Taylor
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Consistency of Sample Mean in Time Series Data
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A few years later... here's a solution that does not involve invoking Cesaro's Lemma. It follows from the following result: If $v_n \to \ell$ as $n \to \infty$, then $\bar{v} := n^{-1}\sum_i v_i \to \...

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Singular matrix with dummy variables
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1 votes

I'm picking up on some confusion about a few things. Linear dependence in the columns of your design matrix $\mathbf{X}$, or equivalently singularity of $\mathbf{X}^\intercal \mathbf{X}$, is bad ...

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Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution
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I do not know how to use the factorization theorem You have to write the likelihood as a product of two functions. One function is allowed to have parameters in it, and one is not. The one with the ...

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Event space (probability)
1 votes

Hint: consider the definition of random variable, then try to write the set $\{\omega: \xi(\omega) = \eta(\omega)\}$ in terms of sets you know are in $F$. The definition of sigma-algebra/sigma-field ...

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Modeling a time series of ordered vectors
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Here's a model: $$ \mathbf{x}(t) = \mathbf{A} \mathbf{x}(t-1) + \mathbf{v}(t) \tag{1} $$ $$ \mathbf{x}^o(t) = g(\mathbf{x}(t)) \tag{2} $$ where $g$ is the ordering function, $\{\mathbf{v}(s)\}_s$ are ...

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Show $J(I-H) = 0$
1 votes

Going for the shortest answer here: just look at the first row of $\mathbf{X}^\intercal \mathbf{H} = \mathbf{X}^\intercal$.

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Can the horseshoe prior be expressed analytically?
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3 votes

The normal conditional density's normalizing constant is not free of lambda, so Wolfram is integrating the wrong thing. $$ (2\pi)^{-1}\int_0^\infty \underbrace{(2\pi\tau^2\lambda^2)^{-1/2}}_{\text{...

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Conditional Expectation : How is E[E[xy|x]]=E[xE[y|x]]?
1 votes

You can reduce this question to this: why is $$ E[XY|X] = XE[Y|X] $$ (with probability 1)? If this is true, then you can just take expectations on both sides. The answer is "because of the (...

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Can I create an own p.d.f to apply it in a Monte Carlo study?
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1 votes

There are a few things you can do with a distribution: a. write it down as an unsolved integral, b. write it down as a solved integral, c. evaluate its normalized version on a computer at different ...

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Best way to combine MCMC inference with multiple imputation?
-1 votes

I wrote a paper arxiv.org/abs/1907.09090 that describes how the pseudo-marginal approach can impute missing data. 400 covariates sounds tough, though, to be completely honest. Depends on what kind of ...

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Using eigendecomposition to transform state vector in linear Gaussian state space model
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2 votes

Must we make an assumption that $\mathbf{Q}$ is known in order to do what the authors are suggesting? No, this is a matter of identifiability, and things like this are commonly done to make the model ...

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Inaccurate parameter estimates for state-space models?
2 votes

"Is this normal?" I don't use this software, and I'm not sure what you mean by "normal," but nothing jumps out at me. A few thoughts: Asymptotic distributions of MLE estimates ...

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Why the sum of two absolutely-continuous random variables isn't necessarily absolutely continuous?
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18 votes

Let $X$ be a standard normal random variable, and let $Y = -X$ (pointwise). Then both are a.c., but $X+Y$ is $0$ everywhere.

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Which notation and why: $\text{P}()$, $\Pr()$, $\text{Prob}()$, or $\mathbb{P}()$
2 votes

This makes me think of Meyn and Tweedie's book. They use $P$ to denote the transition kernel for a Markov chain, and $\mathsf{P}$ for the law of the entire chain on $\mathsf{X}^{\infty}$. This answer ...

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Characteristic Function and Quantile Function?
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2 votes

The inverse of a cdf $F : \mathbf{R} \mapsto [0,1]$ is usually $$ F^{-1}(p) = \inf\{ x : F(x) \ge p \}. $$ The way you invert a c.f. $\phi$ to get measures of intervals is $$ \mu([a,b]) = \lim_{T\...

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Inverting a moment generating function
1 votes

A few comments: Regarding @BruceET's comment, MGFs don't uniquely define random variables all the time, though. Practically, yeah you can identify many/most common random variables by their mgf. But ...

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Particle filter for likelihood evaluation
1 votes

The problem is: It works very poorly. If I deviate a little from the true parameters, the likelihood gets, as expected, smaller. However, there are always alternative constellations which give me a ...

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What does the pmf of a discrete random variable look like if it can take on the value $\infty$?
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0 votes

If you accept $\{\tau < \infty\} = \bigcup_{k\ge 1} \{\tau = k\}$, then $$ P(\tau < \infty) = \sum_{k=1}^{\infty}P(\tau = k). $$ by countable additivity. To add to the intuition, take ...

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convergence of an average of consistent estimators?
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By the triangle-inequality \begin{align*} 0 &\le \left|\frac{1}{nm}\sum_{j=1}^m\sum_{i=1}^nX_i^j - E[\mu]\right| \\ &\le \left|\frac{1}{nm}\sum_{j=1}^m\sum_{i=1}^nX_i^j - m^{-1}\sum_{j=1}^m\mu^...

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Parameters identifiability / estimation in Bayesian linear state-space models
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2 votes

Adding to @Jarle Tufto's comment, the likelihood can be written as $$ p(y_{1:t} \mid r, \sigma_x, \mu, p) = \int p(y_{1:t} \mid x_{1:t}, \mu, p) \overbrace{p(x_{1:t} \mid r, \sigma_x)}^{{\text{AR(1)}}}...

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Trying to understand linear state-space notation
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2 votes

SSMs aren't always identifiable without restrictions on the parameter space. There is a common distinction between centered and uncentered parameterizations. What you have written would be called the ...

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Planar Flow in Normalizing Flows
1 votes

The equation $$ \mathbf{w}^T\mathbf{z_1}+b=0 $$ defines a (hyper)plane. The vector $\mathbf{w}$ is the normal vector. For a refresher on multivariable calculus, see here. So what happens if you have ...

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What is this projection matrix doing?
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3 votes

Multiplying $X$ by $W$ gives you transformed data. Multiplying $X$ by $P$ gives you $X$ again ($P$ is the identity matrix because the eigenvectors are orthogonal). Observe that $$ \text{Var}\left[W^...

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Applying Bayes' rule in a more technical way when densities don't exist
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1 votes

Define \begin{align*} S &= \{ x : Ax = y, x_2 \le s\} \\ &= \{ (x_1, x_2) : a_1 x_1 + a_2 x_2 = y, x_2 \le s \} . \end{align*} We can say that \begin{align*} p(x \in S \mid y) &= \frac{...

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How can I derive the variance of the half-normal distribution?
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3 votes

Hint: if $X \sim \text{HalfNormal}(\sigma^2)$ and $Y \sim \text{Normal}(0,\sigma^2)$, then for any symmetric and integrable $f$ $$ E[f(X)] = E[f(|Y|)] = E[f(Y)|Y \ge 0] = 2 E[f(Y) 1(Y \ge 0)]. $$ I ...

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Where is the error in my computation of the wrapped normal distribution density?
1 votes

Observe that \begin{align*} \log \psi(x) &= \log \sum_{k\in\mathbb Z}\phi(x+k) \\ &= \log \sum_k \exp\left[ \log \phi(x+k) \right] \\ &= m + \log \sum_k \exp\left[ \log \phi(x+k) - m \...

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Intuition Behind $O_p(\frac{1}{n})$
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2 votes

$X_n = O_p(\frac{1}{n})$ means it's not terrible to think of $X_n$ as something like $$ Y / n. $$ This is a single random variable over a changing nonrandom constant. Indeed $Y/n = O_p(\frac{1}{n})$ ...

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Subscript in expected value notation
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1 votes

What do the subscripts of the expectations mean here? They are the distribution you are taking the expectation with respect to. They are the "weights" you're using to calculate the weighted average. ...

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Calculating the observation density
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2 votes

In the context of an SIS particle filter with the transitional prior as the proposal density $q(x_k|x^i_{k-1},z_k)= p(x_k | x_{k-1},z_k)$ Two things: one, SIS stands for "sequential importance ...

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Particle filter: Evaluating Optimal importance density
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3 votes

You're mistaken about a few things (and that's okay!). In this case the optimal proposal density...is [available]. I believe this is only true if $f$, the state transition is Gaussian. It can be ...

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