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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6 views

Interpreting random slope for a dataset with missing data in mixed model

I am struggling to understand the meaning of random effect for the dataset with missing data based on mixed model, I am appreciated if anyone can help. Here is an example. let us say we have 20 ...
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0answers
13 views

Estimating Gamma MLE with left truncated data (using R and maxLik)

I'm trying to find the maximum likelihood estimation of the parameters of a Gamma distributed random variable using maxLik. The following code explain what I did: ...
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8 views

Maximum likelihood and regression [on hold]

Can someone help me understand what we are trying to do when estimating regression parameters with MLE. In the method of maximum likelihood, we pick the parameter values which maximize the ...
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7 views

Observed information matrix in Cox model with constant baseline hazard

I am trying to explore properties of Cox model with (parametric) constant baseline hazard function. So the hazard function for the model is $\lambda(t|Z_i) = \lambda_0(t) ...
3
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0answers
27 views

Is unbiased maximum likelihood estimator always the best unbiased estimator?

I know for regular problems, if we have a best regular unbiased estimator, it must be the maximum likelihood estimator (MLE). But generally, if we have an unbiased MLE, would it also be the best ...
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11 views

What are the extra outputs in this fitdistr function? [on hold]

I have the following code in order to find the scale and rate for the Weibull distribution: ...
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13 views

checking the expectation of the maximum likelihood estimator $\mathbf{\Sigma}$ for the multivariate gaussian

I am trying to find the expectation of the MLE for $\mathbf{\Sigma}$ for the multivariate gaussian. $E(\mathbf{\Sigma}_{ML}) = E\left (\dfrac{1}{N} \sum (\mathbf{x}_n - \mathbf{\mu})(\mathbf{x}_n - ...
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2answers
57 views

Markov chain Monte Carlo (MCMC) for Maximum Likelihood Estimation (MLE)

I am reading a 1991 conference paper by Geyer which is linked below. In it he seems to elude to a method that can use MCMC for MLE parameter estimation This excites me since, I have coded BFGS ...
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8 views

How to write the log-likelihood function of a multi-row and multi-parmeters model in R?

I am a newbie of R and are currently studying how to write the log likelihood function of a multinomial logit model to pass into something like optim or bbmle. As far as I could I am still scatching ...
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1answer
35 views

Exact maximum likelihood estimation of MA(1)

I want to calculate the MLEs of the MA(1) model and for this purpose I have written the exact likelihood for the same. I built a programme in R for the log-likelihood, but it seems some problem in it ...
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26 views

Name of maximum of integrated likelihood?

What do people call the maximum of the integrated likelihood function (i.e. marginal likelihood function)? This is, suppose that $x_i\stackrel{iid}{\sim} f(\vert\theta)$, $\theta=(\alpha,\beta)$, and ...
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22 views

Regression smoothing

I have made a linear model, based on 10000 observations and 10 variables. However when trying to estimate the maximum likelihood, the covariance matrix becomes very large. Hence I thought I would do ...
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30 views

What is the difference between Restricted Maximum Likelihood (REML) and Maximum Likelihood (ML)? [duplicate]

I am a first year graduate student in biostatistics, and I have somewhat of an idea of the difference between REML and ML. However, I want a more in-depth understanding of each estimation method, ...
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20 views

Likelihood ratio test, given distribution

I am trying to find the implementable form for a likelihood ratio test and ran into a problem. Could anyone look at it an give me a hint? I'm stuck. Here's the problem... $X1,...Xn$ are distributed ...
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2answers
50 views

Maximum Likelihood Estimator for equicorrelation model

Consider the equicorrelation model for multivariate normal. Let $X_1, \dots, X_n\sim \mathbf{N_p(\mu, \Sigma)}$, where $\mathbf{\Sigma}=\sigma^2((1-\rho)\mathbf{I_p}+\rho\mathbf{J_p})$ where ...
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0answers
37 views

Definition of asymptotic variance

Upon studying the ML estimator this concept still confuses me. First define an asymptotic covariance matrix for the MLE estimator (just as an example, we have two parameters $\beta$ and $\sigma^2$, ...
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61 views

Asymptotic covariance matrix of the ML estimator

I am trying to get a grasp on the ML estimator and the presentation of the asymptotic covariance matrix is really confusing to me. First, it is stated that the matrix is inverse of the information ...
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14 views

MLE estimate of saturated binomial model with single observation

Consider the single response variable $Y\sim$Bin$(n,p)$. The MLE estimate of $p$ is given by $\hat{p} = \frac{y}{n}$. I want to find the deviance: $$ 2[l(p_{\text{max}}) - l(\hat{p})] $$ where ...
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1answer
34 views

Are MVUEs and MLEs always functions of a minimal sufficient statistic?

Is it the case that both minimum variance unbiased estimators (MVUEs) and maximum likelihood estimators (MLEs) are always functions of a minimal sufficient statistic? If so, how do we know? If not, ...
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1answer
36 views

Confidence interval for MLE estimator?

I have an MLE estimator which is asymptotically normally distributed with mean $\beta$ and variance $\beta^2/n$. How do I get an approximate confidence interval for this estimator? I know usually two ...
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21 views

Maximum Likelihood Ratio Test for Bivariate Normal Distribution

Suppose that $(X_{i}, Y_{i})$ for $i=1,\dots,n$ is a random sample from a bivariate normal distribution with $E(X_{i})=\mu_{1}$, $E(Y_{i})=\mu_{2}$, $Var(X_{i})=Var(Y_{i})=\sigma^{2}$ and $Cov(X_{i}, ...
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1answer
68 views

MLE estimate of $\beta/\sigma$ - Linear regression

I have a question regarding Maximum Likelihood Estimate in linear regression model without intercept. I have a model: $$Y_i=\beta X_i +\epsilon_i, \ \ i=1,...,n$$ where $\epsilon_i$ are i.i.d. $N(0, ...
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1answer
26 views

problem with the relationship between log linear and logistic regression models

I am supposed to fit a logistic regression model and the find the log- linear model which correspond to it, fit that model and show the correspondence between parameters. But it is not working, I am ...
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16 views

Newton-Raphson Maximum Likelihood derivation [duplicate]

I am stuck at the derivation of the Newton-Raphson method for finding the optimal value $\theta$ of a Likelihood function $L(\theta)$. The derivation is as follows: Let $\theta$ be the true parameter ...
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1answer
18 views

MLE classifier of Gaussians

Setup There are two Gaussians $G_1,G_0$ with parameters $(\mu_1,\sigma_1^2)$ and $(\mu_0,\sigma_0^2)$ respectively, and $\mu_1>\mu_0$, $\sigma_1 > \sigma_0$. I am classifying draws $x$ from ...
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40 views

confidence interval uniform distribution $U[1,N]$

You have a lottery ticket and it has the number 320 on it. Show confidence interval for the number of lottery tickets (to $\alpha=0.05$). Assume there are N tickets total numbered 1 to N. I'm trying ...
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1answer
34 views

Log likelihood in EM Algorithm

I try understand the log likelihood in weka. I read about that is a probabilistic metric, but i cant understand, if is better when have low value or high value? How i can get the likelihood value, ...
4
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1answer
39 views

Finding the MLE for a mixture of random variables which are discrete and continuous

I came up with the following situation: I have n i.i.d: $X_i \sim U(0,1)$ and another $Y_i = I_{(X_i < p)}$ (where $0<p<1$). Now obviously $Y_i \sim \mathrm{Bern}(p) $. I want to find the ...
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1answer
53 views

MLE for joint distribution

I have a joint distribution of $(X,Y)$ where $Y$ is Bernoulli with $P(Y=1)=p=1-P(Y=0)$. The conditional distribution of $X$ given $Y=y$ is Normal with mean $\mu_y$ and variance $\sigma^2_y$: that is, ...
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1answer
24 views

MLE for discrete distribution and finite parameter space

One observation $x$ is sampled from a distribution with probability mass function $f(\cdot;\theta)$, where $\theta\in\{1,2,3\}$ and $f$ is given by $$ f(i;j)=a_{ji} $$ where $A=(a_{ji})$ is ...
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1answer
54 views

How are calculations done for REML?

I've read a few questions on this site (e.g., http://stats.stackexchange.com/a/48676/46427), but they rarely go beyond intuitive explanations. I am particularly interested with how to calculate an ...
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8 views

Training Restricted Boltzmann Machines according to the Likelihood Function

Is it possible to chose the parameters of a RBM to maximize the likelihood of the observed data? (I follow the notation of the deeplearning tutorial ). Denote the observable data by $x$, hidden data ...
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3answers
60 views

Why doesn't multiplication by constant affect MLE? [duplicate]

I have an example that derives the MLE for Binomial. Since there's that factorial term $n_i \choose x_i$ in front of the Binomial p.d.f. and it's a constant, the example claims that one can merely ...
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29 views

Maximum likelihood of coin toss of different type?

I was self-studying EM (Expectation Maximization) algorithm, where I came across this example given by the paper. In this paper, there are two types of coins A, B with unknown parameters $θ_A$ and ...
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30 views

Maximum likelihood estimate custom pdf in matlab [migrated]

I have a custom pdf that has 3 parameters (X,n,k), where X represents the data (vector) and n,k are two scalars. I want to calculate the mle for this custom pdf, so I wrote this in matlab: ...
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1answer
34 views

Comparing the distribution fits of a bivariate and a univariate model

Suppose I've done an experiment and I have a distribution of observations $x$ that vary between $-\pi$ and $\pi$. Now suppose each $x$ is associated with a second observation $y$ that may or may not ...
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10 views

The principle of getting the error bar of the MLE of the mean of some univariate Gaussian

I'm reading the book Information Theory, Inference and Learning Algorithms. In Section 22.1, the author gives an example of finding the MLE of the mean of an univariate Gaussian, and then obtaining ...
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15 views

Gaussian QMLE in estimating GARCH model

I am having some troubles understanding the estimation of a CCC-GARCH model (where the univariate GARCH models are GJR-GARCH(1,1)) by the means of Gaussian QMLE with the likelihood function of ...
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25 views

fitting t distribution with lighter tails

I am trying to fit a t distribution in R using the fitdistr function in the MASS package as follows ...
2
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1answer
18 views

Information decomposition in GLM MLE derivation

I am trying to understand the derivation of the MLE estimates of $\beta=(\beta_1,\dots,\beta_p)$ in Generalized Linear Models. The elements of the Information matrix are given by: $$ ...
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113 views

Restricted maximum likelihood with less than full column rank of $X$

This question deals with restricted maximum likelihood (REML) estimation in a particular version of the linear model, namely: $$ Y = X(\alpha)\beta + \epsilon, \epsilon\sim N_n(0, \Sigma(\alpha)), $$ ...
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18 views

Find Maximum Likelihood function for the parameters of a random effects model

Could someone explain me how to get the maximum likelihood function for the parameters of a random effects model? Besides what the assumptions of this model are?
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0answers
22 views

How to estimate the parameters of the following log-likelihood function?

I would like to estimate the parameters based on the famous Merton model used probability of default modelling: Suppose firms' logarithmic returns are following the standard normal distribution and ...
3
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1answer
125 views

How do you fit a Poisson distribution to table data?

I've been given a table of $x=(0,1,2,3,4,5,6)$ and $y=(3062,587,284,103,33,4,2)$, which are such that the number of $x_i$ tells an amount of children that all $y_i$s have. I'm asked to fit a Poisson ...
2
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1answer
35 views

Working out the expectation of a function of iid random variables

I have found the maximum likelihood estimator $\hat{\sigma}$ of a iid r.vs $X_1, ..., X_n$ which all have normal distribution with known mean $\mu$ and unknown variance $\sigma^2$. So $\hat{\sigma}$ ...
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23 views

Problem with initial guess in Newton-Raphson iteration method

I'm working on estimating the four parameters of Exponentiated Modified Weibull Extension Distribution introduced by Sarhan and Apaloo (2013) with the Maximum Likelihood Estimation (MLE). Because the ...
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25 views

Derivation of GARCH student t log-likelihood

I got the general PDF of the student t distribution, that is: $\frac{\Gamma[\frac{(\nu+1)}{2}]}{\Gamma(\frac{\nu}{2})}\,\frac{1}{\sqrt{\pi\,\nu}}\,\bigg[1 + \frac{x^2}{\nu}\bigg]^{-(\nu+1)/2}$ ...
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26 views

MLE for Negative Binomial self study [closed]

Question 3 p(x)=((x_i+r-1)¦x_i )θ^r〖(1-θ)〗^(x_i ) L(x)=∏n_(i=1)((x_i+r-1)¦x_i ) θ^r〖(1-θ)〗^(x_i ) l(x)=∑n_(i=1)〖[log((x_i+r-1)¦x_i ) 〗+rlog(θ)+x_i log⁡(1-θ)] (∂l(x))/∂θ=〖∑n_(i=1)(r/θ〗-xi/(1-θ)) ...
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0answers
16 views

Maximum Likelihood for constrained multivariate normal

Consider a sample of size $n$ from $\text{N}_p(k\mathbf{\mu_0}, \Sigma)$ where $k,\Sigma$ are the unknown parameters, and $\bf \mu_0$ is known. Now we have to find their MLE's. I have solved this when ...
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24 views

MLE for Negative Binomial Distribution [closed]

Let x1,...,xn be iid sample from a Negative Binomial Distribution where r>0 is an integer number, θ∈(0,1) and is the binomial coefficient. Suppose that r is given, find the MLE of θ. hey guys I ...