# Questions tagged [log-likelihood]

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

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### Finding the Cramer Rao bound

Let $x=(x_1,\dots,x_n)$ be a sample of i.i.d random variables with pdf $$f(x;\theta)=(1-\theta)\chi_{[-1/2,0]}+(1+\theta)\chi_{[0,1/2]}$$ where $\theta\in(-1,1)$. Find the Cramer Rao bound. So to do ...
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### Computing the likelihood function

Let $x=(x_1,\dots,x_n)$ be a sample from $X_1,\dots,X_n$ independent identically distributed random variables with pdf $$f(x)=(1-\theta)1_{[-\frac{1}{2},0]}(x)+(1+\theta)1_{(0,\frac{1}{2}]}(x)$$ where ...
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### What is the meaning of $p_{g_i}$ in this equation?

I am reading "The element of statistical learning" and having some question regarding equation 2.36. The book stated that: "A more interesting example is the multinomial likelihood for ...
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### Maximum likehood estimate

I have a question regarding the correction of my exercise: Exercise 6. Let $Y_1,\dots,Y_n$ be i.i.d. such that $Y_i$ equals $1$ with probability $p$ and $-1$ with probability $1-p$, for all $i\in[n]$....
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### What is the log-likelihood formulation for a combined normally distributed dataset?

Consider a sample (i.i.d) $C_{1},...,C_{n}$ that is normally distributed ~$N(µ,1)$ and a sample (i.i.d) $D_{1},...,D_{n}$ ~$N(\theta,1)$, we say $µ\neq\theta$. Now I want to formulate a log-likelhood ...
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### Function proportional to the log likelihood for the Gaussian distribution

The following question has been crossposted to CrossValidated upon recommendation from the community and a lack of responses here. Consider the following problem from a course on statistical ...
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### Does almost sure convergence of $\frac{1}{m}L_m$ imply almost sure convergence of $\frac{1}{m}\max\limits_{k\leq m}L_k$?

Assume we have a distribution function which is controlled by a parameter $f(x;\theta)$. If we sample $m$ i.i.d samples from this distribution We can define the sum log-likelihood ratio of the $m$ ...
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### MLE of Erlang Distribution

For sample $x_1,...,x_n$~Erlang, $f(x_i|k,\lambda)=\frac{\lambda^kx^{k-1}e^{-\lambda x_i}}{(k-1)!}$ Population has expectation $k/\lambda$ and variance $k/\lambda^2$. You may assume second derivative ...
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### How can I prove the Hessian of the log likelihood of the Generalized Error Distribution is negative definite?

I'm working with the multivariate generalized error distribution to model some data. (The parameterization that I am working with follows Graham Giller's work here: https://www.researchgate.net/...
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### Likelihood equation of 2 samples geometric distribution.

Context Given there are 2 groups that can be modelled as a geometric distribution as follows: \begin{align*} f(x_i;p_1) &= p_1(1-p_1)^{x_i - 1} \; x_i = 1,2,... \; 0<p_1 <1 \\ f(y_i;p_2) &...
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### Using a Bin$(k,\theta)$ random sample ($k$ known), determine the posterior distribution for $\theta$ using a Jeffrey's prior.

My question here is, am I misinterpreting a binomial random sample? Doing some research on this I'm thinking perhaps I have misunderstood something - however my answer "feels" okay ...
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### How to find the value that give the global maximum of a function?

I have got the following function: $$L(x,y) = \frac{(16.2x +0.9y+5.2)^{24} e^{-(16.2x +0.9y+5.2)}}{24!} \cdot \frac{(2.1x +4.2y+0.9)^8 e^{-(2.1x +4.2y+0.9)}}{8!} \tag{1}$$ and I am attempting at ...
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### Simplifying $-\log\left(\left( \frac{1}{2\pi\sigma^2}\right)^{n/2} \exp \left( -\frac{1}{2\sigma^2} \sum_{i=1}^{n}(\mu-y_i)^2\right)\right)$

I'm not the most mathematically minded, but I'm doing my best to learn MLEs. We were provided the following original likelihood function of a Normal Distribution and I am having trouble understanding ...
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### Negative Log likelihood and Derivative of Gaussian Naive Bayes

I am trying to derive negative log likelihood of Gaussian Naive Bayes classifier and the derivatives of the parameters. So there are class labels $y \in {1, ..., k}$, and real valued vector of $d$ ...
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### All solutions to $f(a,b)+f(b,a)=0$

I am looking for all functions $f:(0,1)^2\to\mathbb{R}$ that satisfy $f(a,b)+f(b,a)=0$ for every $(a,b)\in(0,1)^2$. I know that $f\equiv 0$, $f(a,b)=c(a-b)$, and $f(a,b)=\pm\log(a/b)$ are such ...
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### Likelihood of p and q given Y

Suppose that Jon has a coin with a probability of landing on heads p and Ann has a coin with a probability of q landing on heads. Each time they flip a coin and they write the sum of the outcome ...
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