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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Calculating Log-Likelihood from Simulated Distribution

I want to perform some sort of model evaluation of a multivariate distribution with the property that it is difficult/impossible to calculate the likelihood (of the whole model, you can do it for ...
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2answers
59 views

Self-study: Finding the maximum likelihood estimates of the parameters of a density function

Consider a random sample $x_1,x_2,...,x_n$ from a newly-generated distribution, whose probability density function is given below \begin{equation} f(x|\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( ...
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1answer
22 views

Finding maximum likelihood estimates of parameters of multiple normal populations

I've just started studying maximum likelihood and likelihood ratio tests. I've calculated the maximum likelihood of a normal population with unknown mean and variance. However, I've been given this ...
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12 views

Confidence interval of a function of Maximum Likelihood estimates

I have a probability density distribution for a variable $x$. This distribution is a function of another variable $Y$, that is $P(X=x \space |\space Y)$. I observe one $x$ value in nature (the ...
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11 views

Different quantiles of a fitted GPD in different R packages?

I am performing an extreme value analysis for meteorological data, to be precise for precipitation data available in mm/d. I am using a threshold excess approach for estimating the parameters of a ...
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15 views

Unsuccessful ML estimation of a modified Weibull model's parameters

I'm running this code in R to find the ML estimators of a modified Weibull model. ...
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1answer
12 views

Verify accuracy of asymptotic variance of estimator

The asymptotic variance of a maximum likelihood estimator can be obtained from the inverse of the Hessian of the log-likelihood function at the MLE, and the variance of derived quantities can be ...
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29 views

Inferring the likelihood function when given a maximum likelihood estimate [closed]

Someone gave me an analytic formula $f$ and told me that it is the maximum likelihood estimate for some likelihood function $\mathcal{L}$. Is there a procedure for then constructing an $\mathcal{L}$ ...
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1answer
49 views

Gradient of Log-Likelihood

Considering the following functions I'm having a tough time finding the appropriate gradient function for the log-likelihood as defined below: $a_k(x)=\sum_{i=1}^D w_{ki}\cdot x_i$ $P(y_k|x) = ...
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6 views

max margin vs max posterior/likelihood advatages

I am working on some parameter learning approaches for image classification. What is the differences between the following two for image classification? max margin methods maximum likelihood/ ...
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24 views

LS vs MLE for Gaussian Conditional Random Field estimation

Is there such a thing as Least Squares estimation for the conditional mean and covariance of a conditional gaussian random field? I'm looking at this paper by Wytock and Kolter 2013, in which they ...
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45 views

Finding sampling distribution of normal MLE and likelihood

I'm reviewing old exams in preparation for a statistics final, and I'm stuck on a particular question: Suppose that you have n independent random variables $Y_i$, with each distributed normal with ...
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1answer
32 views

Probability of a model given an image

I would like to write the likelihood function for an image with respect to theoretically predicted values. Assuming uniform Gaussian noise, the pixels are statistically independent, and we can write a ...
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1answer
114 views
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computing the posterior of two Gaussian probability distributions

I am a bit confused how to solve a Bayesian statistics problem. I have a parameter $\epsilon^s$ which is defined as following: $$\epsilon^s=\frac{\epsilon-g(\pi,z)}{1-g^*(\pi,z)\epsilon}$$ where ...
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1answer
37 views

In Max. Likelihood the expected score is zero for the true values. Is it also true for any other values?

The usual proof of the Expected score in ML expected score being zero goes 'similar' to this: $f(z;\theta)$ is the density function, for data $z$, and parameter vector $\theta$, so $\int ...
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0answers
27 views

Guidelines to estimate using MLE from this definition of error function

Consider a stable causal, single-input/single output, linear time-invariant, discrete-time system. The noisy output is $y[n] = \sum_{i=0}^{p-1} c_i d[n-i] + w[n]$ where $c_i$ is the real-valued ...
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9 views

Obtain the Beta of a regression using MLE Matlab

I am trying to find an alternative way around OLS to find the Beta coefficients so that they give me $\epsilon$ My depedent variable (y) is a time series, and my independent variable (x) is just a ...
2
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1answer
32 views

Maximum likelihood estimation from 2 exponentially distributed sample

$X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_n$ are independent samples from the exponential distributions with parameters $\lambda$ and $\frac{1}{\lambda}$. What is the MLE for $\lambda$? I used the ...
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17 views

Confidence intervals for maximum likelihood estimator with constraints

Let us suppose I have a maximum likelihood estimator for a multivariate parameter $\vec{\theta}$. The parameter is subject to the following constraints: $\theta_i \in [0,1]$ $\sum_i \theta_i = 1$ ...
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1answer
28 views

Maximum Likelihood Estimation and Standard Errors

Suppose, I have the following model: $$ Y = X^T\beta + u_t $$ where $u_t$ ~ GARCH(1, 1) with Gaussian mixture as error distribution (or even something more weird, like normal-inverse-gaussian and ...
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29 views

When to use a bootstrap in MLE

Suppose I have a data set of $n$ observations with the dependent variable sample $\mathbf{Y} \in \mathbb{R}^{n \times k}$, and independent variable sample $\mathbf{X}\in \mathbb{R}^{n \times l}$ such ...
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1answer
147 views

Beginner learning resources : Pdf and likelihood function for non-Gaussian time series model

I am struggling with exercise problems related to blind system identification where the knowledge about the source input is assumed to be known using maximum likelihood estimation of univariate time ...
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3answers
57 views

What is the variance of an MLE for a trinomial distribution?

I am playing with the following trinomial (multinomial) distribution which can get values (a,b,c) with the probabilities: $(\theta^2, 2\theta(1-\theta), (1-\theta)^2)$. Say I have n observations from ...
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156 views

Maximum likelihood estimation involving both probabilities and probability densities

Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version. In general my question regards how to compute likelihoods in mixed ...
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10 views

To demean or to use dummies in maximum likelihood

I have a dynamic panel data with T=20 and N=1500 and I use a maximum likelihood estimation (more precisely its a VAR). Using a dummy variable approach to account for fixed effects introuduces an ...
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16 views

Understanding Multivariate Normal Distribution in simple terms

While reading about the proc Tcalis procedure in SAS for SEM, I came across the statement: "For maximum likelihood (default) and generalized least squares ...
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2answers
112 views

Unsure how to find the MLE for this model

The question I am trying to answer is confusing me as I don't know where to start to find a likelihood estimation The Question $y_i = μ + e_i $ where the $e_i$ are independent variables ...
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155 views

Maximum likelihood estimator of joint distribution given only marginal counts

Let $p_{x,y}$ be a joint distribution of two categorical variables $X,Y$, with $x,y\in\{1,\ldots,K\}$. Say $n$ samples were drawn from this distribution, but we are only given the marginal counts, ...
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17 views

Pooled likelihood in R, how to code it? [migrated]

I don't really know how to construct the following likelihood in R. I have panel data for several individuals. Each individual is observed over time and I have several observations for each ...
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22 views

Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
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13 views

Softmax maximum likelihood problem: arbitrary constant

I'm doing a multiclassification with a softmax function. The probability of a sample $j$ belonging to class $k$ is given by the softmax: ...
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26 views

Maximum likelihood of multivariate t-distributed variable with scaled covariance

I am trying to estimate the covariance of a iid multivariate t-distributed random variable, where I define the multivariate density as in the Statlect textbok, which is the same as the wikipedia page. ...
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48 views

Explain log likelihood behaviour

(This question is related to a previous one I made, here) I have a set of 2D observations (measured data) of sample size $N$: $$O = \{(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)\}$$ I also have a model ...
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1answer
38 views

Lemma 3 of the note on the consistencty of the maximum likelihood estimate by A.Wald

Can someone explain to me how Lemma 3 in this article follows from (13) and (14) and the fact that the functions are decreasing? Below is my transcription of the Lemma. Let $X_1, X_2,...$ - i.i.d. ...
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1answer
112 views

Deriving the maximum likelihood for the parameters in linear regression

Notation: $\textbf{w}$ is an M-dimensional vector of parameters (including the bias parameter), $\textbf{x}_n$ is an M-dimensional vector of the features of each training example, $\textbf{t}$ is an ...
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13 views

Joint significance testing w/ large samples

long time listener, first time caller. Unfortunately I can't show code as the computer the analysis is done on has rather tight security. I need to test for joint significance in a logit model that ...
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1answer
76 views

Maximum likelihood estimator for minimum of exponential distributions

I am stuck on how to solve this problem. So, we have two sequences of random variables, $X_i$ and $Y_i$ for $i=1,...,n$. Now, $X$ and $Y$ are independent exponential distributions with parameters ...
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1answer
33 views

MLE for two-parameter exponential distribution

I have to find the parameters of a two-parameter exponential distribution using the MLE. But imposing FOC I do not find enough conditions to found both the paramenters. Any hint? Thanks!
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2answers
119 views

Maximum Likelihood Estimation

If $V_1, V_2,\ldots V_{n_1}$ and $W_1, W_2,\ldots,W_{n_2}$ are independent random samples of size $n_1$ and $n_2$ from normal populations with the means $\mu_1$, $\mu_2$ and the common variance ...
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3answers
175 views

Inconsistency between R and SAS for MLE on Weibull

I have the following data Y which I want to get an MLE estimate for the parameters using a Weibull distribution in R. 1468, 1872, 475, 1372, 3830, 1849, 978, ...
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1answer
55 views

Unable to calculate the density function for AR

The model is an AR(p) process excited by a white Gaussian noise $\epsilon_t$, \begin{align} Y_t = &c+ \phi_1Y_{t-1} + \phi_2 Y_{t-2}+ \ldots+ \phi_p Y_{t-p} + \epsilon_t\\ \epsilon_t = ...
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51 views

How to Construct a Maximum Likelihood Estimator for Parameter from a Beta Distribution?

I have a question from my Stats Homework that is worded as follows: Based on a random sample size n from the Beta(a,2), construct the MLE of a I really don't ...
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24 views

Exponential distribution of interval censored data MLE using optim on R error

I have a set of data which is interval censored and I wish to calculate its maximum likelihood error using optim. This is my code: ...
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1answer
70 views

Likelihood ratio test for comparing two exponential distributions

I am trying to use a likelihood ratio test to compare the parameters of two exponential distributions. by this thread Likelihood Ratio for two-sample Exponential distribution I found that I can use ...
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1answer
38 views

Is there any method to quantify parameter estimation uncertainty of method of moments fitting technique?

If I want to fit a distribution (let's say we can be certain about the type) to observations using maximum-likelihood method, I have many options to express the parameter estimation uncertainty due to ...
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28 views

Tuning a Maximum Liklehood estimation

This is a follow up question to this question on Cross Validated to which I unfortunately didn't get any replies. The context stayed the same but the method I want to use is completely different so I ...
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1answer
191 views

Estimating parameters for a binomial

First of all I'd like to precise that I'm not an expert of the subject. Suppose to have two random variables $X$ and $Y$ that are binomial, respectively $X\sim B(n_1,p)$ and $Y\sim B(n_2,p),$ note ...
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29 views

K-means and maximum likelihood!

Is there any relation between k-means and the maximum-likelihood estimate in unsupervised learning? Any references would be appreciates! Thank you!
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Likelihood function for MANOVA

I'm trying to get a handle on MANOVA and Repeated Measures ANOVA. My confusion is around what the likelihood function looks like. Here's how I would formulate the likelihood function Notation: The ...