New answers tagged

0 votes

limiting the log-likelihood function for Weibull distribution

I believe that the issue is about the sufficiency of the solution to the first order condition with respect to the global optimization problem. As you showed that the log-likelihood function is ...
  • 3,935
0 votes

Maximum Likelihood estimator with cubic function

Your expression rearranging the likelihood is not quite correct. $$ L (\theta) = \prod_{i = 1}^n \lambda \cdot 3x_i^2 \cdot e^{-\lambda \cdot x_i^3} = \lambda^n \cdot 3^n \cdot e^{-\lambda \cdot \sum ...
  • 145k
0 votes
Accepted

Show that the maximum likelihood estimator aims to maximize the probability of a given event

First, knowing that $f_{\theta}(x) \leq f_{\hat \theta}(x)$ does not imply that $\int_a^b f_{\theta}(x) \, \textrm d x \leq \int_a^b f_{\hat \theta}(x) \, \textrm d x$. The MLE maximizes over ...
0 votes

Does an Maximum Likeihood estimator always unique? It is true that an MLE is always unique for exponential family of probability distributions?

There is an article at The American Statistician addressing the issue you are concerned about: Sudhakar Dharmadhikari & Kumar Joag-Dev (1985) Examples of Nonunique Maximum Likelihood Estimators, ...
  • 3,935
2 votes
Accepted

Using exponential family to find the derivate of the partition function

It sounds like $$ H(\theta) = \log \int \exp \left[\log P(y|z) - \frac{z^2}{2v} + \frac{mz}{2v}\right]dz $$ where the integration is over a range in which $z$ is defined. Let's write this more shortly ...
  • 5,131
0 votes

Maximum Likelihood and method of moment estimation

The maximum likelihood estimator is $\hat \theta=\max(X_1,X_2,\ldots,X_n)$ as @Aaron Montgomery has correctly shown in his answer. As for the method of moments estimator, notice that $$E(X)=\int_0^\...
  • 3,935
1 vote

Maximum Likelihood and method of moment estimation

The reason you usually differentiate the log likelihood function is that it's typically the easiest way to do what you're really trying to do, which is to maximize the likelihood function. So, let's ...
3 votes
Accepted

Compute the MLE of variance for $f(x) = 3x^3 /\theta^3$

By the invariance property of MLE, if $\widehat{\theta}$ is a MLE of a parameter $\theta$ and $g: \mathbb{R} \to \mathbb{R}$ is a function, then $g(\widehat{\theta})$ is a MLE of $g(\theta)$. Since ...
  • 1,624
0 votes
Accepted

Maximum Likelihood Estimation on a standardized normal variable

Ignoring the MLE part of the question, the change in density function for a location-scale change is not difficult, though with a normal distribution you have to be careful not to confuse the variance ...
  • 145k
1 vote

Not understanding Maximum Likelihood last steps

The result given in the book is based on a property that is called "functional equivariance". Or functional invariance. See So as you have correctly calculated $\hat{\beta}=1.6225$. Now ...
1 vote
Accepted

Not understanding Maximum Likelihood last steps

The key insight is that you are not interested in estimating the parameter $\beta$ itself, but rather, a function of that parameter, which the solution calls $h$. Getting an estimate of $\beta$ is ...
  • 115k

Top 50 recent answers are included