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Questions tagged [maximum-likelihood]

For questions that use the method of maximum likelihood for estimating the parameters of a statistical model with given data.

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Maximum a posteriori, type-II error

I've been trying to derive the marginalized likelihood, which I think I have done successfully. The problem is that I've ended up with an expression that I can't really explain, my expression is like ...
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Monte Carlo Markov Chain - Metropolis-Hastings - Estimation of parameters

I have 6 parameters to estimate : $p=(\theta=[a,b]$, $\nu=[r_0,c_0,\alpha,\beta])$ with Bayesian and MCMC methods : $$\text{PSF}(r,c) = \bigg(1 + \dfrac{r^2 + c^2}{\alpha^2}\bigg)^{-\beta}$$ and the ...
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Maximum likelihood estimation in a Poisson convolution

Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $\nu$. The parameter $\nu$ is unknown and we would like to estimate it. We only are given the single data point $$ X-Y. $$ What ...
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31 views

MAP estimate of Erlang distribution

I have a hard time approaching this problem. I understand how to find the MAP estimate for common distributions but from the given problem below I have totally confused. I have a set of $N$ ...
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25 views

Why is the number of non zero eigenvalues equal to $x^T \Sigma^{-1} x$

I've been reading this code and I found that the number of non-zero eigenvalues of the estimated covariance is equal to $x_i^T \Sigma^{-1} x_i$. I want to know how to arrive at this result. Some ...
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Expectation Maximization (EM) : find all parameters from a PSF (Point Spread Function)

I have the 2 parameters arrays : $\theta=[a,b]$, $\nu=[r_0,c_0,\alpha,\beta]$ with the distribution (a point spread function = PSF = response of a Dirac) : $\text{PSF}(r,c) = \bigg(1 + \dfrac{r^2 + c^...
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17 views

Maximum Likelihood Estimate of Angle Measurements

Find the maximum likelihood estimate of the angle $\theta$, given that you have two independent noisy measurements, $c$ and $s$, where $c$ is a measure of the cosine, and $s$ is a measure of the sine, ...
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Detection Theory Test: ML, MAP, Bayes, Neyman Perason

I am looking at a couple of question based on the image below: Question Scenario The question then requires I select all thresholds (vertical lines) which could be used for Maximum Likelihood ...
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42 views

Simplifying a likelihood function [on hold]

I'm trying to simplify following equation: $\log L(\theta|M)=\sum_{d=1}^D\log\bigg((1-\alpha)(\exp(-\epsilon_b)\frac{\epsilon_b^B}{B!}\exp(-\epsilon_s)\frac{\epsilon_s^S}{S!}+\alpha(1-\delta)(\exp(-(...
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Relating posterior to the least square estimator of W

I'm currently working on an assigment and I'm currently stuck and could really use some help, I've been given the fact that my prior over my parameters W is given by a gaussian pdf, likewise is the ...
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Distributional assumptions in Maximum likelihood estimator (MLE) and least squares estimator (LSE)

Many textbooks do mention that MLE does need some distributional assumptions but I could never find which they are. LSE on the opposite, doesn't need distributional assumptions but when missing ...
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49 views

Maximum Likelihood Methods

Given $\hat{\theta}=\frac{-n}{\sum\limits_{i=1}^nlog(X_i)}$ is the mle of $\theta$ for a $\beta(\theta,1)$ distribution and $W=-\sum\limits_{i=1}^nlog(X_i)$ has a $\Gamma\left(n,\frac{1}{\theta}\right)...
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Parametric Estimation with Different Distribution Populations

I am looking for some reference for a parametric estimation problem with different populations. Suppose the parameter we are interested in is $\theta$, and we have data, say, from two probability ...
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31 views

Cramér-Rao Lower Bound for estimator of mean in Exponential distribution

Let $X_{1},...,X_{n}$ be a random sample of size $n\geq3$ from the exponential family with mean $1/\theta$. (1) Find a sufficient statistic $T(X)$ for $\theta$ and write down its density. (2) Obtain ...
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Estimation of 2 parameters with Maximum likelihood and a function depending on 2 random variables

I have the following PSF (Point Spread Function) (Moffat PSF) : I want to estimate the parameters $\alpha$ and $\beta$ ($\theta=[\alpha,\beta]$ represents the vector of parameters to estimate) with ...
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1answer
41 views

Minimum variance estimator Maximum likelihood (ML) vs Least Squares

According to the Gauß Markov theorem, the least squares estimator is the best linear unbiased estimator, given some assumptions. The maximum likelihood estimator however, has asymptotically minimal ...
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Likelihood by normal distribution, no variance given

So I'm currently working on a report for homework but I've ran into trouble. Ive been given the fact that: $$\mathbf{t}_i = \mathbf{W}\mathbf{x}_i + {\epsilon}$$ Where $\epsilon \sim \mathcal{N}(0,\...
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Finding standard errors of maximum likelihood estimates [migrated]

Suppose we use Maximum Likelihood estimation to estimate certain parameters in a model. Furthermore, suppose that the log likelihood function can not be solved analytically and thus must be optimised ...
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14 views

Graphing MLE as a function of variance with some fixed $n$.

I am given an maximum likelihood estimator of $\sigma ^{2}$ which is $MLE=\frac{2\sigma ^{4}}{n-1}$. When $n=10$, I have to graph this formula as a function of $\sigma ^{2}$. Does that mean I create ...
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31 views

Rigorous derivation of MLE in two-parameter case when second derivative test fails

I have a query regarding derivation of Maximum Likelihood Estimator (MLE) when there are multiple parameters to estimate simultaneously in a distribution with support depending on parameter, in ...
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Estimate unknown sum of iid random variables

Let $X_1, X_2, \dots$ be a sequence of independent and identically distributed discrete random variables with common mass function $f_X(x)$ defined for when $x \in \{0,1,\dots,N\}$ and $N$ is known. ...
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Likelihood Estimation from Transfer Function

Given a sequence of observations presumed to be produced by an autonomous linear dynamical system, is it possible (and if so how) to determine the likelihood of the sequence without inferring the ...
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MLE of $\sigma_x^2$ with $W=X-\delta$

Consider $X_1,X_2,...,X_n$ a random sample of an statistic population that is modeled by the density function:$$f(x)= \frac{1}{\lambda}e^{-(X-\delta)/\lambda} I_{[\delta,\infty)}$$ Calculate MLE ...
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Maximum likelihood for $\lambda$ and $\delta$

Consider $X_1,X_2,...,X_n$ a random sample of an statistic population that is modeled by the density function:$$f(x)= \frac{1}{\lambda}e^{-(X-\delta)/\lambda} I_{[\delta,\infty)}$$ Obtain maximum ...
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Finding maximum likelihood estimator.

Can someone please let me know if my solution is okay? I feel as though I got it wrong, so here is my work: $L(\theta, \mathbf{x})=\prod _{i=1}^{n}(\theta + 1)x_{i}^{-\theta -2}$ $=(\theta + 1)^{n} ...
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Maximum Likelihood Estimation - Symbol Change

I'm trying to understand why when simplifying the exponential likelihood function, the symbol changed from a product of sequences to summation and why isn't the theta in the exponential to the power ...
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Hypothesis testing of Generalized Likelihood Ratio Test

I try to understand generalized likelihood ratio test (GLRT) from this paperGLRS. However I not understand how to obtain the threshold value d?
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25 views

Find the MLE of an exponential distribution with parameter 1/$\lambda^2$

I have to find the MLE of an exponential distribution with parameter $\lambda^2$. I did it like this: $$$\frac{1}{\lambda^2}e^{-\frac{1}{\lambda^2} x} = log(\lambda^{-2n}) + log(e^{-\sum_{i=1}^{n}\...
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Showing the Likelihood ratio test is invariant to changes in the scale of the variables

Say I have a model: (1) $y_t = β_1 + β_2x_{2t} + β_3x_{3t} + e_t$ with normally distributed errors $e_t$ ~ $N(0, σ^2)$ and fixed regressors and I suppose this model is estimated in dollars. Next,...
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Finding the Maximum Likelihood Estimates for Normal Distribution

I am having trouble finding the maximum likelihood estimate for $\sigma ^{2}$. Find the maximum likelihood estimates of $N(\mu, \sigma ^{2})$ My work so far: $f(x,\mu , \sigma^2)=\frac{1}{\sqrt{2\...
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MLE of Laplacian Distribution

Suppose $f(r\mid \theta)$ is double exponential distribution with pdf as $$f(r\midθ) = \frac{1}{2\sigma}\exp\left({−\frac{|r − \mu|}{\sigma}}\right)$$ , where $\theta= (\mu,\sigma)\,, −\infty<\mu&...
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Weird visual result for MLE of normal distribution

I am trying to visualize the log likelihood estimator for normal distribution via python but in vain. I am not sure if its a python issue or formuala issue in the code. Can you kindly have a look? ...
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Calculating MLE for unknown parameters

Given a random variable $X$, assume it takes on the values $\{1,2,3\}$ with probabilities $Pr[X=1] = p_1$, $Pr[X=2] = 2p_1$, and $Pr[X=3] = p_2$. $p_1$ and $p_2$ are unknown parameters we must ...
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Can I view maximum likelihood as finding the closest probability mass function (pmf) to the pmf with all its mass on the observed value

I took a deep dive into logistic regression recently. I was bothered by the fact that the likelihood formula often explicitly incorporates the values of the coding (0 or 1) into the calculation. In ...
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2answers
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Confusion in Product of pdfs for MLE

Let $X_1, X_2, \cdots, X_m$ be random samples from a normal distribution $N(\theta_1, \theta_2)$. Then, $$ L(\theta_1,\theta_2) = P(X_1=x_1;X_2=x_2;\cdots;X_n=x_m) = \prod_{i=1}^{m} \dfrac{1}{\...
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Confusion in MLE for continuous distribution

Suppose I have a Bernoulli distribution. It is discrete, so the semantics of derivation of MLE as a joint pmf is clear. For sample set $X_1, X_2,\cdots,X_m$, $$ L(p) = P(X_1=x_1;X_2=x_2;\cdots;X_n=...
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1answer
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Deriving max. likelihood estimate of β for a logistic model of two classes with a single binary regressor

I have the log-likelihood function: $$l(\overrightarrow\beta)=\sum_{i=1}^n [y_i log(p(\overrightarrow x_i;\overrightarrow\beta))+(1-y_i)log(1-p(\overrightarrow x_i;\overrightarrow\beta)] $$ where $...
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Estimation in presence of signal dependent noise

Given a model as below: $$y_1 = x + \eta_1$$ $$y_2 = x + \eta_2$$ where $n_1 \sim N(0,\sigma_1^2)$ and $n_1 \sim N(0,\sigma_2^2)$, $N$ denotes a Gaussian distribution and $\sigma_1^2$ and $\sigma_2^2$...
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76 views

Maximum likelihood when usual procedure doesn't work

I am trying to get the maximum likelihood estimate for the parameter $p$. The distribution is the following: $$ f(x\mid p) = \begin{cases} \frac{p}{x^2} &\text{for} \ p\leq x < \infty \ \\ 0 ...
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104 views

Maximum Likelihood Estimate of a Piecewise pdf [closed]

Suppose $X_1$, . . . , $X_n$ are i.i.d random variables having pdf $$ f(x\mid\theta)= \begin{cases} \frac{4}{\theta}-\frac{4x}{\theta^2} & \frac{\theta}{2} \lt x \lt \theta \\ \frac{4x}{...
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About MLE of $\sigma$ with Normal Distribution

I tried to find answers the questions below, but I could not get clear answers for them. For a random sample of size n, $x_1, x_2, ..., x_n$ from a Normal distribution where $\sigma^2$ is unknown. ...
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Find likelihood for time series

There is time series defined with the equation (ARIMAX model) $$X_t - 1.5X_{t-1} + 0.7X_{t-2} = u_{t-1} + 0.5u_{t-2} + \epsilon_{t} - \epsilon_{t-1} + 0.2\epsilon_{t-2}$$ where $\epsilon_{t}$ is ...
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1answer
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Conjugate Prior for Gamma Distribution

This is very basic, but I have been stuck on this problem for a while. Suppose $Y_1, \dots, Y_n|\alpha,\beta\sim Gamma(\alpha, \beta)$ is iid with $\alpha$ known. I want conjugate prior for $\beta$ ...
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50 views

Why can't we use calculus in finding the M.L.E. of Uniform(-theta, theta)?

I've understood that we use maximum/minimum of x's as MLE of theta. But no one so far has explained the reason why differentiation won't work. Please explain.
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Maximum likelihood Pareto

Assume k is known. > $$ f_Y(y;\theta) = \theta k^\theta \bigg(\frac{1}{y}\bigg)^{\theta + 1} \;, \quad y \ge k; \quad \theta \ge 1$$ This one, I'm not sure about these domain conditions, if ...
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Maximum likelihood - uniform distribution on the interval $[θ_1,θ_2]$

Based on a random sample (6.3, 1.8, 14.2, 7.6) use the method of maximum likelihood to estimate the maximum likelihoods for $\theta_1$ and $\theta_2$. $$f_y(y;\theta_1, \theta_2) = \frac{1}{\...
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1answer
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Maximum likelihood for Poisson, having trouble with frequency table

this one has an answer in the back of the book, but I cannot understand it really. For the Major League Baseball season from 1950 through 2008, there were fifty-nine nine-innings games in ...
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Covariance matrix for a d-dimensional gaussian

I am given that the second derivative of the log-likelihood to be $$ \frac{d^2\ell}{d\boldsymbol{\mu}^2} = -\Sigma^{-1}*n $$ $\Sigma$ is the covariance matrix and $n$ is the number of random ...
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1answer
31 views

Maximum likelihood geometric distribution

This one i'm stuck on because I am not sure where the relative frequencies fit in: The following data show the number of occupants in passenger cars observed during one hour at a busy ...
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1answer
49 views

Log likelihood with exponential function

I'm trying to find the maximum likelihood of this function. I have samples in this question as follows: (0.77, 0.82, 0.94, 0.92, 0.98) $$f_Y(y;\theta)=\theta y^{\theta-1} ;, \quad 0 \le y \le 1 ;, 0 \...