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6

The essential features of this question are: It does not make strong distributional assumptions, lending it a non-parametric flavor. It concerns only tail behavior, not the entire distribution. With some diffidence--because I have not studied my proposal theoretically to fully understand its performance--I will outline an approach that might be practicable....

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A couple of comments: Generalized Linear Mixed Models (GLMMs) have the following general representation: $$\left\{ \begin{array}{l} Y_i \mid b_i \sim \mathcal F_\psi,\\\\ b_i \sim \mathcal N(0, D), \end{array} \right.$$ where $Y_i$ is the response for the $i$-th sample unit and $b_i$ is the vector of random effects for this unit. The response $Y_i$ ...

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Let's find the case we couldn't WIN. Call the individual probabilities as $p_w,p_i,p_n$, and call it $k$ trials: $$P(\text{WI})=(1-p_n)^{k}, P(\text{WN})=(1-p_i)^{k}, P(\text{NI})=(1-p_w)^{k}$$ $$P(\text{W})=p_w^{k}, P(\text{N})=p_n^{k}, P(\text{I})=p_i^{k}$$ Via Inclusion-Exclusion Principle: $P(\text{WIN})=1-(P(\text{WI})+P(\text{WN})+P(\text{NI}) - P(\... 3 Hint: if$X\sim \Gamma (a,\lambda)$so$E(X^{k})=\frac{\Gamma(a+k)}{\Gamma(a)}\lambda^k$so$X\sim \chi^{(2)}_{(n)}=\Gamma (\frac{n}{2},2)$so$E(X^{k})=\frac{\Gamma(\frac{n}{2}+k)}{\Gamma(\frac{n}{2})}2^k$$$F_{(n-1,m-1)}=\frac{\frac{\chi^{(2)}_{(n-1)}}{n-1}}{\frac{\chi^{(2)}_{(m-1)}}{m-1}}$$ $$E(F_{(n-1,m-1)})=E\left( \frac{\frac{\chi^{(2)}_{(n-1)}}{n-... 2 I am not aware of specific work in this direction. You will likely need to "roll your own" approach. I would indeed use a NegBin distribution, with a time-varying mean for each series (whether you want to model this separately for each series, or connect the series in modeling via a hierarchical, panel or other approach is a separate question). In terms of ... 2 This is a nice and neat simulation that I can demonstrate with R code. I would start off by defining the initial values: population <- 100 consumption_rate <- 5 Next, we could define a few functions to vary the population parameters, as specified in the 3 steps: ## Take a uniform value in 0.01-0.02 as a percentage of our current population ... 1 It's not analytically available. You can use numerical approaches/software. What you want to find is actually inverse CDF of quantile function in other words. In Matlab you can use gaminv, in R, you can use qgamma, or in python you can use ppf in scipy.stats. It's not easy to find a table but here is one with unit scale, with varying shape parameters (upto ... 1 I don't know about the GEE but for random-effects: They are two functions for fitting random effects wthin a GAMLSS model, random() and re(). The function random() is based on the original random() function of Trevor Hastie in the package gam. TIn our version the function has been modified to allow a "local" maximum likelihood estima- tion of the ... 1 Here are three reasonably doable approaches: you could do numerical convolution, e.g. via fast Fourier transform With a large sample size like 500 you could use a normal approximation you could do simulation (as you mention) Here's a quick simulation, done in R: x=c(0,1,5,10,-1) p=c(.05,.1,.15,.08,.62) rsum=replicate(10000,sum(sample(x,500,replace=TRUE,... 1 The case n=1 can be solved by algebra, I get$$ F^{-1}(p) = \frac{\alpha}{\exp\left( \frac{\log((1-p)/p}{\beta} \right)}$$which can be used as a test for a numerical solution. For$n \ge 2only a numerical solution is practical, with R the function uniroot is helpful, there must be something similar in python. Some R code: makeF <- function(k, ... 1 To answer your question directly... \begin{align*} \text{erf}\left(c_1 x + c_2\right) &= \frac{2}{\sqrt{\pi}}\int_0^{c_1x+c_2}e^{-t^2/2}dt \\[1.2ex] &= \frac{2}{\sqrt{\pi}}\int_0^{c_1x}e^{-t^2/2}dt + \frac{2}{\sqrt{\pi}}\int_{c_1x}^{c_1x+c_2}e^{-t^2/2}dt \\[1.2ex] &= \text{erf}(c_1 x) + \frac{2}{\sqrt{\pi}}\int_{c_1x}^{c_1x+c_2}e^{-t^2/2}dt. \... 1 It depends on your need. In fact, the AR(p) model can be estimated using OLS and without assuming any distribution. However, more complex models such as ARMA you need a specific distribution for the past inovations (errors). So, the assumption of the distribution is necessary because you need some extra information to estimate the parameters. It is the ... 1 I would suggest to fit different distributions on your observations, and to perform model selection to find the distribution that fits your observations the best. Exponential and Pareto distributions seem to be the best candidates given your hypotheses (positivity, monotone decrease). Once you have fitted these candidates distributions, model selection ... 1 Correspondence to a waiting time problem There is an alternative way to look at this. We can switch between the 'number of words' as being the variable and the 'text size' as being the variable. Imagine assembling the text or book untill you reach some fixed number,x$, of words. Then consider the length of the text,$m\$, as the variable. We can relate ...

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