coming out incredibly worrying. I am ...
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1answer
22 views

Minimise computational cost for given level of MSE

I am trying to understand how to minimise cost of a Monte Carlo implementation for a given value of MSE/RMSE. Please see the notes attached...I do not follow the second line. I would be grateful if ...
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1answer
40 views

One way ANOVA by hand

I am trying to do an ANOVA by hand to the following data. Summer 83 85 85 87 90 88 88 84 91 90 Shoulder 91 87 84 87 85 86 83 Winter 94 91 87 85 87 91 92 86 I ...
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34 views

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
65 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
16 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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17 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
59 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
66 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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25 views

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
31 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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14 views

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
72 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
154 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
56 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
390 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
85 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
785 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
226 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
63 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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0answers
25 views

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
92 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
23 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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110 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
33 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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123 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
518 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
164 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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78 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
155 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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61 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
750 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
110 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
234 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
237 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
83 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
661 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}\...