# Tag Info

Accepted

### Difference between anova () and summary ()?

There's a lot going on here. First, you don't seem to be conducting a repeated-measures anova but a multilevel regression model, which you then test using anova functions. Second, anova() and Anova(), ...
• 2,548

### What does it mean to say that a statistical model is a location family?

In order to have a formal treatment of location families of distributions, one must reckon with how a group of transformations acts on a random variable with certain distribution to generate a family ...
• 9,567

### What is the intuition behind the factorization theorem? (Sufficient statistics)

The factorization theorem states that $S(X)$ is sufficient if and only if the likelihood function of $\theta$ for data $X$ can be factored into the product of a function of $X$ (constant in $\theta$) ...
• 66

### Persistence of the sum of two AR(1) Processes

After reading Granger and Morris (1976) (thanks, @mlofton), it appears that $Y$ does not follow an AR(1) process, but instead follows an ARMA(2,1) process. Further, it takes some algebra, but I can ...
• 508
1 vote

### Can we conclude joint convergence in distribution from marginal convergence in distribution for two (not independent) sequences of random variables?

A counterexample: $X_n\sim N(0,1)$, $Y_n=(-1)^n(X_n)$. Then for even $n$, $(X_n,Y_n)$ converges in distribution to $(X,Y)$ with $Y=X\sim N(0,1)$ and for odd $n$ $(X_n,Y_n)$ converges in distribution ...
• 43.3k
1 vote

### How can I prove that two algorithms for weighted sampling without replacement are equivalent?

Yes, the two schemes are exactly equivalent (ignoring computational costs and numerical stability concerns). We can prove this by showing that both schemes perform $m$ draws from the weighted ...

### How does the reparameterization trick for VAEs work and why is it important?

Lots of great answers already; I'd like to offer a high-level answer: reparameterization is useful when you want to express the gradient of an expectation as an expectation of a gradient. It is useful ...
• 2,490
Accepted

### Does $|X_n|+|Y_n|=o_p\left(1\right)$ imply $|X_n|=o_p\left(1\right)$?

It is correct. $|X_n|+|Y_n|\geq |X_n|$, so for any $M$ and $\epsilon$, $|X_n|+|Y_n|<M$ implies $|X_n|<M$ and so $$P(|X_n|+|Y_n|<M)>1-\epsilon$$ implies $$P(|X_n|<M)>1-\epsilon$$ ...
• 43.3k

### Does the MLE converge in mean-square?

An example of an MLE that converges in probability but not in mean square is the ratio of two binomials. Let $X_n\sim Bin(n,p)$ and $Y_n\sim Bin(n,q)$, then the MLE of $(p,q)$ is \$(\bar X_n, \bar Y_n)...
• 43.3k

### Any relation between two KL divergences?

This might better be done with a different method to gauge differences between distributions. As Kullback-Liebler (KL) is a divergence, the triangle inequality need not hold. Thus comparisons like you ...
• 97.5k
Accepted

If you are just interested in setting up the integrals for the expectation, then maybe the following does that: $$\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\frac{\sum _{i=1}^J e^{u_i} u_i}{\... • 4,195 6 votes ### Mathematical Theory of Monotone Transforms It is often pointless to estimate such a transformation: extremely few statistical procedures require the underlying distribution to be Normal and any estimate will be imprecise anyway. But, as a ... • 329k 1 vote ### How is summation by parts technique used in this derivation? Ok since whuber commented, I decided to look back into it and figured it out. I think the following expression from wikipedia makes it easier to understand:$$ \sum_{k=0}^n f_k g_k = f_0 \sum_{k=0}^n ...
• 33

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