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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17 views

Standard Error of estimate of variance

I am not understanding how is the standard error of estimates of two-factor factorial random effect model calculated ? For example , In the book Design and Analysis of Experiments , by Douglas C. ...
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1answer
13 views

R: MLE estimation of transition probability density (CIR)

I am trying to replicate the model Zhang and Zhu (http://onlinelibrary.wiley.com/doi/10.1002/fut.20209/epdf) used to get the parameters of a stochastic volatility model. Therefore, I determinded the ...
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1answer
29 views

How can I obtain the log-likelihood value in a OLS estimation?

So, in some papers using panel data, I noticed that in the estimate results inherent a pooled OLS regression, they report the value of the log-likelihood. I was wondering how this is possible, in ...
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1answer
30 views

How to prove the identifiability of a likelihood

Consider the likelihood function for parameter vector ...
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6 views

CRF Training: Max-margin vs max-likelihood

I'm trying to use PyStruct's CRF implementation. In its user guide, it says the following: I call these models Conditional Random Fields (CRFs), but this a slight abuse of notation, as PyStruct ...
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2answers
37 views

How can maximum likelihood be used to estimate parameters for a Weibull distribution? [duplicate]

Consider the following survival data (cumulative): Month 0 - 100% Month 1 - 50% Month 2 - 33% Month 3 - 25% Month 4 - 20% (meaning 20% of all initial units have survived by the end of Month 4) ...
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9 views

Instances of this Gaussian variance model with non-convex log-likelihood

On page 4 of Thomas Minka's Beyond Newton's Method, he mentions the following Gaussian variance model with a non-convex maximum likelihood objective. What are some examples/instances of models where ...
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1answer
36 views

Maximum likelihood estimation for Cauchy noise

What is the maximum likelihood estimator of the covariance matrix for a given vector in the presence of Cauchy noise? How can we calculate it given that the Cauchy distribution has infinite variance? ...
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24 views

Conditional versus Unconditional MLE parameters estimation?

I have read about both conditional and unconditional MLE parameters estimation for ARIMA models but I have problem understanding the concept from the statistical books . Does conditional MLE mean that ...
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13 views

why use weighted sum of component densities in unsupervised parametric estimation?

In unsupervised parametric estimation why do we take $p(x|\Theta)$ as $ \sum_j P(w_j)p(x|w_j ,\Theta_j ), j = 1 - c$? That is weighted sum of component densities. where $p(x|\Theta)$ is mixture ...
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27 views
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11 views

convergence assumptions for expectation maximization

in andrew ng lectures notes for expectation maximization, i believe the only assumption invoked for the convergence of the EM algorithm is the jensen inequality that operates on the function Log(x), ...
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11 views

Mean of sample means from samples of different variances

Suppose I have three samples $x_i \sim \mathcal{N}(\mu, \sigma_b^2)$. I cannot measure the $x_i$ directly, so instead I estimate the value of each one by averaging $n_i$ draws from a distribution ...
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28 views

Need for iid in MLE

I am studying about parametric estimation in supervised learning using maximum likelihood estimation. Here is what I learned: Separate our training data according to class; i.e., we have c data sets ...
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10 views

Given an transition matrix what is the likelihood an observed markov chain was derived from this matrix

To give a bit of background, I'm creating a MLE of a transition matrix from a set of empirical data. I'm then creating a simulation of the system that also produces a markov chain. I am looking for a ...
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12 views

Estimate betabinomial distribution parameters from weighted observations

Suppose we observe $N$ independent Bernoulli sequences of different lengths. Let $k_i$ be the number of trials in each observation, and $o_i$ be the number of "successes": ...
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1answer
39 views

Variance of the $\hat{\sigma^2}$ of a Maximum Likelihood estimator

Given some normally distributed observations $x_1,x_2,...,x_n$ $\forall i\ x_i\sim\mathcal{N}(\mu, \sigma^2)$ the ML estimator decides that the variance that maximizes the likelihood function is ...
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1answer
27 views

Maximum Likelihood Estimation for Conditional Random Field parameters

I have a custom potential function for a Conditional Random Field (CRF) very similar to Fei Fei Li's work. In this work, the parameter learning is done by Maximum Likelihood Estimation. I would like ...
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3answers
60 views

Estimates of the variance of the variance component of a mixed effects model

Say $y=X\beta+ Zu +\epsilon$ is our mixed effects model where $u=(u_1,..,u_r)$ and $u_{j} \stackrel{i.i.d.}{\sim} N(0, \sigma^2_{a})$ for $j=1,...,r$ and $\epsilon=(\epsilon_1,...,\epsilon_n)$ are ...
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50 views

Distribution of sum of order statistics

The question is from a problem I am trying to solve in Robert Hogg's introduction to Mathematical Statistics 6th version problem 7.2.9 in page 380. The problem is: We consider a random sample ...
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1answer
36 views

Citation for ML vs. REML

Quick question: can anyone give me a citation that I can use to justify using ML when doing model comparisons? Background: I am fitting some multilevel models in R using lme4, and I do a series of ...
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7 views

R examples and code for estimating temporal discounting functions with maximum likelihood?

Does anyone know of any examples that show and teach how to estimate temporal discounting functions using maximum likelihood in R? I really appreciate any help/suggestions anyone can provide. Thanks!
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1answer
50 views

A question about a Theorem about MLE

I have a question about a proof (about MLE) but let me firstly give you appropriate context. An estimator T is called maximum likelihood estimator of $\theta$, if: ...
3
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1answer
74 views

Find the MLE of the proportion of employees falling in $[I_1,I_2]$

Suppose the incomes of the employees in a firm follow a Pareto distribution as follows:$$f(x)=\dfrac{cA^c}{x^{c+1}}$$ where $x\geq A>0$. Suppose you take a random sample of the incomes ...
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1answer
26 views

MAP and MLE estimation

i stumbled upon the following formula that describes making predictions based on the MAP estimate. i understand that we "plug in" in the MAP estimate in the predictive posterior, but i do not ...
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12 views

ARIMA Fitting: CSS vs CSS log likelihood

Can someone explain to me: in estimating the parameters for an ARIMA model, what is the advantage of optimizing the CSS log likelihood(pg 2), versus optimizing CSS (pg 3)? Are there any ...
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2answers
97 views

Modelling parameters in maximum likelihood

Often times I'll write down an idea and have no idea if (1) It's a good idea, and (2) If it is a good idea, how much has it been applied / studied. I'm asking this question in hopes to get insight on ...
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94 views

Maximizing log-likelihood in R

I am trying to perform ML estimation for a continuous distribution given by \begin{align} F(x;\Phi)=\left(1-e^{-\eta^{k}(x)}\right)\left(1+\lambda e^{-\eta^{k}(x)}\right), \quad x > ...
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46 views

4 cases of Maximum Likelihood Estimation of Gaussian distribution parameters

Let $x_1,x_2,...,x_n$ some normally distributed observations. So $\vec{x}=\begin{bmatrix}x_1 & x_2 & ... & x_n\end{bmatrix}^{T}$ In the context of my research I am trying to estimate ...
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1answer
21 views

How to estimate which distribution a new observation belongs to, when the distributions are given by a set of obervations

The question is that I have several probability distributions. For each distribution $P_i$, I don't know what it is. Instead, I have a set of observations $\{x_i^j\}_{j=1}^{n_i}$ drawn from each of ...
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2answers
47 views

Maximum likelihood estimation for Gaussian mixture

When doing maximum likelihood (ML) inference on a Gaussian mixture model (GMM), Bishop notes in PRML that if there is more than one mixture component in the GMM and the mean of one Gaussian collapses ...
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1answer
33 views

derivative of loss function

If we have a paralyzed loss function of the form of: \begin{align} L(\beta)& =\frac{1}{2}(y-X\beta)^T(y-X\beta)+ \lambda \beta^T f(\beta) \end{align} where $X_{n\times m}$ and $\beta_{m \times ...
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1answer
31 views

Is there a method to form a CI based on an MLE which is not the mean

I know how to find confidence intervals for parameters based on sample means. I want to know whether there exists a method for finding confidence intervals based on MLE's which are not sample means. ...
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1answer
58 views

Why should one use EM vs. say, Gradient Descent with MLE?

Mathematically, it's often seen that expressions and algorithms for Expectation Maximization (EM) are often simpler for mixed models, yet it seems that almost everything (if not everything) that can ...
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20 views

MLE of CTMC parameters

Let the data set be $$D = \{(s_0, t_0), (s_1, t_1), ..., (s_{N-1}, t_{N-1})\}$$ where $N=|D|$. Each $s_i$ is a state from the state space $S$ and during the time $[t_i,t_{i+1}]$ the chain is in state ...
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42 views

How to use variation in MLEs to construct a new PDF?

My simulation has 8 uncertainties and 1 output quantity of interest: Satellite mass. Each uncertainty is characterized by a Gaussian distribution. For a large number $N$; I feed these $8 \times N$ ...
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1answer
58 views

A maxdet problem with three variables - maximum likelihood estimation in Mathlab/Mata/R

Problem Given $n$ known vectors $y_j = (y_1, y_2, ..., y_{K_j})', \forall j=1,n$ and a constant $K$; determine three vectors ($\alpha, \beta, \theta^2)$ that maximise: $$ \text{T} (\alpha, \beta, ...
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2answers
259 views

Finding the maximum point of probability density function

This might be a stupid question,but i'm curious about why we always find mle using the first (partial) derivative without checking the end points or singular point or the second (partial) derivative? ...
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12 views

Find a likelihood function from varying data for Bayes Theorem

I'm not sure how to model a likelihood function for the following problem: Assume I've got a sensor producing the following raw values (normalized to an interval $\pm$1): In reality there are more ...
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54 views

Histogram binning approach to find the pdf and log-likelihood expression?

There is a paper : Presles, B., Debayle, J., Cameirao, A., Fevotte, G. and Pinoli, J.~C., Volume estimation of 3D particles with known convex shapes from its projected areas, IEEE International ...
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18 views

The score of a dynamic model is a martingale difference sequence

I am going to write down some parts of Dynamic models for volatility and heavy tails by Andrew Harvey (2008) with my comments in bold and then ask for an alternative explanation of the final part. ...
2
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1answer
48 views

A simple question about MAP and MLE

I recently got this simple question from a friend. But I am quite confused about it. Suppose we toss a coin $N$ times, and got heads $m$ times. Assume the binomial distribution with $p$ which is the ...
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17 views

Relation between bootstrap mean and parameter value estimated via maximum likelihood

I have an observed data set $O$ and a synthetic model $S(\theta)$ which attempts to describe it. By fixing $\theta$ to different values I can generate $M$ synthetic realizations of the model: $$S_k ...
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1answer
50 views

Finding the Maximum Likelihood Estimate in R [closed]

How do I estimate the parameters through MLE for a Likelihood function like. Note that the function has 2 multiply signs and there is a integration term in the second one.
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1answer
40 views

Likelihood ratio for normal distribution with known variance

Let $X_1,...,X_n$ random sample of $X$~$N(\mu,\sigma^2)$ with known $\sigma^2$.Take $a=.05$ find the expression for power function of the likelihood-ratio test $$H_0:\mu\leq 0\space vs\space ...
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1answer
51 views

Likelihood-ratio test

I was studying on this subject and I got some questions. Lets take the test $$H_0:\theta\in\Theta_0 \space vs\space H_1:\theta\in\Theta_0^c$$ where $\Theta$ is the parametric space and ...
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10 views

multiple imputation - likelihood base

in nonignorable mechanism, selection model or pattern-mixture model is a multiple imputation method or a likelihood-base method? I am confused, i know in MI, missing data were filled in and then We ...
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1answer
62 views

Do IID experiments maximize Fisher information?

Let $x$ and $y$ be random variables whose probabilities depend on an unknown parameter $\theta$. I am specifically interested in the case that both $x$ and $y$ are Bernoulli, but the question below ...
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1answer
44 views

Maximum Likelihood, normal distribution

Let $Z$~$N(\mu,\sigma^2+1)$, find the maximum likelihood estimator for $\mu$ and $\sigma^2$. I did but I want to check that this right actually ...
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1answer
79 views

Modelling time varying volatility when GARCH(1,1) coefficients sum to value greater one

First of all I have to admit that I am not a time-series expert so any help is highly appreciated. I have a financial time-series (a fixed income total return index measured on a weekly basis) for ...