Questions tagged [log-likelihood]

For questions that use the natural logarithm of a likelihood function.

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38 views

Find MLE for $\Phi, f(x;\Phi)=\frac{2x}{1-\Phi^{-1}}$

The probability distribution is given as $$f(x;\Phi) = \begin{cases} \frac{2x}{1-\Phi^{-1}}& \Phi <x <1\\ 0& \text{otherwise} \end{cases}$$ How do I go about ...
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40 views

Likelihood ratio tests and pseudo R2 for four-parameter logistic regression model (dose response)

I am using a four-parameter log logistic function to fit curves to dose response data. The underlying equation I use is the following: $$f(x)= c+ \frac{d-c}{1+ ((log(x)-log(e))^b}$$ Where b is the ...
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1answer
28 views

Finding MLE estimator for given density $f(x, \alpha, \beta)$

I'm having trouble with the following example problem of MLE: Let $X = (X_1, ..., X_n)$ be a trial from i.i.d r.v. with density: $$ g(x) = \frac{\alpha}{x^2}\mathbb{1}_{[\beta, \infty)}(x) $$ where $\...
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23 views

Why can we treat Cox's partial likelihood as a full likelihood?

I am doing some self study on Cox regression, and am trying to figure out how we can derive the partial likelihood for the Cox model from the full likelihood. Generally, I know that to get a partial ...
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1answer
54 views

LR Test in Beta($\theta$, 1) with $H_0 = {\theta_0}$

I'm trying to obtain $\alpha$-level LR Test where $(X_1, ... X_n)$ are from Beta($\theta$, 1) with $H_0 = {\theta_0}$ and $H_0 \neq \theta_0$. I'm looking for $$ \lambda(X) = \frac{\sup_{\theta \in \...
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1answer
52 views

Sufficient statistic by factorization theorem

Suppose we have a random sample $X_1,\dots,X_n$ of $X$, where $X$ has the following pdf: $$f_{\mu,\sigma}(x)=\left(\pi\cdot\sqrt{(x-\mu)(\mu+\sigma-x)}\right)^{-1}$$ where $x\in(\mu,\mu+\sigma),\mu\...
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24 views

Finding Marginalize the product of p(x|z,μ) and p(z|π)

Consider now n i.i.d. observations of the vector data $({x_1,...,x_n}).$ Using the pdf, we can write the log-likelihood expression: $$l(\boldsymbol x)=\sum_{i=1}^nln(\sum_{k=1}^K\pi_kp(x_i|\mu_k))$$ ...
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1answer
42 views

When and why do formulae involving sums over $x_i$ change to formulae involving $X$ in statistics? Specifically when dealing with likelihoods.

I've been reading up on stats recently and a question I'm working through involves calculating the log-likelihood of a distribution w.r.t a parameter $\beta$. From my understanding, for some ...
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33 views

Simplifying Log Likelihood equation

I am reading through a paper (https://www.mitpressjournals.org/doi/pdf/10.1162/0891201053630273) where they describe logloss as a ranking function and can be simplified to the margin of the training ...
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20 views

Well-behaved log-likelihood of von Mises-Fisher distribution

I am trying to write a numerically stable version of the log-likelihood of the von Mises-Fisher distribution, specifically for the 2-sphere in $\mathbb{R}^3$. The PDF can be written as $$ f(\mathbf{...
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29 views

Negative Log-Likelihood Loss with Gibbs distribution for beta approaching infinity

TL;DR: What happens with Gibb's distribution when $\beta \to \infty $ and why? $$ \lim_{\beta \to \infty} \frac{\exp(-\beta E(W, Y^i, X^i))}{\int_y \exp(-\beta E(W, y, X^i)) } \ = \ ? $$ Full ...
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28 views

Fisher information and score

We know that the Fisher information for a probability density function $f_{{\theta}}(\textbf{X})$ with a given true value of the parameter ${\theta}$ and an observable random variable $\textbf{X}$ is ...
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34 views

How to compute the stably adjusted profile likelihood given in Barndorff-Nielsen and Cox (1994)?

I'm interested in computing a modified profile likelihood and came across the stably adjusted profile likelihood proposed by Barndorff-Nielsen and Cox in their book 'Inference and Asymptotics' which ...
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14 views

Can anyone explain how does the likelihood function of different model came?

In every textbook, they used to write a likelihood function without explaining it. I am thoroughly confused, so much so that I am cramming different likelihood functions. For eg., while calculating ...
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1answer
52 views

Find asymptotic distribution of $l(\hat{\theta_n}) - l(\theta_0)$

Where ${\hat{\theta_n}}$ are the asymptotic efficient roots for $\frac{\partial}{\partial \theta} l(\theta,\underline{X}) = 0$ I have really no clue how to begin this, but I was given Hint: a good ...
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14 views

Can the Fisher information matrix at the true underlying value be expressed with the MLE?

I found the following information that could answer my question but I don't see how the last line makes sense when the Fisher score vanishes at the MLE? Fisher introduced the concept of information ...
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1answer
30 views

Maximum Likelihood Estimation of a bivariat uniform distribution

I have to find the size and position of a window given four light points. The light points are i.i.d. from a uniform distribution with the parameter $\theta := (x_{min},x_{max},y_{min},y_{max})$. Each ...
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22 views

Understanding how to check a certain condition for composite null hypotheses

A generalization of the Wilks Theorem for a simple hypothesis requires certain conditions on the log-likelihood ratio statistic $W(\theta) = l(\hat{\theta}) - l(\theta)$ with $\theta$ a vector of ...
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28 views

Express the Fisher information matrix at the true underlying parameter with the vector of unknown parameters $\theta$ or with $\theta^*$

Suppose we would like to analyze an expression like $$l(\theta) - \theta^T\Sigma\theta$$ where $l$ is the log-likelihood function, $\theta$ is an unknown vector of parameters and $\Sigma$ is the ...
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18 views

PMF of an awkward combination of Poisson processes

I have two independent Poisson processes A and B, and have a model I wish to fit to the combination A+2B using a maximum likelihood method. I understand that this combination is no longer a Poisson ...
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1answer
32 views

Monotonic transformation preserves extrema

I have heard that given a function $f$ which has certain relative extrema, if $g$ is monotonic, then $gf$, $g$ composed with $f$, will have the same relative extrema as $f$. Is this true, and if so, ...
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47 views

mathematical statistics- find likelihood

We let $X_1, . . . , X_n$ be independent identical normal distributed stochastic variables hvor $X_i$ ∼ N(θ, $θ^2$) with an unknown θ > 0. You can for example estimate θ using the mean: $θ_n$ = X = S/...
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1answer
52 views

What is the log-likelihood function and MLE in uniform distribution $U[\theta,5]$?

For uniform distribution $U[\theta,5]$ with sample size $n$, Likelihood function is: $$L(y;\theta) = (5-\theta)^{(-n)}$$ Log-likelihood function is: $$\log(L(y;\theta)) = -n.\log(5-\theta)$$ For MLE ...
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1answer
19 views

MLE positive and negative root how to choose?

After I solve a quadratic equation which is the derivative of log-likelihood function to find MLE, I get one positive root and a negative root. Which one do I choose? Sample space is between $$0<y&...
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15 views

Manually Calculating Log-Likelihood for MLM Models with Different Distributions (in order to compare AIC)

We are running different models - some have ordered factors as the response variable and others have continuous (duration of use) or count data (frequency of use) as the response variable. I am ...
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42 views

Relation with Hessian and Log-likelihood

I make following the post demo about the Hessian of Log-Likelihood since I have not managed to demonstrate the equation $(1)$ below in the general form of the Log-likelihood : $$E\Big[\frac{\partial \...
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2answers
26 views

Likelihood formulation

I try to understand the relation of likelihood to cross-entropy by reading cross-entropy. The problem is I cannot understand the formula for the likelihood in the article. The likelihood is defined ...
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13 views

Proving that the Normalised Log Likelihood is not complete

Setting: N iid normalised random variables ~ N(μ, 1). The normalised log -likelihood is a function-valued statistic. It is: $S(X)= \cfrac{N \mu^2}{2}+ \mu \sum_{i=1}^{N}{Xi} $. It is easy to prove ...
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27 views

likelihood of a line on a plane

Assume we have an observation(one point) of a line segment on a plane and the distribution of the observation is a 2D Gaussian distribution. The line in the figure is a possible line segment according ...
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1answer
34 views

Differentiation of a log likelihood function

I am trying to maximize a particular log likelihood function but I am stuck on the differentiation step. Given: $ \Theta_1 + ....... + \Theta_k = 1 $ The likelihood function is: $f_n(x|\Theta_1,......
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1answer
25 views

Show that the likelihood ratio test statistic is 34.7. [closed]

A question from my class: Ask students whether they are vegetarian. Of n=25 students, y=0 answered "yes". For testing Ho: p=0.5 and Ha: p <> 0.5. Show that the likelihood ratio statistic equals 34....
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31 views

Maximum likelihood estimator for fishes in a pond [duplicate]

I've been stuck with this question for quiet a while now and I don't even seem to understand it right as I'm not sure about what to do or even approach the problem so any hint would be highly ...
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1answer
23 views

How to maximize this likelihood function to find the MLE? [duplicate]

Let $X_1,..., X_n$ be iid with pdf $f(x|\mu) = e^{-(x-\mu)}I_{[\mu, \infty)}(x)$ where I is $1$ if $x \in [\mu, \infty)$ and $0$ otherwise. Then the likelihood and log likelihood functions are given ...
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13 views

Comparing the likelihood of Gaussian errors and student-t errors

I am looking at the solution of a question that asks us to briefly comment on how we can compare two linear models with different distributed ϵi. For the first linear model ϵi is t-distributed and for ...
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18 views

Need help with GLR question

So I'm trying to find the GLR statistic $-2log(\Lambda) $ of a log likelihood function (1) $ln(L(\tau))=\tau \sum x_i + 2nln(1-e^\tau)$ to test $H_0: \tau=\tau_0$ vs $H_1:\tau\neq \tau_0$ where $\tau=...
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1answer
93 views

Log-likelihood of multinomial(?) distribution

If i have 3 probability groups of $p_1(\theta):p_2(\theta):p_3(\theta)=(1-\theta):(1+2\theta):(1-\theta)$ with $n_1=31,n_2=47,n_3=22$ how would I find the log-likelihood of above and MLE of $\theta$? ...
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21 views

Choosing a threshold based on the likelihood ratio test

I am stuck with a task of ratio test. Please help me with some advice. We are studying the Linear Discriminant Analysis. After projecting all the points on to "best" line the entire 2D dataset becomes ...
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1answer
47 views

Finding the distribution of the test statistic under the null hypothesis $H_0$

Suppose we have a random sample $X_1, \dots, X_n$ from $N(\mu , \sigma^2 )$, with known mean $\mu$. For the hypotheses $H_0: \sigma^2 = \sigma_0^2$ and $H_A : \sigma^2 = \sigma_1^2$, where $\sigma_1^2 ...
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15 views

Maximum Likelihood Estimator for MA(0) process with white noise that follows Laplace distribution

Assume we have the MA($0$) process: $X_t = \mu + \epsilon_t$ where $\mu$ is a constant and $\epsilon_t$ is independent white noise with the following Laplace distribution density function: $f_{\...
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1answer
23 views

(-2)*loglikelihood ~ Chi-square?

I just vaguely remember that I have learned something like "(-2)*loglikelihood asymptotically follows Chi-square distribution." However, I failed to find the relevant theorems in the textbook. Does ...
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1answer
141 views

Maximum Likelihood Estimation - Demonstration of equality between second derivative of log likelihood and product of first derivatives

I am faced to a problem of demonstration, about Maximum Likelihood Estimation, summarized on this image : Indeed, I don't know how to prove the following equality between : (1) $$\begin{aligned} \...
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1answer
47 views

Maximum Likelihood Estimation of this strange density

Let $X_1,\dots,X_n$ be independent random samples from distribution with density $$f(x) = \frac{\theta e^{\theta(x-\mu)}}{(1+e^{\theta(x-\mu)})^2}$$ for $x$ real, and $\mu$ from $\Bbb R$, $\theta >...
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1answer
55 views

Computing the LRT

Suppose we have the data $(Y_i, x_i),\dots,(Y_n,x_n)$ such that $Y_i \sim N(\theta x_i,1)$. We can compute the MLE, which will yield \begin{align*} \hat{\theta}_{ML} = \dfrac{\sum_ix_iY_i}{\sum_ix_i^2}...
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1answer
20 views

Is this the right equation to compute the likelihood of a normal distribution at a point?

This wiki gives this equation of maximization of the log-likelihood function. ${\displaystyle \ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(...
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21 views

Why is the expected gradient of a density not parallel to the expected gradient of the log density?

I'm cross posting from Stats Stackexchange, in case this community has more insight - hopefully that's okay! I'm confused by a seemingly counter-intuitive property of the interaction between ...
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10 views

Likelihood ratio test for a subset of the parameters

In the likelihood ratio parametric test, I was taught that the test is used when I want to test if the last q parameters are equal to some specific values or not. Can I also use likelihood ratio ...
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5 views

Estimation of point with highest relative likelihood in known Gaussian Mixture Model

In my current situation, I have a Gaussian Mixture Model with known mixture weights $[\omega_1,...,\omega_n]$, known means $[\mu_1,...,\mu_n]$ and known covariance matrices $[\Sigma_1,...,\Sigma_n]$. ...
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19 views

Unclear step in simplification of log likelihood for logistic regression

In my self-study on logistic regression I would like to show that the log likelihood of logistic regression is a concave function. For this purpose I would like to simplify the log likelihood from the ...
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1answer
26 views

What is the maximum likelihood estimation for a binomial distribution with zero successes in the data?

Say a random variable X is distributed as binomial(10, theta). In the data, out of 10 trials, there are 0 successes (i.e. 10 ...
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20 views

Curve of function is inaccurate (R)

I'm trying to plot a log-likelihood function given the certain parameters. I've written the function and the output it gives me is correct; however, when I plot it the plot is not correct. ...