# Questions tagged [log-likelihood]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...