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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2answers
150 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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2answers
355 views

least square adjustment of resection

By setting up at an unknown point, and measuring the horizontal angles between three points with known coordinates, it is possible to calculate the coordinates of the unknown point. This process is ...
1
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1answer
36 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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1answer
155 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 off ...
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1answer
42 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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1answer
20 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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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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1answer
33 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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1answer
113 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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1answer
56 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
51 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
117 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, $\sigma^2$) independently....
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1answer
145 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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1answer
62 views

differentiating MSE

I have a error signal which I want to minimize using MSE. This error signal at time $k$ is a vector of length $3$: $e_k = C^{T} R_k - B^{T} A_k = [c_0, \ldots, c_{N_c-1}] \begin{bmatrix} r_{2k}\\ \...
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1answer
40 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
3
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0answers
240 views

root mean square distance between two simplices

As the title says, I want to compute the root mean square distance between two n-dimensional simplices. Say I have two surfaces $S$ and $S'$, the mean error is $$ d_m(S,S') = \frac{1}{|S|} \int\int_{...
2
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0answers
88 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 $$ ...
2
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0answers
32 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 $$c\sum_{...
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0answers
87 views

Measuring Model Bias

If given the choice between two statistical models (for argument's sake, let's say Model 1 is $y = \beta_0 + \beta_1 x_1 + \epsilon$ and Model 2 is $y = \beta_0 + \beta_1 x^2_1 + \epsilon$), is there ...
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0answers
40 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
114 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 $\theta$, $\hat{\...
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0answers
235 views

Calculating MSE for two different size matrixes

I have two $2$-column matrixes, one of the has $467$ rows while the other one has $61468$ rows. Both them are trajectory paths of same robot, the big matrix is kind of raw data and the smaller one is ...
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0answers
45 views

Can convergence in distribution say anything about mean-square convergence rate?

Suppose I have a sequence $\{x_n\}$ that I already know converges in the mean-square-sense ($\lim_n E |x_n|^2\to 0$). Suppose further I know that the sequence $\{x_n\}$ converges in distribution to ...
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0answers
38 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them (for ...
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0answers
83 views

MSE For a Single Calculation (intel processor errors)

This is the question, from a practice final for a stat course: The Intel Pentium Processor chip has been discovered to make small errors occasionally; that is, errors of +1 or –1 (in $10^{-4}$ ...
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0answers
197 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
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0answers
1k views

Hypergeometric Distribution Probability (mean, variance, Std Deviation)

The distribution of the number of children per household for households receiving aid to dependent children (ADC) in a large eastern city is as follows: 5% of ADC households have one child, 35% of ADC ...
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0answers
60 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments via ...
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0answers
179 views

Getting the correct error for a mean calculation

A constant k needs to be calculated including its gaussian error. $k = f_{(u,t)}$ $k_i$ can be calculated with the values and errors of $u_i$ and $t_i$ and their respective errors. Main issue is ...
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0answers
19 views

minimization problem: finding smallest deltas that satisfy equation.

After long derivations to find a better backpropagation algorithm for neural networks, I got this elegant optimization problem. Index $i=1..n$ given constants $c_i \in R, w_i \in R$ variables to ...
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0answers
25 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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0answers
10 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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0answers
37 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 }^{\infty}...
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0answers
22 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
22 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
13 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 $1....
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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
15 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 = \frac{V_3}{...
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0answers
39 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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0answers
12 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
17 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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0answers
24 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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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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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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0answers
44 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 f=\sqrt{\sum_{k=1}^n\Big(\frac{\...
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0answers
18 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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0answers
199 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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0answers
48 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 ...