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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19 views

Restricted Maximum Likelihood (REML) Estimate of Variance Component

Let, $$\mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i,$$ where $\mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf ...
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41 views

Bayesian inference via approximate data likelihood

Suppose that we have a very large i.i.d. sample $x_1,...,x_n$ and a data likelihood defined by $$p(x | \theta,\beta) = \prod_ip(x_i | \theta,\beta)$$. Further suppose that $\theta$ is the parameter ...
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2 views

Generalized Likelihood Ratio of 2 signals with various unknowns

I am attempting to derive the likelihood ratio of the 2 expressions where S and C are known, a and $\sigma$ are unknown. This is the paper (1st link), equation no. 54. I am able to derive (51). If ...
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7 views

Adjust data to some distributions [on hold]

I have some data that looks like this: > str(nidd) 'data.frame': 154 obs. of 1 variable: $ Nidd: num 32.24 124.02 3.84 7.21 12.26 ... What should I do to ...
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2answers
62 views

Student t-distribution parameter/s and MLE

So I always thought of the Student t-distribution as having only 1 parameter, v, the degrees of freedom (as described by wikipedia). When I searched however on how to find the MLE of v I keep coming ...
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11 views

Composite-likelihood ratio test

I am interested in a number of articles (such as Kim and Stephan 2002 and follow-up articles) that use composite likelihood ratio tests to infer selection pressure on linked (phased) genetic data. I ...
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0answers
43 views

What is the exact log-likelihood of an AR(2) model?

Let's say we have the following AR(2) model: $y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+e_t, \; e_t\sim N(0,\sigma^2_e)$ with T observations in total. Working out the conditional log-likelihood is ...
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1answer
31 views

Estimation of the local trend models (State Space) through ML

Tsay, R. S. (2010), Analysis of Financial Time Series, 2nd Edition, discusses on page 504 the estimation of local trend models (state space). The measurement and the transition equations are as ...
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8 views

Goodness of fit for complex valued curves (i.e. frequency responses in frequency domain)

I'm not a hero at statistics, som my apologies for perhaps the stupidity of this question. Presume that one has the frequency response $Y_{data}(k)$ and also has the synthesized response $Y_{syn}(k)$. ...
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22 views
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3 views

What is the effect of using the partial or maximum likelihood upon estimation sample size?

I'm running a series of different models on some longitudinal data with repeated waves. I've noticed that a Cox proportional hazards with time-varying covariates drops entire participants even where ...
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3answers
24 views

fitting distribution with zero values [on hold]

Can I estimate the claim severity distribution with R if the data contains zero claims? or i must remove them from the data?
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13 views

Optimization with a part of gradients

I am exploring some gradient-based optimization functions. For my complex objective function, it is easy to calculate gradients for some parameters, but very hard for some other parameters. I am ...
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42 views

Critical region of likelihood ratio test

This is problem # 5 from RSS's 2014 Graduate Diploma Module 2: $$P(X_j=k) = \begin{cases} (1-p)^3 & k=0\\ 3p(1-p) & k=1\\ p^3 & k=2\\ 0 & \text{otherwise} \end{cases}$$ $Y_k = ...
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0answers
12 views

MLE of dispersion parameter in Beta-Binomial and Beta-Poisson models goes unboundedly to infinity

Background I am interested in estimating the $n$ as well as the $p$ parameter in a Binomial distribution to investigate under-reporting in the data available to me. I found sources accomplishing this ...
2
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1answer
35 views

Estimate transition matrix from many short Markov chains

I have a situation where data from the following process is observed: For $i = 1, \dots, n$ let $(X_{i,1}, \dots, X_{i,m_i})$ be a sequence of $m_i$ random variables coming from a discrete-space ...
3
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1answer
33 views

Measure-Theoretic Definition of MLE

This question really boils down to the following: Under what conditions can we refer to pointwise values of a probability density function? Obviously continuity of the pdf suffices, but because of the ...
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7 views

Differences between ML and kernel Gaussian classification strategies

The Gaussian distribution can be used in "parametric" and "nonparametric" classification for new test points. One way to do parametric classification using a Gaussians is to estimate ...
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0answers
11 views

Why does the likelihood of ML ancestral states change when tree is scaled?

I realize this might not be the best place to have this question answered, but I figured I'd try anyway. My question concerns the calculation of log likelihoods for ancestral state estimates using the ...
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4answers
2k views

Likelihood - Why multiply?

I am studying about maximum likelihood estimation and i read that the likelihood function is the product of the probabilities of each variable. Why is it the product ? Why not add ? I have been trying ...
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0answers
16 views

How to fit a distribution of censored data? [duplicate]

I have a right censored data set and want to fit a distribution to my observations. How does maximum likelihood estimation work on censored data and are there any issues that I need to consider?
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29 views

Maximum likelihood with special constraints in R? [closed]

I want to find maximum likelihood estimation of parameters under special constraints. My density have 3 parameters $(a,b,c)$ with bounds $$-\infty<a<\infty,\; 1<b<\infty,\; ...
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0answers
21 views

Instructive figure to explain Maximum Likelihood confidence bands / standard errors

I am trying to better understand and explain maximum likelihood estimation. To explain the intuition of many ML aspects I find it easiest to explain them graphically, like for example the ML-based ...
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1answer
108 views

Covariance matrix of parameters in logistic regression

Let $$f(y)=\prod_i \binom{n_i}{y_i}\pi(x_i)^{y_i}(1-\pi(x_i))^{n_i-y_i}$$ where $$\pi(x_i)=\frac{e^{\sum_j x_{ij}B_j}}{1+e^{\sum_j x_{ij}B_j}}$$ then the likelihood is $$L\propto ...
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2answers
56 views

Uniform distribution MLE

Just a quick question: I know a $U(0, A)$ with density of $1/A$ has as MLE of $X_{max}$, but would a $U(1,1+A)$ have the same MLE that of $X_{max}$? I'm assuming so but just for clarity. Thanks in ...
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55 views

Markov model parameter concentration and Fisher Information Matrix

For iid data, the posterior on the parameter $$ p(\theta \mid x_{0:T}) = \prod_{t=0}^T p(x_t \mid \theta) p(\theta) $$ is known to become independent of the prior which is the Bernstein-von Mises ...
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1answer
21 views

MLE for Mu and SD assuming Normal Distribution

In my university material I have the following summary question which I believe is broken into two parts, it goes as follows: Define the heights of the male student population as a random variable ...
3
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41 views

Maximum Likelihood of n

Let $X=\{x_1,x_2,...x_m\}$ is a rv with density $f(x)=cx^{2n}$ if $-1<x<1$, $0$ therwise and $n \in \{ 1,2,3,4\}$. Find fe mle of n. I tried to find the Likelihood function : ...
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7 views

Estimating parameter of a distribution for some generate random number [migrated]

I just new in R for solving my statistical problem. Currently I'm working to estimate the parameters of a distribution using 200 random numbers (RN) that I generate using R. I generate 200 RN in 100 ...
2
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1answer
40 views

Is the Jackknife estimation better than Maximum Likelihood Estimator?

I'm trying to estimate distribution parameters with Maximum Likelihood Estimator (MLE) and Jackknife estimator based on it. The estimation statistic is mean. Jackknife estimator is considered to be ...
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0answers
45 views

Maximum Likelihood Estimation of Coefficients for Linear Regression Model

I am reading up about using Maximum Likelihood to estimate the parameters of a linear regression equation. I came across this video (https://youtu.be/_-Gnu498s3o) and I thought it explained it very ...
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4 views

parameter estimation of fractional factorial design

I've been asked to estimate the parameters of fractional factorial design model which is normally estimated using least square method in R (code is lm). I want to know that, is it possible if I change ...
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19 views

Can we calculate the MLE of $\mu$ and $\sigma^2$ of normally distributed data using the profile likelihood approach?

My definition of profile likelihood is that given a vector of parameters $(\theta_1, \theta_2)$, with $\theta_1$ the parameter of interest, and $\theta_2$ a nuisance parameter -- If $L(\theta_1, ...
2
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1answer
75 views

Bayesian derivation of unbiased maximum likelihood estimator

I was recently reading an old NIPS paper by Bishop and Qazaz where they claim that an unbiased estimator for variance, based on $N$ Gaussian $\rm i.i.d.$ samples with unknown mean and unknown ...
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14 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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33 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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11 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) ...
4
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0answers
58 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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14 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 - ...
3
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2answers
81 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 ...
2
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1answer
41 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 ...
3
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0answers
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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24 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 ...
1
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0answers
31 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, ...
2
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0answers
24 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 ...
3
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2answers
57 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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39 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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65 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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0answers
18 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 ...
4
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1answer
36 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, ...