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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5
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1answer
1k views

Linear MMSE estimate of MMSE estimator

This question is prompted by a recent discussion about the relationship between conditional expectation and covariance. Suppose that $X$ and $Y$ are zero-mean unit-variance random variables with ...
4
votes
1answer
636 views

Minimize combined variance of multiple measurements with known (but varying) variance

I have multiple measurements for the same property but with different but known uncertainty (variance). And I would like to combine that measurements in a way that I get as close to the real value as ...
4
votes
1answer
647 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: ...
4
votes
1answer
223 views

Straight line through data by eye - least squares? [closed]

I heard an interesting fact a while ago about how people draw a line through a cloud of points on a scatter plot. Usually, when calculating lines of best fit, we use the minimal the sum of squares of ...
4
votes
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 ...
3
votes
3answers
386 views

Why root mean “square” error?

Root-mean-square error is frequently used in for calculating the error between a predicted value and actual value. The formula for RMSE is given below: $\mathrm{RMSE} = ...
3
votes
0answers
235 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|} ...
2
votes
1answer
525 views

Minimizing Mean Squared Error for Exponential Function

I have a function that I'm trying to model using an exponential function and I'm trying to determine the constants for the exponential. I know I could optimize it using trial-and-error in R or another ...
2
votes
0answers
86 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
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 ...
2
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0answers
86 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 ...
1
vote
2answers
364 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
1
vote
2answers
4k views

What is the purpose of subtracting the mean from data when standardizing?

What is the purpose of subtracting the mean from data when standardizing? and What is the purpose of dividing by the standard deviation?
1
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2answers
504 views

Given a function, calculate MMSE and LMMSE

Let $X = \frac{1}{1+U}$ where $U$ is uniformly distributed over $[0,1]$. I need to evaluate $E[X\mid U]$ and $\hat{E}[X\mid U]$ and the calculate the MSE, $E[(X-E[X\mid U])^2]$ and ...
1
vote
2answers
297 views

Least squares fit with a trick

Fitting a circle with least squares is easy once you get the trick for $c = r^2 - a^2 -b^2$ and you got a linear set of equations. my problem is as follows: $$z = \alpha x + \beta y + \alpha \beta$$ ...
1
vote
2answers
121 views

Linear Regression to quadratic function

What is the optimal linear regression (w and w/o y-intercept) for a quadratic curve w.r.t. mean square error. Mathematically speaking: Given, $$y = x^2$$ for $$x = [-a,a]$$. What is the best ...
1
vote
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) ...
1
vote
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 ...
1
vote
1answer
214 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 ...
1
vote
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 ...
1
vote
1answer
283 views

Improving the mean squared error of the ordinary least squares solution

I have the following relationship that holds: $y = a x_1 + b x_2$ Now I would like to determine constants $a$ and $b$ by performing tests. Each test results in 10 values for $y$, $x_1$ and $x_2$. ...
1
vote
1answer
962 views

Minimizing mean squared error

I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
1
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1answer
27 views

Calculate the Mean earnings of a person

There are $12$ employees who earn $\$750$ collectively. I need to calculate the mean salary earned by a single employee. My workings so far: I tried to divide $750 / 12$, but I am not sure if this ...
1
vote
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 ...
1
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1answer
123 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 ...
1
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0answers
34 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 ...
1
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0answers
76 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
208 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
44 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 ...
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0answers
82 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
919 views

What is the difference between RMS and RMSE?

I did not unserstand what is the difference between root mean square (RMS) and root mean square error (RMSE). In some sources RMS term is used for error analysis, in others RMSE. Can you explain ...
1
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0answers
978 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 ...
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0answers
173 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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2answers
1k views

Step by step correlation calculation

I must understand how I can calculate the correlation for the following probability variables. ...
0
votes
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. ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
30 views

Error of function with errors on arguments and size of arguments

EDIT: my question is not how to derive the formula below (I think the derivation is more or less what I guessed, like the answer below supports), but whether it can be made valid for the case where ...
0
votes
1answer
293 views

MMSE (Minimum min square estimate) problem

I have a problem as follows. As of now, I cannot provide the definition of X and Y but can anyone provide a rough overview of what needs to be done ? An experiment consist of rolling a single ...
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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: $$ y = Xc + e $$ $y, e \in \mathbb{R}^n$, $c \in \mathbb{R}^m$, $X \in \mathbb{R}^{n \times m}$, $e \sim \mathcal{N}(o,\sigma^2)$, $n>>m$ ...
0
votes
1answer
39 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 ...
0
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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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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 ...
0
votes
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
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 $g(Y)= aY+b :$ $min_{\text{a,b}} E[(X-aY-b)^2].$ $b^*=E[X-aY]=E[X]-aE[Y]$ ...