# Questions tagged [maximum-likelihood]

a method of estimating parameters of a statistical model by choosing the parameter value that optimizes the probability of observing the given sample.

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### Find asymptotic variance of the moment estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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### How can I get the Binary Cross Entropy from the Cross Entropy function for GANs

I got the definition of log-likelihood by Goodfellow's Deep Learning book: \label{eq:loglikelihood} \theta_{ML} = {argmax}\sum_{i=1}^{m} \log p_{model}(x_i; \theta). \end{...
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### Diffusion race model with censoring. Help to verify overall logic

I conducted a Go-noGO experiment in which the subject had to press a button if the stimulus on the screen was an orange ($O$) and had to refrain from pressing it if he saw an apple ($A$). The ...
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### The probability/cumulative density function for inequality of two random variables

I have two random variables X and Y which came from different inverse gaussian (IG) ...
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### How can I estimate the variance of the error terms in a conditional/multinomial logit model?

Conditional/multinomial logit models(CML) can be esimated by the Maximum Likelihood Estimation(MLE). The likelihood would consists of choice probabilities: P_{ij}=\frac{e^{V_{ij}}}...
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### Why is it said that maximum likelihood becomes intractable if there are latent variables?

It is said that EM algorithm helps in cases where direct MLE cannot be carried out due to missing/latent variables. However, I could not understand why direct MLE cannot be carried out when there are ...
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### mgcv GAM models in R package caret - how to interpret output

I am attempting to evaluate two GAM models I developed in mgcv via leave-one-out cross validation in the caret package. I am a newbie to both GAMs and cross-validation. For the purposes of this ...
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### Maximum Likelihood Estimation Uniform Distribution [duplicate]

A sample is drawn from the uniform distribution: $X_{1},X_{2},...,X_{n}:U(0,2\theta )$ I am interested in finding the MLE for $\theta$. My intuition say that $\hat{\theta }=max(X_{i})/2$ How do I ...
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### Is there a good package to implement latent profile analysis in R, which allows use of FIML for missing data

We are looking to conduct latent profile analysis in R, using a large number of continuous variables (physical activity level at different times of day). We have objective measurement of physical ...
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### Using restricted maximum likelihood on marginal residuals

In a mixed-model setting, I want to estimate the variance components and the pertaining random effects of a random-intercept/random-slope model. The coefficients of the fixed effects have already been ...
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### Maximum Likelihood Estimation for data with non normal distribution [closed]

I have a set of data (results) that does not follow Normal distribution. In this case, wow can I get Maximum Likelihood Estimation? Thank you very much
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### GPD and GEV Fitting: Maximum Likelihood vs. Least Squares

I am trying to build a model based on real world data which involves fitting generalized extreme value distributions and generalized Pareto distributions. Most literature immediately turns to the ...
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### How to estimate Pearson type-IV-GARCH using MLE in R?

I am just wondering whether there is an R program that can be applied to run GARCH specifications with Pearson types IV distributions. If you are familiar with any or can guide, it is greatly ...
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### t-Copula MLE on nu (DoF) only - log-likelihood function possibly convex?

I am working with t-Copula's to generate random synthetic data eventually. The paper I use as the foundation is Benali et al., 2021. To determine the best fitting t-Copula, they propose determining ...
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### emmeans, should I use ML or REML fitted models [duplicate]

I see a lot of examples out there which use ML or REML fitted models as an input for emmeans. If I understand correctly for model comparison of fixed effects for "fixed effects nested models"...
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### Self-Study: Score Statistic of a Poisson

I have been attempting a question on Generalised Linear Models. I thought I understood it but doing exercises myself now, proves otherwise. I have attached an image of the question here, struggling ...
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### What is the limiting distribution of $$R_n=2(l(\hat{\theta})-l(\tilde{\theta}))$$ under null model $H_0$ for Poisson?

If $X_1,\dots, X_n\sim Poisson (\theta_1)$ and $Y_1,\dots, Y_n\sim Poisson (\theta_2)$, and $X_i$ is independent of $Y_i$, we want to test $$H_0: \theta_1=\theta_2, \, H_a: \theta_1\neq \theta_2$$ ...
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### Determine the asymptotic distribution $\sqrt{n}(\hat{\theta}_n-\theta)$ for trinomial distribution over the group sizes $(x,y,z)$

A random sample of $n$ individuals are classified into three groups, with probabilities $\theta^2$, $2\theta(1-\theta)$, and $(1-\theta)^2$ respectively, yielding the trinomial distribution over the ...
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### Is the likelihood of a discrete binomial variable same as it probability? Like in the case of tossing a coin lets say 12 times

I am working on a probability project where we first generate random variables for a given Binomial experiment and then we generate a PMF for 10 coin tosses using the list of random variables we ...
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### Is the likelihood for Gaussian mixture models still multimodal when Y is partially observed?

In discussing Gaussian mixture models (GMMs), https://normaldeviate.wordpress.com/2012/08/04/mixture-models-the-twilight-zone-of-statistics/ highlights the issue of Multimodality of the Likelihood. ...
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### MLE for sample mean estimation in practice

I usually use median for a truncated normal distribution mean estimate. In theory the best approach is to do gradient descent optimizing the mean for data generation. Does anyone have a good example ...
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### Proportion data: Logistic with MLE vs. OLS with logit-transformed response

This is an expansion of @Beethoven_90's comment on this question. Suppose I have proportion data $Y_i$ computed from a binomial; $Y_i = \frac{S_i}{N_i}$ where $S_i \sim Bin(N_i, p_i)$ and $p_i$ is the ...
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### Why is it said, that GEE is not likelihood based, IF the estimating equations are derived exactly thru d(log(L)/dB?

I found this article: https://sakai.unc.edu/access/content/group/2842013b-58f5-4453-aa8d-3e01bacbfc3d/public/Ecol562_Spring2012/docs/lectures/lecture22.htm#choosing and most of it is showing the ...
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### Parameter Estimation on S1 model

I have the following model of creating a random graph on the circle: First, N nodes are uniformly distributed on the circle of radius $N/(2\pi)$ to give a node density of $1$. We sample the expected ...
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