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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How can I estimate unknown parameters in nonlinear forward model?
I have a table in $Z=0$. Above the table in $(YZ,ZC)$ I have an instrument, C.
Think of the instrument as a monocular rotating around the X axis.
With the instrument I can measure at which angle, $\...
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Does "independent variables X and dependent variable Y are jointly gaussian" means "the residual term has 0 conditional mean"?
I saw the following statement in my lecture note:
"The data generation process is $y = x+\epsilon$, whereas in the regression we run y on x so the regression model is $y = \beta_{OLS} x+e$, the ...
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Posterior Distribution and James Stein Estimator
Assume $\mathbf{\mu }\sim N\left( 0,I_{r}\right) ,$ where $I_{r}$ is a $%
r\times r$ identity matrix, and $\mathbf{y}|\mathbf{\mu }\sim N\left(
\mathbf{A\mu },I_{T}\right) ,$ where $\mathbf{A}$ is a $...
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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 $X_i\sim N(b\cdot k_i,\sigma^2)$ ...
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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 ...
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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 ...
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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
\...
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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 ...
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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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True value of a parameter in statistics
Let us consider a statistics model
$X_1,X_2,\cdots,X_n\sim^{i.i.d} f(x,\theta)$,
where $f(x,\theta)$ is a probability density function with a parameter $\theta$.
Let $\theta_0$ be a true value of $\...
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Finding MVUE for $p^k$ - cross check solution
(Warning, long text) I've been given the following instructions;
Let $\boldsymbol X = \mathit X_1, X_2, ..., X_n$ be a random sample from binomial distribution $b(N,p)$, with PMF $f(x;p) = {N\choose ...
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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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Finding the cumulative distribution function of a maximum function, based on a uniform distribution [duplicate]
Suppose I have a uniform distribution
$X$ ~ U($0, θ$).
I have an estimator which pertains to the random variable of the largest value among my random sample. We can express this as
max$(X_1, X_2...X_n)...
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Unbiased estimator from sorted sample [closed]
5 samples are drowned from uniform distribution with range [0, 2B].
The 5 samples are sorted: ($x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$).
Which of the following is an unbiased estimator for B and why?...
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Why can we replace $p$ with its estimate $\hat{p}$ and not lose the normality of the distribution?
This is from 'Introduction to Mathematical Statistics' by Hogg et al (8th Ed):
Example 4.2.3 (Large Sample Confidence Interval for $p$). Let $X$ be a
Bernoulli random variable with probability of ...
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Show biasedness of estimator of discrete uniform distribution
Let be $X:\Omega\to[0,1,2,3,\dots, \theta]$ a discrete random variable which obeys a uniform distribution. We don't know $\theta$ and estimate it by the maximum of the sample $(x_1,x_2,\dots,x_n)$, ...
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Define statistical model and find unbiased estimator
A call center wants to analyse customer behavior and counts on $n$ days incoming calls. Which statistical model is adequate in this situation? Define an unbiased estimator regarding incoming calls.
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Using the Kelly criterion, what is the maximum amount you should wager when the odds are unknown?
Thinking from a general, layman's perspective, when one cannot properly assess the risks of a particular situation, but still wants to apply probability to maximize chance of gains, how can one use ...
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Reference request for exponentially smoothed moving average estimators for the diffusion matrix in continuous time
Currently, I am working on a project that involves the estimation of the diffusion matrix $\Sigma(t)$ of a continuous time Markov chain. The model can be described by the stochastic differential ...
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Construct a consistent estimator for asymptotic variance.
Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with mean $\mu \in \mathbb{R}$, variance $\sigma^{2} \gt 0$ and $E\{X^4\} \lt \infty$. For the sample variance $S^{2}_{n}$, we have ...
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A parametric model for inverse of quadratic equation
I have data $\left\{ q_i, p_i \right\}_{i = 1}^{N}$ from a model $p_i = a q_i^2 + b q_i + c$:
I know data is always in the positive quadrant as $q_i \geq 0, \, p_i \geq 0$ and I also know $a > 0$ (...
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Joint distribution of bivariate normal and bernoulli
Let (X,Y) be a bivariate normal and Z follow Bernoulli distribution and be independent of (X,y). Its mean $(X,Y)\sim N(\mu,\sigma I)$ and $Z \sim Ber(p)$. How can I find the joint distribution of them?...
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Need a sufficient unbiased estimator of a parameter be unique?
In a recent multiple choice examination I encountered a "select all that apply" type question, which had this statement among others:
If a sufficient estimator exists, it is always unique.
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Estimate parameters in model
Consider the following model:
$\hat y = f(x_1, ..., x_n) + \delta$
$\hat x_i = x_i + \varepsilon _i, i=1,..,n$
where $\delta, \varepsilon _1, ..., \varepsilon _n$ are i.i.d. random variables with ...
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MSE of estimator method of moments
Let $X_1, \dots, X_n$ a sample from the density distribution:
$$ f(x) = 1/\theta \quad \theta < x < 2\theta, \quad \theta>0 $$
Found $ \hat\theta = \frac{2}{3}\bar X$ with method of moments,...
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Truncated lognormal distribution calibration with MME
To estimate the parameters of a truncated distribution (lognormal for example), we can use the Maximum Likelihood Estimation or Method of Moments.
For the Method of Moments Estimation, one needs to ...
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Calculate the bias of the estimator
given is $c(t,s) = E[(X(t) - \mu(t))(X(s) - \mu(s))]$ and the estimator $\hat{c}(t,s) = \frac{1}{n} \sum\limits_{i = 1}^n (X_i(t) - \hat{\mu}(t))(X_i(s) - \hat{\mu}(s))$ with $\mu(t) = E[X(t)]$ with $...
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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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How do you infer the model of a car based on prior information?
Sorry if this does not quite make sense as I am still wrapping my head around it as well. Suppose I have j car models (i.e. different brands, builds etc.) such that $\textbf{m} = {m_1, m_2, . . . , ...
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Estimating one of the means of a bivariate Gaussian when the two means are unknown
Suppose we want to estimate a single mean $\mu_1$ of a bivariate Gaussian, whose covariance matrix is known, but the means $\mu_1$ and $\mu_2$ are unknown.
Let $N$ be the number of joint samples. If ...
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Find the maximum likelihood estimator for θ
We have a simple random sample of size n from a distribution with pmf 𝑝(𝑥) = $\theta{(1-\theta)}^{x-1}$ for 𝑥 = 1,2, …. Find the MLE[𝜃]
My try:
$
L\ =\ \theta{(1-\theta)}^{1-1}\times\theta{(1-\...
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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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Maximum Likelihood Estimation with degenerate functions
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 \...
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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 < ...
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What are the conditions for convergence for the following problem
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)$. ...
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Find a method of moments estimator for Uniform distribution ($\theta_1, \theta_2$)
Let $X_1, X_2, . . . , X_n$ be independent and identically distributed random
variables with probability density function uniform ($\theta_1, \theta_2)$. $-\infty < \theta_1 < \theta_2 < \...
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Intuition on bias corrected estimator in statistics
Say I have my original objective function $\|Y - X \beta\|_2^2$, and for some reason (other motivation), I want to add a penalty term and obtain a new objective function.
The target estimator $\beta_0$...
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I need a Excel formula to find the coordinates of the point in the new bent cylinder.
In the beginning, I have a straight cylinder where I have the value of the distance of plane 2 from the beginning of the cylinder and an angle value of the point on the surface.
Then, I will have a ...
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Estimating Negative Binomial parameters from subsampled data.
Suppose I need to estimate parameters $n$ and $p$ (or alternatively $\mu$ and $\sigma^2$) for a count data, that follows Negative Binomial distribution. However, I do not observe the raw counts, but ...
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$O_p$ notation in asymptotic proof
I am currently reading a proof in Pesaran's 2006 paper (Lemma A2 Eq:A11).
There is a quantity $f(N,T):=\frac{F^T\bar{U}_w}{T}$ and the aim is to show that
\begin{align}
f(N,T) = O_p\left(\frac{1}{\...
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Unbiased Estimator of $\sigma^4$
$\{X_i\}_1^n$ is random sample from $N(\mu, \sigma^2)$ with unknown parameters. Find an unbiased estimator of $\sigma^4$.
My first thought was to estimate kurtosis (as it has $\sigma^4$ term) and the ...
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