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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7 views

What r parameter is used in a negative binomial regression?

From my understanding of the negative binomial regression, we have $Y_i|X_i; \theta$ is distributed $Neg Binom (r_i, p_i)$, where $r_i$ is known and fixed (analogous to a fixed $\sigma^2$ when we ...
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49 views

Finding MLE by R code [on hold]

I would like to find the maximum likelihood estimation (MLE) of the parameters of following distribution and desnity function : ...
2
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1answer
32 views

Removing undefined (NaN) values during log-likelihood maximization

I am trying to maximize a Poisson likelihood function for an array of values, of the form llhij = -mij + kij*ln(mij/kij). (where I use the numerical approximation ln(k!) ~ k*ln(k)). For my ...
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20 views

Specifying the initial values of likfit function of geodata in R

I have a geostatistical data. For analysis, I am using geoR. I want to estimate the parameters. But likfit or ...
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0answers
19 views

Consistency and asymptotic normality of two-dimensional parameter

In a textbook exercise, for a sequence of iid variables, I have calculated the score function to be $$\begin{bmatrix} - \frac{n}{2\lambda} + \sum_{i=1}^n \frac{( x_i - \mu)^2}{2\mu^2 x_i}& \sum_{i=...
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14 views

Expectation of infinite sum of random wishart matrices

I have problem figuring out how to find the expectation for an infinite sum of random matrices. More explicitly, my problem is: Let $\mathbf{S}_i$ be the maximum likelihood estimator of the sample ...
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0answers
10 views

Maximum likelihood of gaussian with right censoring

I'm trying to fit the mean $\mu$ of right-censored gaussian data ($n$ samples) in a toy example (let's assume $\sigma^2=1$ is known), and the censoring happens always at the same value $s$. As far as ...
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37 views

Find Bayesian estimator with random sample from poisson and prior distribution from exponential distribution

Let $X_{1}$ .....$X_n$ be a random sample from Poisson distribution, prior distribution with parameter $\pi(\lambda) = \beta e^{-\beta\lambda}$. Find Bayesian estimator I couldn't take integration ...
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1answer
56 views

Properties of conditional maximum likelihood estimators

I am trying to find a source that describes the properties of conditional likelihood estimates like those obtained from conditional logistic regression?
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24 views

mle for the binomial distributed data

For example i have folowing dataset of number of boys in families that have 5 kids: 0 boy - 34(number of such families) 1 boy - 128 families 2 boys - 233 families 3 boys - 267 families 4 boys - 144 ...
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27 views

Maximum likelihood estimation for non-stationary time series

I want to estimate how the taxes influence the retail price of alcoholic beverages. The price function is tricky because in EU countries there is excise duty and also VAT. The non-linearity (which is ...
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1answer
19 views

What is the mode of Dirichlet-Multinomial (Polya) distribution?

What is the ML estimate of the parameter $e_i$ for the Dirichlet-Multinomial (Polya) distribution defined below? $p(\mathbf{x}|\mathbf{e}) = \frac{N!}{\prod_i^d x_i!}\frac{\Gamma(A)}{\Gamma(N+A)}\...
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2answers
33 views

Minimising MSE of $\sigma^2$ estimator of specific form

I have found a past exam question for a statistics course and can't seem to find the required result. Part A is fine but my working for part B must be incorrect [see below]. Can anyone figure out ...
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0answers
25 views

Model comparison via AIC or BIC for different likelihood maximization procedures

Maximum likelihood estimation of different models (which all model the same variable and assume the same likelihood function) is done by a different method for each model. Simple numerical ...
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10 views

MLE estimators given score and information

I want to find the MLE estimators of $\lambda>0$ and $\mu > 0$ in a distribution with score vector $$\left[ \frac{-n}{2\lambda} + \sum_{i=1}^n \frac{ (x_i - \mu)^2}{2\mu^2 x_i} \ \ \ \ \ \ \...
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7 views

How to test if the difference between two logit-function-fits is significant or not?

I am considering a binomial experiment where each trial can only have two possible outputs (0,1). I vary the value of a predictor (experiment parameter) and repeat the experiment several times for ...
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21 views

Uniform Distribution on (a 0) MLE

If we have X~U(a,0) where a<0 what is maximum likelihood estimation of a ? I tried to find on internet but I could not find any resource about (a,0) interval, there is resources only for (0,a). Is ...
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1answer
58 views

Using MSE instead of log-loss in logistic regression

Suppose we replace the loss function of the logistic regression (which is normally log-likelihood) with the MSE. That is, still have log odds ratio be a linear function of the parameters, but minimize ...
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1answer
40 views

Maximum Likelihood Estimations examples

I am trying to find the MLE of the following functions but I'm getting stuck. I know the method and steps to follow but Pi notation is confusing for me. 1) f(x) = øx^(ø-1), 0 < x < 1 and 0 < ...
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17 views

Maximum Likelihood Estimates using Negative Log-Likelihood

My query is for an assignment where the first question is to determine if the rate of failure on mechanical equipment is different after a change in manufacturer. In order to answer this I have done a ...
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0answers
14 views

Maximum Likelihood Estimation on Zero-Inflated data using Constraint

I have written a function that evaluates the log-likelihood of Zero-inflated Beta Binomial data: ...
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1answer
15 views

Fitting and comparing distributions based on diverse summary statistics

I have a bunch of samples, about 35, drawn from a fat-tailed distribution. I think it is reasonable to assume that the samples are all drawn from distributions from the same distributional family, ...
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40 views

Maximum Likelihood estimation and the Kalman filter

I know the Kalman filter recursions and can derive these but what I don't really get is how to estimate the hyper parameters using maximum likelihood. I understand that when running the Kalman filter ...
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2answers
151 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 G\...
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46 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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2answers
66 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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18 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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51 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
38 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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10 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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25 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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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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44 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 = \sum_{...
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19 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
39 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
35 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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8 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 class-...
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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 ...
12
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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
17 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,\; 0<c<\...
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22 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 ...
2
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1answer
112 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 \prod_{i}\pi(x_i)^{...
2
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2answers
59 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 ...
2
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0answers
57 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 ...