Questions tagged [parameter-estimation]

Questions about parameter estimation. Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured/empirical data that has a random component. (Def: http://en.m.wikipedia.org/wiki/Estimation_theory)

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14 views

How can I derive OLS predicted error term $\hat{e}_i$ as a function of $e_i$?

First of all, I'd like to say that any kind of help would be really helpful, whether it's a hint or a good grad/undergrad book. Right now I'm working with Econometric Analysis of Cross Section and ...
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1answer
46 views

Calculating the asymptotic normality result of a MLE from a skew-logistic distribution

Suppose we have $X$ with cumulative distribution function $F_X(x) = (1-e^{-x})^\frac{1}{\theta}$ where $x \geq 0, \theta > 0$. How can one find a MLE for $\theta$ from this and the asymptotic ...
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45 views

Mixture of two densities

Suppose we have densities $f_1, f_2$ from the random variables $W_1$ and $W_2$ where $W_i$ has known mean $\mu_i$ and variance $\sigma_i$. Consider the mixture of the two densities $$ f(x;\theta)=\...
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25 views

Variable transformation for a multiple linear regression model

I can only transform C) in a way that leaves me with only constants for the parameters. In all other functions I either end up with $ln(\beta_1)$ or need to take the square root of $\beta$, etc. I'm ...
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23 views

Absolute value equation with parameter p

I know how to calculate absolute functions without parameter: 𝑥 ⋅ |𝑥 + 6𝑝| = 36 The question is how parameter p changes the number of solutions. I tried to somehow calculate the discriminant for ...
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28 views

Method of moments estimator for lognormal distribution

Let $X_1,\cdots X_n$ be identically and independently distributed lognormally. I want to find the method of moments estimators for $\mu,\sigma^2$. We know that $E[X]=e^{\mu+\frac{\sigma^2}{2}}$, $E[X^...
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22 views

Under or over determined linear system vs inconsistent linear system

We know the following: If there are more number of observations than that of unknowns, the linear system term as an over-determined system. If there are less number of observations than that of ...
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17 views

Simplest optimization method for specific nonlinear function

I want to find the best parameters for a model given a set of measurements. The model has the function $y = a + b \cdot \frac{x-c}{x+c}$, where $a,b,c \in \mathbb{C}$ are the parameters that must be ...
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27 views

MMSE estimator using pmf table

We have the probability mass function as follows, $$\begin{array}{c|c|c|} & \text{Y=0} & \text{Y=1} \\ \hline \text{X=0} & \frac 15 & \frac 25 \\ \hline \text{X=1} & \frac 25 &...
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21 views

Variance estimator from PCA

I have done a fit of the inverse Covariance-matrix of a multivariate Gaussian distribution. In order to assign an estimate of the standard deviation of the parameters, which includes the correlations ...
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38 views

Solving a parameterized matrix equation with specific structure

Consider the equation: $\mathbf{y} = \mathbf{\Lambda_{\epsilon}C\Lambda_{\epsilon}^{\dagger}h}$, where $\mathbf{y}$ and $\mathbf{h}$ are $n$-length complex-vectors, $\mathbf{C}$ is a circulant $n\...
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54 views

Large sample properties of classical estimator for scale parameter

I've also post this question on Stats Stackexchange as advised in the comment. Suppose $X=(X_1,X_2,\ldots,X_n)$ are non-negative and have a joint probability density $$\frac{1}{\sigma^n}f\bigl(\frac{x}...
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41 views

An incorrect application of the Rao-Blackwell theorem

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1") Note:- here $Y_1$ is a sufficient ...
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24 views

Estimating parameters using 0-1 matrix system of equations

I have the following set of $n$ equations: $ \hspace{4cm}\boxed{y^{(1)}=h_1x_1^{(1)}+ h_2x_2^{(1)} +\ldots+ h_kx_k^{(1)} +w_1\\ y^{(2)}=h_1x_1^{(2)}+ h_2x_2^{(2)} +\ldots+ h_kx_k^{(2)}+w_2\\ \vdots \\...
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62 views

Probability of probability confusion

I'm a little lost regarding the concept of estimators of probability in regard to the two following examples. Suppose I have $n$ marbles in a bag, and I sample them many many times and find that $20$ ...
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51 views

MLE of number of colors

I'm looking at this question and the solution given and I understand it, but I'm unable to see where I'm going wrong. The question states that there are $k$ equally frequent colors and we do not know $...
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1answer
31 views

When should I use a kalman filter, if the observability function is known?

Assume that we have a nonlinear dynamical model, e.g called transition function $$\hat x = f(x, u)$$ And $y = x$ as our observability function. According to Mathworks Kalman filters are used to ...
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10 views

EM algorithm and Akaike information criterion

I wonder if there is a relation between the Expectation Maximisation algorithm and the Akaike information criterion, EM is used for estimating missing variables (latent variables), but what is the ...
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1answer
20 views

Why do we need Wald's confidence interval to estimate p in a Bernoulli distribution?

I'm studying statistics and I'm a bit confused about why Wald confidence interval is needed to estimate the p in Bernoulli distribution. Let's say, I am modeling some phenomenon with a Bernoulli ...
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1answer
100 views

Parabolic Interpolation with three data points and measurement noise

The question is as follows: Have$$z_i = y(x_i) + w_i,\quad i = 1, 2, 3$$ where $x_1<x_2<x_3$ and $z_1<z_2>z_3$. $\space$ Assume the unknown function $y(x)$ can be approximated with ...
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17 views

Estimate random variables given estimated function values

Let $X_1$ and $X_2$ be two positive independent random vectors. Let random variable $Y_1$ depend only on $X_1$; and random variable $Y_2$ depend only on $X_2$. If I have the following means square ...
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1answer
23 views

Estimate bias determination

A statistical question for me went: Suppose $X_1, X_2, ..., X_n$ are identical and independent variables and follow $uniform(0,\theta)$. Show that $\frac{n+1}{n}X_{(n)}$ is an unbiased estimate of $\...
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7 views

Distributing a fixed number of noisy samples of a function to best estimate that function

I want to learn more about how to best spatially distribute noisy samples of a function to reconstruct the function itself. The name of this field of research or some references to papers or books ...
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19 views

Interest rate modelling: Iterated extended kalman filter with Maximum Likelihood

I recently started to deepen my knowledge about interest rates' modelling, and I am trying to estimate a two-factor Ornstein–Uhlenbeck process using Euro area OIS rates by means of an Iterated ...
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31 views

Method of moments for convolution of uniform and normal distributions

I have 2 distributions $$X ∼ N(\mu, \sigma),\ \ \ \ \ \ \ Y ∼ \text{Unif}(a,b).$$ Density of sum of these distributions is $$f_{X + Y}(x)=\frac{1}{2(a-b)}\left(\text{erf}\left(\frac{b+\mu-x}{\sqrt{2}\...
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1answer
32 views

Maximum likelihood estimator (probability density function given by intervals)

Let $\vartheta \in (0,1)$. Let $X_1,\ldots,X_n$ be a simple random sample with probability density function $f_\vartheta$, $$\begin{align}f_{\vartheta}(x)=\begin{cases}1-\vartheta & \text{if} &...
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55 views

Calibration/parameter estimation of CEV model

The CEV model for a stock price $S(t)$, interest rate $r$ and variance $\delta$ $dS(t)=rS(t)dt+\delta S(t)^{\gamma}dW(t)$ where the volatility for the stock is given by $\sigma(t)=\delta S(t)^{\gamma -...
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39 views

Minimal sufficient statistics for two unknown parameters

I have a problem for finding a sufficient and a minimal sufficient statistics for the next density: let $X_1,...,X_n$ a sample with $$f_X(x;\alpha,\theta)=\frac{\alpha x^{\alpha-1}}{\theta^{\alpha}}I_{...
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1answer
18 views

Assessing bias and consistency of modified OLS estimator

Given the (multivariate) linear regression model $\mathbf y=\mathbf X \mathbf\beta_0 + \epsilon$ and $\mathbb E[\epsilon|\mathbf X]=0$ for $\beta_0 \in \mathbb R^k$, determine if the following ...
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1answer
33 views

Finding a consistent estimator for area under simple regression line

I am trying to solve the following problem: Take the following simple linear regression model, where $x_i \in \mathbb R$: $y_i=\beta_0 + x_i \beta_1 + \epsilon_i$ Given that: $\mathbb E[\epsilon_i]=...
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27 views

Finding the variance of slope parameters?

I have that $s^2 = 2.23$, $$X = \begin{bmatrix}1&-2&4\\1&-2&4\\1&0&0\\1&0&0\\1&1&1\\1&2&4\\1&4&16\end{bmatrix}, \,\,Y = \begin{pmatrix} 67\\...
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1answer
59 views

UMVUE over a discrete distribution

The problem is: let $X$ a random variable such that $P(X=x)=\left\{\begin{array}{cl}2p(1-p)&\mbox{if }x=-1\\p^x(1-p)^{3-x}&\mbox{if }x\in\{0,1,2,3\}\end{array}\right.$ Find, if there exist, an ...
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32 views

What makes inequality true in proof of Gauss Markov theorem

Elsewhere on this site, I found a very compact proof of the Gauss-Markov theorem, seen below. I don't understand the justification for the middle step with the inequality. Specifically, what property ...
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1answer
66 views

Linear regression optimization

I'm trying to solve the linear regression problem but I'm stuck and can't solve the question. imagine you have below form $$ Y = WX-\epsilon $$ and $\epsilon $ is from Gaussian distribution $\epsilon \...
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37 views

What is the geometric interpretation of the normal equations $\mathbf{N}={{\mathbf{A}}^{T}}\mathbf{A}$ matrix when the columns of A are orthogonal?

I have a homogeneous problem with the form $\mathbf{A}_{n\times6}\mathbf{x}_{6\times1}=\mathbf{0}_{n\times1}$ in which the columns of the $\mathbf{A}$ are orthogonal in each row, for example $\mathbf{...
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37 views

Is this model identifiable?

Let $\Phi=\textbf{cov}(\textbf{f})$ is a $m\times m$ symmetric matrix containing $\frac{m(m+1)}{2}$ unique factor variances and covariances, $\Psi=\textbf{cov}(\mathbf{\epsilon})$ is a diagonal matrix ...
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61 views

Prove that $ f(x|\theta) = \frac{1}{4} e^{-\frac{1}{2}|x-\theta|}$ is not an exponential family. [duplicate]

I want to prove that $ f(x|\theta) = \frac{1}{4} e^{-\frac{1}{2}|x-\theta|}$ does not belong to the exponential family. By definition, it is easy to answer such questions as 'prove a given ...
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18 views

Estimation of a model of $X_{n}=ABX_{n}+U_{n}$ given data of $X_{n}$

I am working on a model of the form $X_{n}=ABX_{n}+U_{n}, n=1,...,N,$ where $X_{n}$ is a $T\times 1$ vector of observation, and $A$ is a $T\times k$ matrix, $B$ is a $k\times T$ matrix (assuming $k$ ...
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28 views

Can a biased MLE estimator that is asymptotically unbiased be a asymptotically minimum variance estimator if it approaches the CRLB asmptotically?

I found a mathematical problem which investigated the statistical properties of a MLE. The MLE is biased and asymptotically unbiased and the variance of the MLE tend towards the CRLB at larger sample ...
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20 views

comparing two estimates by the law of large numbers

I would like to make sure that I have enough data to show that the estimate of the mean of a fair dice and a dice with and exchanged dot for two dots (2,2,3,4,5,6) are significantly different. Is ...
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18 views

How marginalization of Normal Inverse Gaussian (NIG) is multivariate student-t distribution?

I am having a problem with some integrations I was reading Deep Evidential Regression paper which describe how to model uncertainty using conjugate prior in regression task. Starting from assumption ...
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1answer
62 views

Is every estimator a sufficient statistic?

It is clear that not any sufficient statistic (s.s.) makes a good estimator (since a monotonic transform of a s.s. is still a s.s.). But is a "good" estimator of the parameter always a s.s.?...
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123 views

Calculate the ML estimate in multinomial distribution

It is the process of calculating the maximum likelihood estimate {$\pi_j$} in a multinomial distribution. The multinomial log-likelihood function is $l(\pi)=\sum_j{n_j}{log\pi_j}$ $\partial l(\pi) \...
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56 views

Characterization of Total Error for Estimation of the Parameters of a System

Suppose that we have a system which can receive different kinds of input (the index $k$ signifies the kind of input $I_k$) and performs a calculation on that input based on the internal parameters of ...
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27 views

How to find priori information in Cramer Rao Lower Bound when the pdf of the parameter is unkown

In order to compute a Bayesian Cramer Rao Lower bound, we need to find the prior information of a random parameter $\theta$ which is a complex scalar parameter. The pdf of $\theta$ is $f(\cdot)$. ...
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38 views

Using parameter estimation for training a neural network

Assume that we have 4 layers in a neural network. $$z_1 = L_1(x, W_1)$$ $$z_2 = L_2(z_1, W_2)$$ $$z_3 = L_3(z_2, W_3)$$ $$y = L_1(z_3, W_4)$$ Where $x$ is the vector input, $y$ is the vector output ...
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1answer
36 views

how to find the smallest parameter for trigonometric equation?

Find the smallest value of parameter $\alpha$ such that equation $${\mathrm{sin}}^{2}x \times  \mathrm{cos}2x + \alpha ({\mathrm{cos}}^{4}x - {\mathrm{sin}}^{4}x) = -10(2\alpha  + 1{)}^{2}$$ has at ...
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24 views

Propagation of Error: Tricky Example

I am having a difficult time finding error bars on a particular quantity. There are two random variables, let's call them $price$ and $color$; color can only be either red or blue. We don't have a ...
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2answers
52 views

Comparing two weird estimators

I have this question that I feel is completely out of the blue. We are studying the first subjects in statistics (bias, mean squared error, consistence) and we have been given the task to compare the ...
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1answer
70 views

Find UMVUE of $\tau=(\lambda-\mu)e^{-(\lambda+\mu)}$ from samples $X_1, \dots, X_n\sim \rm{Pois}(\lambda)$ and $Y_1,\dots, Y_n\sim \rm{Pois}(\mu)$.

Problem Statement We have two independent Poisson samples $X_1, \ldots, X_n$ with means $\lambda$ and $Y_1, \ldots, Y_n$ with means $\mu$. We would like to estimate $\tau=(\lambda-\mu)e^{-(\lambda+\mu)...

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