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2answers
37 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
14 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 ...
-1
votes
2answers
46 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 ...
0
votes
1answer
9 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 ...
4
votes
1answer
103 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 ...
0
votes
1answer
35 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
0answers
22 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 ...
1
vote
1answer
56 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 ...
0
votes
1answer
22 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
37 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 ...
0
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0answers
56 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 $$ ...
0
votes
2answers
75 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 ...
0
votes
0answers
10 views

Difference Between Three Similar Error Reducing Algorithms

I found a Least Square Error Recognition algorithm that finds the least mean square error from a 2-d matrix element by element. Logistic regression from this site, on the other hand, seeks to ...
1
vote
1answer
20 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
votes
0answers
22 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 ...
0
votes
1answer
30 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 ...
0
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0answers
54 views

closed-form solution for this constrained optimization

I want to find a closed-form solution for the vector $w=\left[\begin{array}{c} c\\-b \end{array}\right]$where $c$ and $b$ are column vectors, such that the following MSE is minimized: $\begin{align} ...
0
votes
0answers
14 views

how can I find the resulting MS error with linear estimation?

The problem says: If $\eta_x=\eta_y=0, \sigma_x=\sigma_y=4$ and $\hat{y}=0.2x$ (linear estimate), find $E\{(y-\hat{y})^2\}$. I am doing this, based on Papulis formulas for homogeneous linear estimate: ...
0
votes
0answers
17 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
0
votes
0answers
18 views

How Can I show that a=A in this linear MS estimation problem?

How can I show that if the constants A,B and a are such that $E\{[y-(Ax+B)]^2\}$ and $E {\{[(y-\eta_y)-a(x-\eta_x)]^2 \}}$ are minimum, then $a=A$. I am trying to use this: $e=e_m$ is minimum if ...
0
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0answers
23 views

Multivariate mean square error

Consider a one-dimensional vector: $X = (X_1, X_2, ..., X_n), \ \ \ \ X_i \in \mathbb{R} $ The mean squared error compared to a certain value is: $MSE = \frac{1}{n}\sum_{i=1}^n (X_i - \mu)^2$ But ...
0
votes
1answer
56 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, $\sigma^2$) ...
0
votes
0answers
77 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 ...
0
votes
1answer
68 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 $X_t = Z_t + \theta Z_{t−1}$.
0
votes
1answer
20 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: ...
0
votes
0answers
25 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 ...
0
votes
1answer
57 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}\\ ...
1
vote
0answers
111 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 ...
1
vote
1answer
189 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 $x_2$. ...
0
votes
1answer
26 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
33 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 ...
1
vote
0answers
32 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 ...
3
votes
3answers
242 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} = ...
0
votes
2answers
86 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 ...
1
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0answers
35 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 ...
1
vote
2answers
257 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
0answers
64 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 $10^{-4}$ ...
0
votes
1answer
201 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 ...
1
vote
0answers
183 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 ...
0
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2answers
240 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
vote
2answers
399 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
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2answers
248 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
0answers
712 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
648 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 ...
2
votes
1answer
371 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 ...
1
vote
1answer
727 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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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 ...
1
vote
2answers
115 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 ...
0
votes
2answers
2k 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?