10 votes

Maximum Likelihood with Categorical Variables - Does this Change Anything?

The algorithm used is exactly the same: although the features are not continuous in your example, the coefficients in a regression model still are. We are optimizing the coefficients in the model, the ...
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  • 18k
6 votes

Chi-Square Difference Test for Nested Models with a Continuous Outcome

Chi-square is a distribution, not a test. Chi-square tests are used for things that are chi-square distributed. There are two different things they are used for: The chi-square test of a contingency ...
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5 votes

Chi-Square Difference Test for Nested Models with a Continuous Outcome

The term 'chi-squared test' tends to be applied to almost any test that has a test statistic whose distribution under $H_0$ is (at least approximately) distributed as chi-squared. Something similar ...
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3 votes

Maximum Likelihood with Categorical Variables - Does this Change Anything?

On the one hand, there are features, which are attributes of your data, stuff that you measure, and on the other hand, there are the parameters to your model. Both can be continuous or discrete. If ...
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  • 3,138
3 votes

Estimating Mixture Models with Maximum Likelihood

While the observed likelihood is a well-defined function $$L(\theta|\mathbf x)=\prod_{i=1}^n \{\pi_1\varphi(x_i;\mu_1,\sigma_1)+ (1-\pi_1)\varphi(x_i;\mu_2,\sigma_2)\}$$ it does not offer enough ...
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2 votes
Accepted

The probability/cumulative density function for inequality of two random variables

As mentioned by @whuber, you're better off attempting to obtain the distribution of $X/Y$ than $X - Y$. Because both $X$ and $Y$ are supported on the positive reals one can write $Pr(X>Y)$ as $Pr(...
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2 votes
Accepted

Find the MLE density function of uniform [-\theta,\theta]

A few extra (and less) steps: \begin{align}\require{cancel} F_{\hat\theta_{MLE}}(x) &= \mathbb P_\theta(\text{max}\{-X_{(1)},X_{(n)}\}\leq x)\\ &= \mathbb P_\theta(-X_{(1)}\le x,X_{(n)}\leq x)...
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2 votes

MLE of the Uniform Distribution

In your example $n=3$, $\min x_i = 1$, & $\max x_i = 3$. When $\theta=1$, $$I(\max x_i \leq \theta, \min x_i \geq 0)=I(3\leq 1,1\geq 0)=I(\mathit{false})=0$$ This factor, & therefore the ...
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1 vote

Finding the maximum likelihood solution corresponds to finding the root of a regression function. How?

The connection given in Bishop (2006) (p. 96) is that he is implicitly taking: $$z = \frac{\partial}{\partial \theta} \ln p(X|\theta),$$ which then gives: $$\mathbb{E} \bigg[ \frac{\partial}{\partial \...
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1 vote
Accepted

How to get most likely value with confidence interval or expected value from a set of observations

A key comment above is @Bernhard's statement that you may need to make additional assumptions for a good answer. One reasonable assumption is that the data are normal. Descriptive statistics (fron R) ...
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  • 50k
1 vote

Statistical comparison of (covariance) matrices

My favorite tool for comparing covariances comes from Förstner W., Moonen B. A Metric for Covariance Matrices. In: Grafarend E.W., Krumm F.W., Schwarze V.S. (eds) Geodesy-The Challenge of the 3rd ...
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