8 votes

Seemingly contradiction: probability density function and maximum likelihood calculations for continuous random variable

There isn't any contradiction here. Your confusion is a very common one when dealing with the likelihood function for a continuous random variable. Let's take a step back and start from a discrete ...
  • 6,428
5 votes
Accepted

Observed Fisher information for the binomial: How is $I(\hat{\theta}) = \frac{n}{\hat{\theta}(1 - \hat{\theta})}$ calculated?

Hint: Substitute the value $\hat\theta=x/n$ in $I(\theta) .$ See what happens. Substitute in place of $\theta$ to get $$\frac{n^2x}{x^2}+ \frac{n^2(n-x) }{(n-x) ^2}.$$ Simplify it.
  • 3,694
5 votes

Use of Relative Likelihoods in Statistics?

The relative, or normalized, likelihood (function) $\tilde{\mathcal{L}}$ is defined as $$ \mathop{\tilde{\mathcal{L}}}\left(\theta\right) = \frac{\mathop{\mathcal L}\left(\theta\right)}{\mathop{\...
  • 3,795
4 votes

Is the Jackknife estimation better than Maximum Likelihood Estimator?

Comparing maximum likelihood estimation (MLE) and the jackknife is like comparing apples and oranges. These techniques accomplish different things. Maximum likelihood estimation is a method for ...
  • 7,210
3 votes

Are maximum likelihood estimator robust estimators?

One way to look at it is that there is no strict distinction between robust and non-robust estimators, but rather they can be compared according to their robustness properties (as already pointed in ...
  • 3,099
3 votes
Accepted

Justification of the fixed variational distribution in diffusion models

Under this view the diffusion process (the noise-adding steps, $z_T\dots ←z_t←z_{t−1}←\dots z_1$) defines an approximate posterior distribution $q(z_{1:T}|x)$ This is the forward trajectory, but the ...
3 votes
Accepted

Use of Relative Likelihoods in Statistics?

Apparently, these plots can be used to show that some of these Confidence Intervals contain "implausible values" These 'implausible values' are considered to be values that are outside the ...
3 votes

Observed Fisher information for the binomial: How is $I(\hat{\theta}) = \frac{n}{\hat{\theta}(1 - \hat{\theta})}$ calculated?

You forgot the expectation operator in the definition of Fisher information: $I(\theta) = E_\theta\left[-\frac{\partial^2}{\partial\theta^2}\log L(\theta)\right]$ instead of what you posted. After ...
  • 7,296
3 votes
Accepted

Marginal distribution of an autoregressive process of order one AR(1)

This is a good technical question. Without being given the initial distribution of $Y_0$, the rigorous proof of this well-known result actually requires advanced probability theory. For the sake of ...
  • 7,296
2 votes

Marginal distribution of an autoregressive process of order one AR(1)

Consider the $\sf AR(1) $ model: $$ Y_t= c+\rho Y_{t-1}+v_t.\tag 1$$ For $|\rho|< 1,$ the process is covariance stationary and the stable solution is $$Y_t = \sum_{j\in \mathbb N\cup \{0\}}[\rho^j ...
  • 3,694
2 votes
Accepted

Use of weights in choosing power parameter in Tweedie distribution

"How come that the weights don't play any rule in the function? Shouldn't it be taken into account when computing $\phi_{mle}$?". "... the function optimizes over the function ...
  • 36
2 votes

Use of Relative Likelihoods in Statistics?

In some literature the term "plausible" for parameters is used synonymously with "high likelihood". The idea is in principle the same as the idea behind statistical tests: ...
2 votes

Use of Relative Likelihoods in Statistics?

This is an extended comment, to complement the great answers by @ChristianHennig, @SextusEmpiricus and @statmerkur, with background material from related Cross Validated threads. The theory of ...
  • 7,210
2 votes
Accepted

Efron Hastie CASI exercise 4.3

Rather than a schematic graph of the score function $\dot{l}_x(\theta)$ vs $\theta$, I plot $\dot{l}_x(\theta)$ for an assortment of distributions. Then a schematic can be generalized from those cases....
  • 7,210

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