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Overall, the intuition lies in the fact that Ordinary Least Squares linear regression is not (sort of) symmetric with $\mathbf{Y}$ (the predicted) and $\mathbf{X}$ (the regressor). Linear Regression The basic model we follow here is $\mathbf{Y} = b\mathbf{X} + \epsilon$ (neglect the intercept for a moment), where the error $\epsilon$ is only assumed to be ...


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DWLS (Diagonally Weighted Least Squares), in some articles also called (WLSMV; Muthen, du Toit & Spisic, 1997), is the recommended choice of the estimator for non-normally distributed data in SEM (Finney & DiStefano, 2006; Flora & Curan, 2004; Wirth & Edwards, 2007; Yang-Wallentin, Jöreskog & Luo, 2010). If you want to try an alternative, ...


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Note that $\log(x_1^2)=2\cdot\log(x_1)$. Consequently, $\text{cor}(\log(x_1),\log(x_1^2))=1$. The log of the square adds no value when the log is already there.


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Is using ACF and PACF plots to check whether residuals follow and AR(1) process a valid methodology? I suppose it is valid unless there is something special about this model which I do not know. The plots suggests AR(0), AR(1) or AR(5) depending on how sensitive to marginally significant ACs and PACs you want to be. (An AR(12) could also be considered but I ...


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The CDF can be used in the likelihood. Remember that the likelihood encodes the information about our observations. For one example, if the observation is not known exactly but bounded from below, like in survival analysis, then we represent this information with a CDF in the likelihood. See section 2 of https://cran.r-project.org/web/packages/flexsurv/...


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If, after the occurrence of the event, the "clock" for the next starts, and the arrival time between events is independent, then you have a Renewal Process. Let $t$ be the unknown time passed, $N(t)$ the number of occurrences prior to time $t$, which you observe, $\mu$ the expected time between events, and $\sigma^2$ is the variance of time between ...


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Even your simple suggestion to evaluate a function at random points is often used in cases where a full search is computationally infeasible and the function is not smooth. A famous example is RANSAC for estimating shape parameters from point clouds (here the function is the argmax of an accumulator array in a rasterized parameter space). There are much more ...


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Random search means that you explore the potential hyperparameter values by picking the random combinations of hyperparameters. Marsaglia (1972) invented and algorithm for sampling points uniformly at random in a sphere, this may or may not be how you would like to sample the hyperparameters. There are many different algorithms for generating pseudo-random ...


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The question has two aspects: terminology and validity of the model. Regarding terminology, searching for PC-VAR I find Morana "PC-VAR Estimation of Vector Autoregressive Models (2012). In that paper (and using its notation), PC-VAR amounts to regressing the original variables $x_t$ on lags of some of their principal components $f_{t-j}$ (let us call it ...


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Counter-example: If one considers the case of a sample $(X_1,\ldots,X_n)$ from a $N(\theta,\theta^2)$ distribution, a minimal sufficient statistic is $$T=\left(\sum_i X_i,\sum_i(X_i-\bar X_n)^2\right)$$ Now $\varphi(T)=\bar X_n$ is an unbiased estimator of $\theta$ that is not sufficient. But the conditional expectation $$\mathbb E_\theta[\varphi(T)|\varphi(...


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If you are only interested in one dependent variable, you may look at its equation alone. As Christoph Hanck correctly notes, the other equations of the model do not affect its estimation if you do equation-by-equation OLS (and that is a preferred method for an unrestricted VAR). However the residual diagnostics fail when looking at that equation. Trying ...


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Normally, inverting the likelihood ratio test statistic or Rao's score test statistic is a nice technique to keep the parameter within its parameter space. However, since we have a sample size of 1 and the exact distribution of these test statistics is unknown in this case, these methods will fail miserably. It's times like these when the frequentist ...


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Simulated annealing is a general-purpose optimization algorithm, technically a "metaheuristic", that requires a supplemental function (the "energy function") to optimize. A nice description is here: http://webpages.iust.ac.ir/yaghini/Courses/AOR_891/05_Simulated%20Annealing_01.pdf . This energy function is where your data would come into ...


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BTW the link is broken. If one takes $y_0=0$, and $x$ to be time, $t$, the equation refers to a particular solution to the second order linear differential equation with constant coefficients using maximum initial functional value (at $t=0$). The reason for the simplifying assumption of $y_0=0$ is to formulate the answer as a density function, as a density ...


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Unless you are in the rare situation where X vs Y is linear, the most sensible estimand is the continuous function relating X to a property of Y such as the median or mean. If the relationship were linear then the (constant) slope would be sufficient. The estimator of the regression function would be for example a regression spline.


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Frame challenge: Zeno's Paradox is an illusion, or plausible yet invalid line of reasoning which results by ignoring or failing to recognize the fact that the runner requires successively decreasing time to cover the infinitely decreasing denominations of distance and can reach the goal in finite time at some constant speed measurable with finite distance ...


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Since the problem is a yes/no answer, we can model the outcome as binary (1-- yes, 0-- no). The standard error of the outcome is the estimate of the standard deviation divided by the square root of the sample size. Hence, the standard error is $$ \dfrac{\sigma}{\sqrt{n}} \>.$$ All we need do is estimate the standard deviation. Let's assume our main ...


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The within transformation will not work because of the non-linearity of the logit function. There are some possible solutions: Fit a panel linear probability FE model Conditional logit Unconditional fixed effects logit estimator using dummies Pseudo-demeaning algorithm CRE (Mundlak-Chamberlain device)


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To elaborate on the comment, $T(X) = X^2$ is complete if: $$E_{\theta}[g(T(X))] = E_{\theta}[g(X^2)] = 0 \text{ for all } \theta \in [0, \frac{1}{2}] \implies \\ g(T(X)) = g(X^2) = 0 \text{ with probability 1 for all } \theta \in [0, \frac{1}{2}] $$ As you've shown $$E_{\theta}[g(X^2)] = 0 \text{ for all } \theta \in [0, \frac{1}{2}] \implies g(0) + 2\theta (...


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