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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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How to estimate the parameters of a censored exponential mixture process.

I have a real world scenario that involves a process behaving as follows: there are two kinds of machines and when these machines fail, their recoveries follow exponential distributions. Let's say the ...
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Consistency for maximum likelihood estimator with a single sample

Suppose you have a finite family of probability measures $\{\mu_\theta: \theta \in S\}$ on a finite space $\Omega$ (with respect to the discrete sigma algebra). Let $X$ be a random element of $\Omega$ ...
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Likelihood ratio criterion for testing equality of means in 2 normal populations

Let assume we have the following $$ X_{11}, X_{12},...,X_{1n_{1}} \sim N_{p}(\mathbf{\mu_{1}},\mathbf{\Sigma_{1}}) \\ X_{21}, X_{22},...,X_{2n_{2}} \sim N_{p}(\mathbf{\mu_{2}},\mathbf{\Sigma_{2}}) $$ ...
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Probability of intersection for stars and bars

There is a vector $\vec{x}$ of length $N$, consisting of ones and zeroes. Initially $\sum_i x_i = N^x_I$. Now, each value of $\vec{x}$ that is equal to zero flips to $1$ with probability $p$. Let's ...
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24 views

Finding the condition on $k_1$ and $k_2$ of an unbiased estimator

I'm taking a statistics course and am asked the following : Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the ...
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Weighted maximum likelihood estimator for poisson variables

I have been learning about the E-M algorithm in the Normal case. I would like to create the algorithm with two Poisson variables. To do so, I need to calculate the weighted MLE as seen here in the M-...
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Maximum likelihood estimator of the parameter of randomness in Watts and Strogatz's model (1998).

According to the paper Menezes, M. B., Kim, S., & Huang, R. (2017). Constructing a Watts-Strogatz network from a small-world network with symmetric degree distribution. PloS one, 12(6), e0179120,...
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estimate Gamma parameters based on mean and variance

I am following these two approaches (which are the same)this and this, to estimate the two parameters of Gamma dist based on mean and var. I am not sure why I cannot get the same mean and var from ...
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25 views

Determining MLE from $x$ and $p$.

Currently attempting to better understand Maximum Likelihood Estimators with a sample problem that is a bit outside of my level of understanding. Given the observation $x$, the purpose is to find the ...
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Score statistic of the Pareto function [closed]

I am having trouble with part b (question2) of this question. Would it be possible if any body could help? My specific problem is trying to show the score statistic, s, can be written as how the ...
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Understanding a likelihood as a loss function

Paper Link (IJCAI'18 Yang et al): http://dmkd.cs.vt.edu/papers/IJCAI18.pdf In the following paper, the authors defined Eq.8 as the conditional check-in rate in the Recurrent-censored Regression model ...
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Arriving at Maximum Likelihood Estimates

I am trying to develop a text classifier and I'm reading about MLE to help me understand the process. I came across this example: and I wanted to try this myself. I'm running into a problem and so ...
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20 views

What is the maximum likelihood of a binomial distribution?

i've looked everywhere I could for an answer to this question but no luck ! If I have $X_1 .... X_n$ random variables that are independent and identically distributed such as ∀ $1 <$ $i$ $<n$, $...
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Equivalence of two MLE estimators with a constraint

Consider the following objective function $S\left( \theta ,w|\Lambda \right) .$ The value of $S\left( \theta ,w|\Lambda \right) $ depends on the nuisance parameters $\Lambda .$ However, the first ...
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1answer
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Can a summation be transferred into the denominator?

I include a bit of an introduction, even though my main question is more mathematical. I was tasked with finding the Maximum Likelihood Estimate for $\theta$ in $$\mathrm P(X>x) = \left(\frac ax \...
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EM algorithm for a specific example

We consider $X \in R^{n\times d}_+$ as well as $n\times d$ mutually independent latent variables $z_{ij}$ where $z_{ij}$ has a Poisson distribution of parameter $X_{ij}w_j$ for $i\in \{1,..,n\}$ and $...
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Proof of concavity via matrix differentiation

I have found a maximum likelihood estimator which is given by: $${{argmax }\atop {\vec{\mu} = [\mu_1 \dots \mu_2] \atop \mu_i \geq 0}} -\sum_{j=1}^m \vec{p}_j^T \vec{\mu} + \sum_{j=1}^m y_j ln(\vec{p}...
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14 views

confidence interval with MLE estimator

$f_{\theta}(x) = 2 \theta x e^{- \theta x^2} $ on the interval $(0, \infty)$. T is the MLE estimator of $\theta$. We construct the confidence interval of $\theta$ $(aT, bT)$, where $a$ and $b$ are ...
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19 views

Likelihood Function Given Maximum of data, but not actually data points

I was just wondering how I would go about creating a likelihood function if I have a $N( \theta,1)$ distribution and know $x(n)$ the maximum of n observations, but not the actual observations ...
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how log likelihood's derivative is equal to zero in maximum log likelihood.

if the log-likelihood function is strictly increasing and it has not horizontal asymptote then how it's derivative is equal to zero in maximum log likelihood. Now since it is strictly increasing every ...
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Wald test for variance of normal distribution

Let $y_1, y_2,...,y_t$ follow a $N(0,\sigma^2)$ distribution. [Note that the mean is zero and you know that it is zero]. Derive the LR, LM and Wald test of hypothesis $\sigma^2 = 1$. I have got the ...
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Plot the log likelihood function for X ~ Bin(10, p)

This is the third part of a question I am working on. So after solving for the parameter estimate $p$ by using the MLE method, I've been asked to plot the log likelihood function for X = 5, n = 10. To ...
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Likelihood for model with two data sources

$\theta$ is parameter for model and $D_{1} ,D_{2}$ are two i.i.d data sources. The parameter for model are jointly optimised by MLE. My goal is to just confirm likelihood for the model is right. $ P (...
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Likelihood ratio test at level $\alpha.$ (Verification)

We have an observation $X_1$ distributed with a density function $$f_{\theta}(x)=\frac{\theta}{(x+\theta)^2}$$ for $x\geq 0$, otherwise $f_{\theta}(x)=0.$ Here we assume that $\theta>0.$ We want ...
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Find the MLE of a parameter with a given density function [duplicate]

I have to find the MLE of $\theta$ The density function is given, which is: $$f(x) = \theta x^{-2}$$ And $x \geq \theta, 0 $ for $ x < \theta, \theta > 0 $ unknown I have tried to find the ...
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Finding the MLE of expected value of two successions of normals.

I have $X_1 ... X_{n_1}$ ~ Normal$(\mu, \sigma^2_1)$ and $Y_1 ... Y_{n_2}$ ~ Normal$(\mu, \sigma^2_2)$, with $\sigma^2_i$ known. All random variables are independent. I want to find maximum likelihood ...
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1answer
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How to show that MSE of ML estimator is greater than that of Bayesian posterior mean?

This question is based on problem 9 from chapter 4 of Gelman et al.'s Bayesian Data Analysis. Suppose we observe $y\sim N(\theta,\sigma^2)$ and wish to estimate $\theta$, with $\sigma^2$ known. We ...
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Does the Integral over the product of distributions result in another distribution?

In this post one guy asks why those likelihoods in Maximum Likelihood get multiplied. It's clear why they get multiplied. I have another problem. In my textbook it is state that for each sample the ...
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Likelihood Ratio tests

I am trying to figure out how to calculate an LRT for a question, but my problem seems to start with the fact that I have no idea how this is possible: $$\ln L1 = \ln(100/63) + 63 \ln(1/2) +(100-63)\...
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Determining the generalised likelihood ratio for a test determining whether the $x_{i}$ share a common parameter

I need to determine the generalised likelihood ratio in order to test whether a set of values $X$ are such that each $x_{i} \in X$ are given by a common poisson distribution $Po(\lambda)$ or ...
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Log likelihood of normal time series

Given a time series $\{x_i, i=1, \ldots, n\}$ and define the corresponding series $$y_i = \log(x_{i+1} + a) - \log(x_i + a),$$ where we assume that $y_i \sim N(\mu, \sigma^2)$. The aim is to estimate $...
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What is the log-likelihood for the Cauchy distribution when it is maximized?

For the normal distribution and e.g. for the continous uniform distribution MLE is easy to perform in all details, but MLE is a hard job for the Cauchy distribution. For the normal distribution we ...
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225 views

Which of the following statements are correct regarding MLE

MLE here implies Maximum likelihood estimator. Statements are : $1$. MLEs are always consistent $2$. MLEs are always unbiased $ 3$. MLEs follow normal distribution asymptotically ...
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78 views

find partial derivative of the function

I have this equation which is basically a maximum likelihood equation for EM-algorithm. $$L(\theta) = \sum_{i=1}^n{\ln{(\sum_{j=1}^kw_jp_j(x_i;\theta_j))}}$$ I'm trying to derive a partial derivative ...
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MLE of $\lambda$ Given $f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}$

$f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}\ \ \ ; 0\le x\le1 $ $0$ otherwise What is the maximum likelihood estimate of the parameter $\lambda$ based on two independent observations $...
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Parameter estimation using method of maximum likelihood. What am I doing wrong?

Question from Michael Baron's Probability and Statistics for Computer Scientists, 2nd edition: Data: 3 7 5 3 2. Assume data is produced from Geometric distribution. Estimate $p$, the geometric ...
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least square failure as classifier

I was reading pattern recognition and machine learning by Christopher bishop in chapter 4.1.3 page 186 about least square classification failure I stumbled on this phrase "The failure of least ...
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Find the MLE of $p$ where $f(y;p)=2p^2y^{-3}$

Find the MLE of $p$ where $f(y;p)=2p^2y^{-3}$. Attempt: Method: find the likelihood function, differentiate with respect to $p$ then set to zero and solve for $p$. $L(p;y)=\prod\limits_{i=1}^n [2p^...
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63 views

Likelihood Ratio Test Variance of Normal Distribution

Let $X_1,...,X_n$ be a random sample from $N(0,\sigma_X^2)$ and let $Y_1,...,Y_m$ be a random sample from $N(0,\sigma_Y^2)$. Define $\alpha := \sigma_Y^2/\sigma_X^2$. Find the level $\alpha$ LRT of $...
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76 views

Existence of MLE

I have a problem with MLE's definition: Casella Berger in Statistical Inference and Nitis Mukhopadhyay in Probability and Statistics said that MLE for a parameter $\theta\in\Theta$ is respectively $...
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28 views

Why are the local extrema of a log-transformed function equal to local extrema of the original function?

I am studying maximum likelihood and to simplify taking the derivative of the likelihood function, it is often transformed by the natural log before taking the derivative. I have read in other posts ...
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Example of a maximum likelihood estimator that is not a sufficient statistic

I am currently researching on providing some bounds on estimation using some information theoretic tools (I won't expend on that here for now, I may make a post about it later) and turns out that ...
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Maximum Likelihood of random variable being Fréchet distributed with multiplicative noise?

I have a random variable being Fréchet distributed with shape parameter $\nu$ and scale parameter $c$ (location $0$). Now I add multiplicative noise, i.e., I set: $$ Y=X \cdot \epsilon \;, $$ with ...
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MLE of $\theta$ in $U[0,\theta]$ distribution where the parameter $\theta$ is discrete

Consider i.i.d random variables $X_1,X_2,\ldots,X_n$ having the $U[0,\theta]$ distribution: $$f_{\theta}(x)=\frac{\mathbf1_{[0,\theta]}(x)}{\theta}$$ , where the unknown parameter $\theta\in\{1,2,\...
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Do I really need to solve 5 nonlinear equations in this Lagrange multiplier problem?

I need to solve the following optimization problem Let $X=\left\{ x_{i}\right\} _{i=1}^{n}$ be an independent sequence of $k$-face die rolls. Where for $j\in\left[k\right]$ we have $p\left(x_{i}=j\...
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Compute likelihood from a uniform distribution with known $\theta$

I'm having a hard time trying to understand how to compute the likelihood from a uniform distribution with pair variables ($X_1$, $Y_1$) ...) i.i.d when $\theta$ is known. Most examples I've found is ...
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1answer
39 views

Taylor expansion of likelihood function

$\require{\cancel}$ ...For large samples, as a consequence of the central limit theorem, the likelihood function approaches a gaussian, whose expected value is equal to the maximum likelihood ...
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MAP/MLE/Neyman Pearson Example Problem

I'm currently attempting a practice problem and wanted help checking my work: The random variable X is such that P(X=1) = 2/3 and P(X=0) = 1/3. When X = 1, the random variable Y is exponentially ...
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1answer
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Find max likelihood estimator for a 2 variable function

I have been given a problem in which I am given a set of independent random variables, X1, X2, ... Xn with the distribution: $f_x(x ; b, m) = \frac{1}{2b}e^{-|x-m|/b}$ for $-∞ < x < ∞$ and I ...
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Censored piecewise exponential maximum likelihood

Let $0=a_0 < a_1 < \dots < a_m = \infty$ be given real numbers and let $\lambda_1, \dots, \lambda_m$ be positive real numbers. Let $T_i$ have hazard $$ h(t) = \lambda_j, \quad \text{if $t \...