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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Comparing performances of distributions

I have a Probability Distribution Function (PDF) of a new distribution given below, \begin{equation} g(x)=\frac{1}{\Gamma \left( \alpha ...
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8 views

Maximum Likelihood Algorithm Implementation

Hello Guys i have to implement Algorithm of Maximum Likelihood using Hyper-spectral data of 170 bands, have to classify 16 classes using ground truth data. The formula for implementation is as follow, ...
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17 views

Multinomial distribution pdf and fisher information matrix

The question is from Robert Hogg etal's Introduction to Mathematical Statistics 6th Edition Page 347 Example 6.4.5. It says: Consider a random trial which can result in one, and only one, of k ...
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22 views

Singular Hessian/Observed information Matrix at optimal solution

I am trying to estimate the standard errors of an maximum likelihood estimate (multidimensional) in R'sfunction optim. I want to ...
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9 views

Is a skill score based on mean squared error appropriate for evaluating regression models based on maximum likelihood estimation?

I would like to evaluate a generalized linear model assuming a gamma distribution of the target variable and with one to several predictors by means of cross-validation. With the observational series ...
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35 views

How can we get “estimated standard deviation” from MLE?

I'm looking for MLE as in the following picture. I have some questions: How can we get "the estimated standard deviation"? Is "the estimated standard deviation" included in the output of the ...
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82 views

Deriving likelihood function for IV-probit

So I have a binary model where $y_1^*$ is the latent unobserved variable and $y_1 \in \{0,1\}$ the observed. $y_2$ determines $y_1$ and $z_2$ is thus my instrument. So in short the model is. ...
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1answer
52 views

Testing a power-law hypothesis from the averaged distribution

A way to test the existence of power-law in the distribution is given in the following paper: http://arxiv.org/abs/0706.1062 The gist of the whole procedure is as follows: Let $\textbf{X}$ be the ...
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10 views

how to write these two inequality? (Asymptotic properties of m estimator)

$T_n^*:=sup\{t|\sum ψ(x_i;t)\gt 0\}$ $T_n^{**}:=inf\{t|\sum ψ(x_i;t)\lt 0\}$ As it's seen in the above figure, $-\infty \lt T_n^{*} \le T_n^{**} \lt +\infty$ Then, how to show that the followings ...
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21 views

Is maximum likelihood a form of data substitution? Or not?

I’m using maximum likelihood with missing data. In this case of missing data, is maximum likelihood a form of data substitution? I’m significantly more familiar with multiple imputation which I ...
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19 views

Papers suggesting ML and Bootstrapping can be used together?

Does anybody have any good research papers for me to read on whether using bootstrapping and maximum likelihood estimation together is a good idea, particularly when ML is being used with a relatively ...
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128 views
+50

Beginner level : Unable to understand an application of Maximum Likelihood estimation from a paper

The paper "Distance Learning for Similarity Estimation" download link talks about finding distance measure for similarity estimation based on statistical analysis of distribution models and distance ...
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33 views

Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2) $. My aim is to calculate the ...
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1answer
48 views

Can we express logistic loss minimization as a maximum likelihood problem?

I have a simple question about the equivalence of loss minimization and likelihood maximization for logistic regression. Say are given some data $(x_i,y_i) \text{ for } ~i = 1,\ldots,N$ where $x_i ...
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24 views

explain the interpretation of the following proposition and how to prove it?

Please explain the interpretation of the following proposition and how to prove it? Proposition: Assume that $\exists t_0 $ s.t. $\lambda(t)\gt 0 $ for $t \lt t_0$ and s.t. $\lambda(t)\lt 0 $ ...
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1answer
72 views

Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used

I am trying to understand a proof of quite a long theorem that I report completely for the sake of completeness. This is From Jensen and Rahbek Asymptotic Inference for Nonstationary GARCH (2004). My ...
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1answer
33 views

Censored Binomial model - log likelihood

I have a dataset with multiple samples of batches of observations (e.g. one batch of 20 obs., one batch of 50 obs,, etc.). There is a probability that the batches have contaminated observations, with ...
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51 views

Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = ...
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54 views

Confidence interval of the mean for a beta distribution when alpha and beta are estimated [duplicate]

I have elementary knowledge in statistics. I'm trying to estimate the confidence interval for mean of a beta distribution as specified in this article using log likelihood estimation given alpha, beta ...
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2answers
57 views

Bernoulli maximum likelihood

Suppose $X_1, X_2, \dots, X_n$ are iid Bernoulli(p) random variables. How do you find the restricted maximum likelihood for p where $0<p<0.5$? My work so far: Write out the likelihood: ...
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21 views

Understanding variance estimation by restricted maximum likelihood (REML)

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
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33 views

Estimation with ML or Bayesian

A marketing department is supposed to find the market share of their product. To answer this question, a survey among 720 representative people is conducted, 696 of which complete the poll with ...
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36 views

How to find maximum likelihood of multiple exponential distributions with different parameter values

Let's say that I have a bunch of independent samples, $X_1, X_2, \dots, X_n$ and that they all follow Exponential($\theta_i$) distributions. (So they all have pdf $f(x_i)=\theta_i\exp(-\theta_iy_i)$.) ...
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1answer
29 views

Defining grad in R's optim for MLE

I have a ML I want to maximize in R's function optim. I am currently using the method BFGS. The optim procedure is quite slow however, and I was hoping to speed up the process by specifying the ...
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23 views

Maximum likelihood estimate of two random samples from poisson distribution with means $\lambda\alpha$ and $\lambda\alpha^2$

I would like to solve what seems to be a simply question on maximum likelihood which I believe can be solved by differentiation of the likelihood function. $X_1,X_2,X_3,...X_{n1}$ are independent ...
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21 views

log multivariate normal differentiation with VAR process

I am trying to estimate a regime switching model with an autoregressive component using the EM algorithm. The process itself can be presented this way: $$ r_{t}= A_{n \times (n+1)} \boldsymbol ...
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53 views

Maximum likelihood estimation beta-binomial distribution with R [closed]

I have seen that there is a possibility to estimate the parameters of a beta binomial distribution with the ML method by the use of the function mle2 . But I'm having some problems maybe because of ...
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1answer
34 views

Restricted Maximum Likelihood

Why don't we use restricted maximum likelihood to estimate parameters in non-mixed models?
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1answer
40 views

Efficiency becomes 0 when sample size becomes big?

I am trying to solve Robert Hogg's mathematical anaysis 6th exercise 6.2.11. The problem says. Let $\bar{X}$ be the mean of a random sample of size n from a $N(\theta,\sigma^2)$ distribution, ...
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1answer
25 views

Log Likelihoods of Exponential Families

How can one derive the log-likelihood of the saturated model of an exponential family in general? Differentiating the log likelihood w.r.t $\theta$ gives $y_i=\hat{\mu_i}$ but I don't think replacing ...
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1answer
36 views

Prove the loglikelihood is strictly concave for ABO allele frequency blood type data

I am working through the problems in Kenn Lange's book Numerical Analysis for Statisticians. I am going to try and do all of the problems in the book, though none of them are specifically assigned for ...
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2answers
29 views

Maximum likelihood estimation of p in a Binomial sample

Assuming I need to find the ML estimator for p, p being the chance of success in a Binomial experiment $Bin(N,p)$, I would expect my density function to be: $$ f(y) = {{N}\choose{y}} p^y(1-p)^{N-y} ...
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28 views

MLE of Poisson Regression in R

Given the following Poisson Regression Model: $$ Y_1,...Y_n \sim^{indep} Poi (\lambda_i)\\ log(\lambda_i)=\beta_0+\beta_1 x_i + \beta_2 x_i^2\\ \text{for }i = 1,...n $$ I am trying to find the log ...
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25 views

Which Hessian to use to compute standard errors

Let that I have a data vector $\textbf{x} = (x_1,x_2,x_3....x_n)$ Say these are realizations of IID random variables having a common density $f_\theta$ Likelihood computed using $i^th$ observation ...
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70 views

Delta Method to Compute Standard Errors of Transformed Variables

I am estimating a a finite mixture model to identify proportions of four behavioral types using an experimental dataset. This dataset has data for 500 individuals and each individual has 30 tasks. In ...
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1answer
95 views

Estimation with MLE and returning the score/gradient (QMLE)

I am estimating a simple AR(1) process by the ML approach. I also wish to compute the Quasi MLE standard errors, which is given by the sandwich form of the Hessian and the Score (see for example the ...
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32 views

Quasi-Newton Accelerator (QN1) for EM Algorithm

I am trying to implement what is called a "very simple to implement" accelerator for the EM algorithm. Specifically I am talking about the QN1 algorithm, described here, and am using a multivariate ...
2
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1answer
80 views

Assumptions and terminology for dynamic regression with endogenous offset ($y_t=y_{t-1}+\beta X_{t-1}+\epsilon_t$)

I'm dealing with a fairly simple time series regression model with the following basic form: $y_t=y_{t-1}+\beta X_{t-1}+\epsilon_t$ I'm assuming that observations of $y$ are known without error. $X$ ...
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3answers
66 views

How to use an optimization solver to get t-stats and p-values for the estimates?

I calculate a data log likelihood (evaluated at a set of parameters to be estimated), and my task is to find the set of parameters that maximize my log likelihood. My problem is: thought there are a ...
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10 views

choice of maximum likelihood over expectation maximisation

Given a probability distribution two common statistical measures are the expectation value and the maximum likelihood (equivalent to mean and mode?). My question is, given a probability distribution ...
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1answer
100 views

Why do you have to use MLE instead of OLS in time series data?

I know it has something to do with the errors being correlated with the variable, but I'm not sure exactly what that means. Could someone please give me a quick simple explanation about why you must ...
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73 views

Sandwich Estimator in Maximum Likelihood Estimation of Logit

I am estimating a discrete choice model using mixed logit using Halton Draws. So everything is basically numerically done. The code is written in MATLAB. I am using MATLAB's fminunc to do ...
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20 views

Full Information Maximum Likelihood, Imputation and Classification

I need to do a classification of a dataset, I have some missing data and I would like to try some "missing data techniques" to achieve the best accuracy. I already tried multiple imputation and ...
2
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0answers
29 views

EM algorithm: With prior vs. not prior

I have a working EM algorithm without prior. I am asking for some advice on how to add prior on latent variables. Define: $t_i \in \{ +1, -1 \} $: variables of interest to be predicted $p_j \in ...
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2answers
49 views

Estimate Number of People

In stats we just finished learning about the theory behind MLEs, so I presume this question has something to do with them ... but there is so little information I have no idea where to begin. You ...
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Does the order of variables in a Markov Regime Switching model matter?

since Ive received feedback that my previous question was not well-recieved Ill just have to give it another shot. I am estimating Markov Regime Switching Models, and I am getting different results ...
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1answer
62 views

Stuck on a maximum likelihood estimation

I am given a random variable with the following density function : $e^{\beta-y}$. I need to find the ML estimator of $\beta$. To find ML estimators, I usually apply the following steps (as explained ...
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46 views

Estimator of The Mean of the Ratio of Uniformly Distributed Variables

Given two random variables, $ X \sim U \left[ {\mu}_{x} - \frac{{l}_{x}}{2} > 0, {\mu}_{x} + \frac{{l}_{x}}{2} \right] $ and $ Y \sim U \left[ {\mu}_{y} - \frac{{l}_{y}}{2} > 0, {\mu}_{y} + ...
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1answer
115 views

What will be the estimator for these parameters

Question: $y_0 = z^d$ is computed from the sum of some recordings by a sensor. Let, there be $k$ sensor nodes. This parameter is calculated by each sensor node and then transmitted to the base ...
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2answers
254 views

Can we use MLE to estimate Neural Network weights?

I just started to study about stats and models stuff. Currently, my understanding is that we use MLE to estimate the best parameter(s) for a model. However, when I try to understand how the neural ...