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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Fitting mixture model on data with duplicate values

What is the correct procedure to fit finite mixture models on data with many duplicate values using EM? Let's say I have N(0,1) distributed data and try to fit a 2 component mixture using EM. There ...
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Why can't regression via Maximum Likelihood shrink coefficients to zero?

Why can't regression via Maximum Likelihood shrink regression coefficients to zero as in LASSO? Does shrinking coefficients to zero not give higher L-likelihood? Does the answer to my question have to ...
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Does the newton-raphson either find the maximum of the loglikelihood function or estimate the MLE and likelihood function? [closed]

Does using newton-raphson method or some other optimization method used in nlme package or mixed effects models actually find the maximum of the log-likelihood function's height or does it simply find ...
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Indefinite Hessian in Negative Binomial log-likelihood estimation

I have the following problem: I am estimating a Negative Binomial regression. In order to estimate the model, I implemented the Newton method from the scratch. I am optimizing the dual of the log-...
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Maximum likelihood and Minimizing Kullback–Leibler divergence to the ecdf?! (i.e.: finite sample statement?)

There is a known relationship stating that finding MLE is asymptotically the same is minimizing Kullback–Leibler divergence (see wiki here), or just the cross entropy. I'm wondering if there is a ...
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Difference Between $L(θ | X=x)$ and $P(θ | X=x) $? [closed]

Although statistically different, I feel like they both say the same thing? $L(θ | X=x)$ = the probability that a sample provides support for particular values of a parameter in a parametric model. ...
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How does likelihood differ from probability? [duplicate]

i.e. Tossing a fair coin $p_H =$ probability of heads $= 0.5$ $L(p_H | HH) = P(X = HH | p_H) $ $= 0.25$ Okay, right-hand of equation I understand, $P(X = HH | p_H) = 0.25 =$ probability of "$X = ...
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When Bayesian estimator coincides with MLE

Assume that $\theta$ is an unknown parameter with prior belief $p(\theta)$ and that $X_1,...,X_n$ is an iid sample from $p(x|\theta)$. What does it say about our prior distribution $p(\theta)$ when ...
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AIC and model selection with multimodel

For a simple example, I am doing a multi-model regression/likelihood estimation. The data is $(y_1,y_2,x_1,x_2,x_3,x_4,x_5)$. The first model (A) consists with two regressions: $y_1=e^{a_1x_1}+e^{...
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why the sigma(volatility) is so small when useing the maximum likelihood estimation for vasicek model

This is my monthly data(in percentile)-2year japan government bond's yield to marturity from year 2001 to year 2019.And my LL function comes from "Maximum likelihood estimation using price data of the ...
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Assumption of error of logistic regression [duplicate]

I know there are debates about whether the error exists and its distribution in the case of logistic regression. Suppose we assume that the error term follows the logistic distribution. Are we ...
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Manual calculation of logit regression [duplicate]

I have been working on Logit regression and I don't understand how to find coefficients by hand. I have seen that its possible to do it using maximum likelihood but I don't have much background in ...
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Fisher matrix with penalty function

I am fitting a parametric model to human tracking data. Because my data is in part corrupted (see below why), as such I had to introduce a penalty function to my optimization algorithm. The Fisher ...
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how to set the parameter constraints in maxLik() in R [closed]

question: how to use maxLik() to set the constraints? For example,the beta and alphh should >0, and sigma should >=0.0001 .The following is my code, and it doesn't work in mle<-maxLik(...),because ...
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Constrained MLE

How to write R-code for obtaining MLEs of a ,b and $\theta$ for the density function \begin{equation} f(x)=\theta(a+bx)e^{-(ax+\frac{1}{2} b x^2)}\left(1-e^{-(ax+\frac{1}{2} b x^2)}\right)^{\theta-1} \...
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Re-writing the Score Test Statistic [closed]

We all know that the formula of the score test is: $$ S = \frac{\{l'(\theta_0)\}^2}{E\{-l''(\theta)\}\bigg|_{\theta_0}} $$ where $l'$ and $l''$ are $log$ $likelihoods$ What I need to do is represent ...
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Show that $E[u(X_1)\mid\sum^n_{i=1}X_i=z]=\Phi \left( \frac{\sqrt n(c-z/n)}{\sqrt{n-1}} \right)$ [duplicate]

Let $X_1, ..., X_n$ be a random sample from the $N(\mu,1)$ distribution. First show that $$E \left[ u(X_1)\, \Big|\, \sum^n_{i=1}X_i=z \right] = \Phi \left( \frac{\sqrt n(c-z/n)}{\sqrt{n-1}} \right)\,,...
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Over “specification” of a statistics model?

For a simple example, I am fitting data with a likelihood function generated by a normal distribution. The first model is the normal distribution with two-parameters. The second competing model is the ...
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Maximum likelihood estimator for difference of parameters

We're comparing two different clinical treatments. For each patient we observe the pair $(Y_i,R_i)$, where $Y_i$ denotes if the treatment was successful ($Y_i=1$) or not ($Y_i=0$) and $R_i$ indicates ...
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Identifying original elements in a permutation of a pattern

In the pattern above, the blue peaks are known and are labeled $p_1, p_2, ... p_N$. The red line is the same pattern of peaks, but altered slightly. Given a blue pattern and red pattern, I want to be ...
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Newton-Raphson method to solve for dof when performing MLE of a multivariate Student-t distribution using EM

I am reading the derivation of EM algorithm to estimate the maximum likelihood of a multivariate Student-t distribution $\mathcal{T}(\mathbf{x} \vert \pmb{\mu}, \pmb{\Sigma}, \nu)$ in Kevin Murphy's ...
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Maximum A Poseriori - Regression - Dependence between feature coefficients and features?

I was going through some lectures on regression when I came across these statements. Wouldn't the expansions, respectively, be $P(w|y,X) = \frac {P(y|w,X)*P(w|x)} {P(y|X)} $ and $ P(y,w|X) = P(y|w,X)...
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Request for Explanation: Deriving Probability Density Function of a Maximum Likelihood Estimator of a Uniform Distribution [duplicate]

I am reviewing some practice problems and have both a question and its solution but am struggling to understand them and am hoping someone can help me. I am struggling to follow the logic for ...
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Maximum Likelihood Estimator with exponential noise

So I need a little help with this please. I'm given N measurements of a signal $Y_{i} = A + v_{i}, i = 1,...,N$, where $v_{i}$ is measurement noise with the exponential pdf $f_{v}(v) = e^{-v}, v \geq ...
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Compare return levels of fitted GPD using MLE in different R packages

This question is related to this post: Different quantiles of a fitted GPD in different R packages? I want to constraint "potvalues" data to be in a period of 6 years, this is, 16 observations per ...
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would it be possible to pick a different likelihood model achieve the same posterior estimation?

Take the coin flipping example. When we decide to use the Bernoulli distribution to model a coin flip, of course with and without a conjugate prior would make some difference for estimation. Would it ...
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Deriving the $p$-dimensional maximum likeliihood estimation

So I'm currently following a statistic course and we are deriving the maksimum likelihood estimator for a $n\times p$-dimensional normal disitribution. So for $\mathbf{X}=\begin{pmatrix} \mathbf{X}_1^*...
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Logistic regression: Equation for marginal effect at the mean

I am estimating the following logistic regression (binomial family) by maximum likelihood: $$ \ln\left(\frac{Y}{1-Y}\right) = \beta_{0} + \beta_{1}D + \beta_{2}X + \epsilon$$ where D is a dummy. ...
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What are some of the robustness checks for the likelihood ratio test?

In the application of statistical methods in social science, one usually does a lot of robustness checks. If I got some publishable findings using LRT test by discrimination two theoretical models, ...
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What does Maximum Likelihood Estimation mean in Machine Learning? [duplicate]

I am wondering what Maximum Likelihood Estimation means to Machine Learning in terms of training a predictive model. I understand Machine Learning uses Maximum Likelihood Estimations for model ...
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Who performed the first Maximum Likelihood Estimation?

I am very interested in the historical development of statistical theories. Here is the research I've done: I've tried to read two old papers of Fisher. I think the first theory paper on MLE should be ...
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What regression/estimation is not a MLE?

I just rigorously learned that OLS is a special case of MLE. It surprises me because the popular and "reliable" sources such as researchgate and this do not mention this most important connection ...
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Expected value and Maximum Likelihood Estimation

I'm doing this exercise about Poisson distribution and maximum likelihood estimation: I have had no problem with points a and b, but I'm struggling with the correct answers of the C part. From my ...
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Can I use maximum likelihood to optimize a scaling factor for a composite independent variable?

Basically, I am wondering if it is possible to use some kind of likelihood approach to optimize a scaling coefficient for an independent variable. I have the following standard linear model: $y = B_0 ...
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Parameter estimation using data and model [duplicate]

For the equation $A=Bw+C(1-w)$, I would like to find the most likely value for $w$. Each coefficient A, B an C has an x and y coordinate and uncertainty (x,y; sd). For example, say I have the ...
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Effect of heteroskedasticity in hierarchical (non-)linear models

Unlike linear models estimated via OLS where heteroskedasticity lead to inconsistency of the variance estimator but not the coefficient estimates, heteroskedasticity causes inconsistency of both ...
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When using L2 regularization outside of linear regression, do the same MAP estimation assumptions hold?

Some context is shared below, and my question is bolded at the end. In the linear regression setting, we learn model weights $\hat{\mathbf{w}}$ to make predictions $\mathbf{\hat{y}}$ from new samples ...
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Statistical estimators distance for close empirical distributions

Is it valid to argue that two empirical distributions $ p_1, p_2 $ having small Wasserstein distance $W_r(\cdot)$ for an order $ r $ will yield close MLE estimators for a statistical model ...
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1answer
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Maximum Likelihood Fit for Non-Linear Regression

I am reading the blog on Bayesian priors and overfitting and it mentions that assume that a data is generated by the following function: $$y_t=\sin\left(\dfrac{x_t}{10}\right)+ \cos\left(z_t\right)+...
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Extend likelihood equation to P(Y>=y) in R

This question involves both math and coding in R. Apologies if this should be on Stack Overflow, but I decided statisticians would ultimately provide better support....
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1answer
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Bayesian random variables VS. MLE fixed variables

"Bayesian inference treats model parameters as random variables whereas frequentist inference considers them to be estimates of fixed, ‘true’ quantities." (Ellison 2004) What does it mean in the ...
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“Explanation power” of a parameter [closed]

Suppose there are two practically similar statistical models with similar "breath": $L(\theta)$ and $M(a,b)$, where $\theta, a, b$ are parameters. When fitting with data, if we assume that $b$ is ...
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MLE $\hat{h(\mu)} = h(\hat{\mu})$ of $h(\mu) = var(Y_1) = \mu^2$

Question: Suppose Y1, · · · , Yn follows an Exponential distribution with $\lambda = \frac{1}{\mu}$. Derive the MLE $\hat{h(\mu)} = h(\hat{µ})$ of $h(µ) = var(Y_1) = µ^2$, and show that $h(\mu)$ is ...
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Logistic regression fitting methods clarification

Each book I read propose a different fitting method for Logistic Regression. The general idea is to maximize this expression. $$ Pr\left(\beta|y,X,M\right) = \frac{Pr\left(y|\beta,X,M\right) Pr(\...
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Parameter estimation when the likelihood function does not exist

The observations $Z_1,Z_2\cdots$ are i.i.d. We have $$Z_k = \sum_{i=1}^\infty \frac{X_{ki}}{2^k}.$$ where the $X_{ki}$'s are i.i.d. with a Bernouilli$(p)$ distribution. If $p=\frac{1}{2}$ then $Z_k$ ...
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MLE vs MAP hypothesis, performance

I have a question regarding the performance of Maximum a posteriori vs Maximum Likelihood Estimation hypothesis. If you only consider the training data. Can you then state that the MAP hypothesis ...
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2answers
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Can an improper prior distribution be informative?

I have just worked through an example where, with an improper prior, the bayesian estimator equals the maximum likelihood estimator, leading me to believe that improper priors are uninformative. But ...
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1answer
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Invariance of maximum likelihood estimates to rearrangements of parameters/constants in the model?

I know that maximum likelihood estimates are invariant to re-parametrization (https://stats.stackexchange.com/a/335368/267430). Is the MLE also invariant to rearrangements of the constants and ...
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1answer
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Question about Casella and Berger's proof of MLE invariance

In Casella and Berger, p. 320, they have a proof of the invariance of the MLE. Let $g: \theta \mapsto \eta$ be a function. They define the induced likelihood as $$ L^*(\eta \mid X) = \sup_{\{\theta: ...
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Two-sided one-parameter likelihood ratio test

Consider a single-parameter distribution and a two-sided test for hypothesis $\theta_0$. Does the likelihood ratio test just reduce to: $$ \frac{\max_{\theta \in \Theta_0} \mathcal{L}(\theta)}{\max_{\...