# 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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### Self-location estimation from three landmarks under conditions where distance cannot be measured

There are all different points $O_1, O_2,$ and $O_3$ with known global coordinates in 3-dimensional space. Let $F, R, U$ be the $x, y, z$ basis vectors of the local coordinate system of point $P$ with ...
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### determine if the variance estimator is consistent and efficient of a normal distribution

I have the stimator $σ^2_{n}=1/n⋅(∑_ {i=1}^{n}(Xi−µ_{0})^2$ from a normal distribution with mean μ (known) and variance $σ^2_{n}$ (unknown) I have to determine if the stimator is: Efficent (Cramer-...
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### Get $b$ value that minimized error

Let's say I have a list of values that I got sampling the normally distributed random variables $[X_i]_{i=1}^{i=N}$ once (as in 1 value from each RV), and I know that $X_i\sim N(b\cdot k_i,\sigma^2)$ ...
1 vote
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### Are we finding the density of $x$ or evaluating the density of $\theta$ at $x$? | Alpyadin Machine Learning

In section $4.4$ The Bayes Estimator of Alpaydin he discusses the use of the prior density of $p(\theta)$ to construct a posterior density for $\theta$. This is standard Bayesian estimation to get a ...
1 vote
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### Reformulation of a parameter estimation problem to use Least Squares Method

the measurement equation of the $i$-th sample is: $0=r_i + 2 x_i^T P \dot x_i$, where $r_i\in\mathbb{R}_+, 0 \prec P\in\mathbb{R}^{n\times n}, x_i\in\mathbb{R}^n,\dot x_i \in\mathbb{R}^{n}$. My goal ...
1 vote
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### What is the first order derivative of the MMSE estimator over the observation?

Consider the observation $y=x+n$, where $n\sim N(0,\sigma^2)$ and the priori distribution $x\sim p_0(x)$. I know the MMSE estimator is given by $\hat{x}=E[x|y]$. What is derivative of the following \...
1 vote
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### Maximum likelihood as correspondence (or, How I hate the poor usage of Mathematics in Statistical textbooks)

Maybe the title was a bit much, but it describes both my question and my sentiment towards (what I perceive to be) the neglect, in statistical textbooks, of mathematics. The preamble to my question is ...
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### Using mcmc to estimate parameters of Dirichlet distribution

We have a probabilistic model with two parameters, $\theta$ and $\eta$, both of which are uniformly distributed between 0 and 1. The model has five possible outcomes, and the probability of each ...
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### Is it possible to change a weight matrix $W$ so it minimize a vector $J$?

Assume that we have a weight matrix $W \in \Re^{n x n}$ somewhere and if it's changing, then a vector $J \in \Re^{m}$ is going to be minimized. The problem is that this is not an ordinary optimization ...
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### Is it possible to estimate $\mathbf{A}$ in $\mathbf{Ax}=\mathbf{b}$ based on initial guess

I am trying to solve $\mathbf{Ax}=\mathbf{b}$, but this time $\mathbf{A}$ is unknown. I do, however, have a pretty good starting guess for $\mathbf{A}$, but it needs some slight modifications. I know ...
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### Integrating x,y gaussian distribution over a square area, expressed in terms of error (erf)

I am struggling to wrap my head around the math of a paper I need to understand for my work. In essence, it has to do with super-resolution microscopy, and determining the expected value of a pixel in ...
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### Estimation of exponential distribution parameter from smallest $n$ out of N observations

I am interested in estimating the parameter $\lambda$ of an exponential distribution based on the smallest $n$ out of a total of $N$ observations. In mathematical terms: let $X$ be distributed ...
1 vote
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### UMVUE for $\frac{p}{1-p}$

Given a random sample of $X_1, X_2, ..., X_n$ of the negative binomial distribution $Nb(N,p)$, I am instructed to find the UMVUE for the parameter $\frac{1-p}{p}$. In general, the mean $\mu$ of the ...
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### How to merge 2 multivariate Gaussians with different probabilities of occurring?

So, have a system that estimates a covariance matrix. The scenario occurs where at time step $t$, I will either have a Gaussian with covariance matrix $\Sigma_A$ with probability $p$ or $\Sigma_B$ ...
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### Best estimator for geometric distribution

I need an estimator for geometric distribution $\text{Geom}(p)$ that best fits my data $X_1, X_2,\ldots$ Is $\widehat{p} = \dfrac{1}{\overline{X}}$ the answer? Both MLE and method of moments yield ...
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### average calibration estimator

i was reading this paper regarding the uncertainty in machine learning. My issue is with a mathematical definition of average calibration and the estimator presented in section 3. Given 2 random ...
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### Fisher Information Matrix singular but unbiased estimator exists? Please help me figure out where I've gone wrong

I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
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### Bain & Engelhardt Ex9.6: Wanted some advice/confirmation if the answer is correct.

Q: Find MLE based on random sample $X_1, . . , X_n$ from the pdf $$f(x;\theta_1,\theta_2)=\frac{1}{\theta_2 - \theta1} ; \theta_1\le x \le \theta_2$$ and zero otherwise A: \begin{align} \log f(x)= -...
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I have data which can be described by a background term, $f(x)$, and a signal term, $g(x,\theta)$. In this particular case, $f(x) = x^a$ is a single power law whose index is known, and $g(x) = A_0 \... 0 votes 0 answers 18 views ### Normalization in Maximum Likelihood Estimation over a restricted data range I have data which can be well described by the sum of two distributions,$f(x,\theta) + g(x,\theta)$, where$\theta$are the parameters to be estimated. However, the data range is limited to$x_0 < ...
Following is a part of a bigger problem that I am trying to solve. Let $\alpha_t , \beta_t, k$ be random variables as a function of time $(t)$, such that $k(\alpha_t +e_1(t)) = \beta_t +e_2(t)$. ...