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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1answer
31 views

Error in average of $x^2$ from error in average of $x$?

Is there an easy way to obtain the error in $\langle{x^2}\rangle$ from $\langle{x}\rangle$ or are they independent? The values of x are from a molecular simulation application, I obtained a set of ...
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0answers
22 views

Number of measurements for least squares and relation to maximum likelihood

I have a simple overdetermined system of equations: $$ y = Xc + e $$ $y, e \in \mathbb{R}^n$, $c \in \mathbb{R}^m$, $X \in \mathbb{R}^{n \times m}$, $e \sim \mathcal{N}(o,\sigma^2)$, $n>>m$ ...
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1answer
38 views

Root Mean Square Error - How did he get this number?

So I am studying for a college final exam, and following a past exam paper at the moment. The lecturer has provided us with solutions to the previous years exam paper, not very clear in some cases ...
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0answers
6 views

PCA: MSE of projecting a set of points to a subspace

Let $\{x_1,\dots,x_n\}$ be a dataset of n vectors in $\mathbb{R}^d$ s.t. $\sum_{i=1}^n x_i = 0$. Let $p_1, \dots, p_k$ be a set of k orthonormal unit vectors and V the subspace that they span. Given ...
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1answer
31 views

how to find mininimum $f(x)$ using $\int_{-\infty}^{\infty} f(x)g(x)dx$?

I would like to know the $f(x)$ which minimizes the $\displaystyle\int_{-\infty}^{\infty} f(x)g(x)\,dx$. Actually, this question start from the MMSE (Minimize Mean square error) ...
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1answer
19 views

Derivation of MMSE from an estimator of two Gaussians

Suppose $X$ and $N$ are independent Gaussian with different variance but N has zero mean. Now $Y = X+N$. I am trying to find out the minimum mean square error estimator for $X$ given $Y$. I set the ...
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0answers
34 views

Minimise Mean square error(MMSE) proof procedure

I am awkard to understand the basic things so I have suffered from the procedure of proving the minimize the mean square error. the mean square error is $$ E[(X-g(Y))^{2}]=\int_{-\infty ...
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1answer
17 views

Is $Φ^T$ a linear operator which transforms simultaneous equations such that we obtain LMS solution?

The below explanation is long winded, if you already know about using pseudo inverse to find the best fit solution to a set of simultaneous equations please go down to the tl;dr The Problem Given a ...
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0answers
19 views

Absolute and relative errors

I am trying to compare two regression models, say $A$ and $B$ and calculate the absolute error measure (MAE, RMSE) and relative ones. But it turns out that both the absolute measures for $A$ are ...
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0answers
16 views

orthogonality condition in Minimum MSE linear estimator

I have questions about orthogonality condition in minimum MSE linear estimator. When estimating $X$ by a linear function $g(Y)= aY+b :$ $min_{\text{a,b}} E[(X-aY-b)^2].$ $b^*=E[X-aY]=E[X]-aE[Y]$ ...
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0answers
12 views

error on the median

I have a set of values ${x_i}, i=1, \dots ,N$ of which I calculate the median M. I was wondering how I could calculate the error on this estimation. On the net I found that it can be calculated as ...
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0answers
21 views

error calculation when the error is not constant

I have to calculate the error on the following quantity: $$f(\epsilon^M,\epsilon^S)= \sqrt{ \frac{1}{N}\sum_{i=1}^N (\log{\epsilon_i^M} - \log{\epsilon_i^S})^2 } $$ Usually I would use this standard ...
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0answers
14 views

Minimizing error of estimation in a differential equation system

I have a system of equations which describe dynamic nature of a system. There is no closed form solution for this system. System of equations is as follow: $$I_c=I_1 + I_2 + I_3$$ $$R_3 = ...
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32 views

How to solve this vector MSE equation?

Let's assume an error at time $k$ is: $e_k = \mathbf{c}^T \mathbf{r}_k - a_{k-d} - \mathbf{b}^T \mathbf{a}_k$, where $\mathbf{c} = [c_0, ..., c_{N_c-1}]^T$, $\mathbf{r}_k = [r_0, ..., r_{N_c-1}]^T$, ...
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1answer
86 views

On normalized error measures

I have function values $f_1,\ldots,f_n$ that are approximated by data $y_1,\ldots,y_n$. I am looking for a measure that describes the error in the data $y_1,\ldots,y_n$ and I want the measure to take ...
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0answers
11 views

Is the weighted mean of residuals over another variable equal to $0$?

I understand how residual errors must sum to zero around in a random sample (e.g. $y$-axis price of diamond predicted by x-axis weight of diamond). However, why must the weighted sum of residuals with ...
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0answers
16 views

What is the best formula to calculate percentage area of low-volume data (i.e. values close to 0)

I initially used the MAPE to calculate the percentage error between the actual data, and the data I got based on a model (forecast). However, near the values close to 0, the graph/percentage error got ...
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1answer
31 views

A way to calculate the error of a model?

I am currently making a model for a set of raw data of sea levels from the NOAA data base. On the site, the sea level is recorded every 6 minutes. Because I wouldn't have time to copy data every 6 ...
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0answers
23 views

Finding estimator with the smallest MSE

There is an estimator $\hat{\theta}$ of $\theta$ which has expectated value $\frac{3n}{3n+1} \theta$ and $E(\hat{\theta}^2)=\frac{3n}{3n+2} \theta^2$. Now I need to pick another estimator $X$, such ...
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1answer
64 views

Negative Mean Square Error

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. ...
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1answer
638 views

Matlab code for finding the curvature of a curve using given data points

I have data points $(x,y)$ for a plane curve, and I would like to find its curvature. Wwhile I was googling to check how could I start, I found this matlab code: ...
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1answer
83 views

How to minimize the minimum mean square error of this difference

I am trying to minimize the mean square error. More precisely, I am trying to minimize the following optimization problem $$\arg \min _{\bf{w_1},\bf{w_2}}\mathbb{E} \,\,[\|{\bf s} - {\bf Wy}\|^2 ]$$ ...
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0answers
18 views

Finding the closest vector to an observation

I have a collection of vectors (a codebook in hand) which are presented within a matrix $A$ $$ A = [ a (\theta_1), \, a (\theta_2), \, a (\theta_3), \, a (\theta_4), \,\cdots]$$ We have obtained ...
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1answer
121 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
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0answers
33 views

Can the error term variance ever be estimated without fitting a regression line in a basic linear regression model?

Can the error term variance ever be estimated without fitting a regression line in a basic linear regression model? I don't understand how this would be possible and why. Because wouldn't you always ...
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0answers
73 views

Mean squared error consistency of estimator

Given is the following distribution: $f_\theta(x)=\frac{1}{\theta}$ if $0<x\leq\theta$, and $0$ otherwise; $\theta<0$. I need to show that the maximum likelihood estimator of $\theta$, ...
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0answers
17 views

MMSE detector for elliptical distribution.

Suppose we have $Y=HX+W$ where dimension of $Y$ is $N$ and $W$ is elliptically distributed $H$ is also elliptically distributed $X$ is uniformly distributed. We want to estimate $\hat{X}$ using MMSE ...
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2answers
49 views

Estimation, bias, and mean square error

Let $X$ be a continuous random variable with pdf $f(x) =\frac{1}{2}(1+ \theta x)$, for $-1 < x < 1$, and $-1 < \theta < 1$ (a) Show that $E(X) = \frac{\theta}{3}$. (b) Given a random ...
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0answers
42 views

Error propagation with dependent errors

I have a function $f(x_1,\ldots,x_n)$ where the variables $x_k$ have errors $\delta_k$. If these errors are independent I can add them root mean square: $\delta ...
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2answers
138 views

What is the Difference between Variance and MSE

I know that Variance measures the dispersion of an estimator around its mean i.e. $\sigma^2=E[X - \mu]^2$ or Second central moment (moment about the mean) But ...
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0answers
17 views

What are some error measures used for fitting PMFs?

I have a given PMF, $f_X(x)$, and am trying to create a fitted PMF, $g_X(x)$, that comes "as close as possible" to it, but am not sure what to use as a measure of fit. Simply minimizing standard error ...
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1answer
46 views

Matrix of vector-by-vector sum-squared deviations of two matrices of column vectors

Context I'm working on a Python program in which I will calculate some number $r$ of matrices $\mathbf{A}^i$ with identical dimensions $m\times n$. For this application, each matrix is probably best ...
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1answer
113 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
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1answer
41 views

Least-squares solution to a linear matrix equation

Let $\\A$ be a matrix of size $\\(m, n)$, $\\b$ a column vector of size $\\m$, $\\x$ a column vector of size $\\n$ and $\\a$ a real number. If $\begin{bmatrix} x \\ a \end{bmatrix}$ is the ...
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0answers
28 views

Cost function for very sparse, real-valued data

Suppose the target output of my data prediction model is an $M\times N$ matrix where $95\%$ of the values are $0.0$ and the other values are anywhere between $0.0$ and $1.0$, what would be a good loss ...
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1answer
212 views

comparing MSE of estimations of binomial random variables

$X$ is a binomial random variable defined over 12 Bernoulli trials with a success probability of $p$ in each (i.e. $X\sim\operatorname{Bin}(12,p)$. Consider $\hat p=\frac X{10}$ Determine the range ...
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1answer
47 views

mean square error comparison

Do you have any idea about how i can solve the question below? $X_1$ and $X_2$ are random variables that satisfy $E[X_1]=E[X_2]=\mu$ and $Var[X_1]=Var[X_2]=1$. Show that when $|\mu - 10| \leq ...
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1answer
140 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
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0answers
85 views

How to fit normal cumulative distribution functions

For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means $$ ...
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2answers
140 views

How to find the bias, variance and MSE of $\hat p$?

If $X_1,\dots,X_n$ are iid $\mathrm{Binomial}(3,p)$, then the maximum likelihood estimator of $p$ is $$\hat p = \frac{1}{n}\sum_i X_i$$ Find the bias, variance and MSE of $\hat p$? We are asked to ...
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1answer
33 views

Standard error of RMSE?

If I want to calculate the RMSE between an estimated value $\hat{x}$ and its reference value $x_{\textrm{ref}}$, let \begin{equation} y_i = \hat{x}_i-x_{i,\textrm{ref}} \end{equation} Since ...
2
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0answers
30 views

minimizing mean square error with type 1 and 2 error weights

Suppose we have a random variable $X$ with a pmf that puts strictly positive probability only on integer values $0,1,2,\dots,n$. The objective is to choose a $z\in\mathbb{Z}$ that minimizes ...
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2answers
149 views

Test for, and compare means of folded normal distribution

I have two datasets of absolute distances to a single point in a 2D space. I have reasons to expect that if I had the sign and magnitude of these distances, my datasets would be normally distributed ...
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1answer
47 views

How to find the minimal MSE?

I'm confused as in how to find $⍴$ in c) and why $σ^2$ gives a smaller MSE than $s^2$ I know $MSE(θ) = E(θ - θ_0)^2 = Var(θ) + Bias(θ)^2 $ and that $ Bias(θ) = E(θ) - θ_0$ But I don't get what θ is ...
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1answer
112 views

linear regression, expectation and mean squared error

Let us assume that data is generated according to a true model $$y_i = \beta_{true}x_i + \epsilon_i$$ for $i = 1, ..., n$ Assume that $x_i$ are fixed, and $\epsilon_i$~ N(0, $\sigma^2$) ...
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0answers
183 views

Finding a relative error measure on a data set proportional to another

I have a set of exact data points $\mathcal{X}=\{X_i\}$ and another approximate one $\mathcal{Y}=\{Y_i\}$ where there is a correspondence between $X_i$ and $Y_i$ for all $i$. If $\mathcal{Y}$ was ...
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1answer
131 views

Finding the best linear predictor

How do I find the best linear predictor of $X_{n+1}$ in terms of $X_{n-1}, X_n$, if $X_t$ is the MA(1) model $X_t = Z_t + \theta Z_{t−1}$.
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1answer
22 views

Getting the average of values with errors.

I have five data values each with an associated error. I want to find the mean of these values but also take the errors into account. How do I do this? Lets say the data values and errors are: ...
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2answers
614 views

How do I find the angle $\theta$ of the line through the origin that matches the given points the best?

How to find angle $\theta$, that the line passing through the origin that is the best fit for the points given below in the mean square sense makes with the horizontal axis. $$x_1=[1\;\; 2]^T$$ ...
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0answers
44 views

Mean square relative error. Some considerations

I'm facing with the following mean square relative error $$\frac{1}{T}\sum_{t=1}^T s_t^2 = \frac{1}{T}\sum_{t=1}^T \left(\frac{a_t - b_t}{b_t}\right)^2$$ There are two circumstances I don't know how ...