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5

The series converges to a distribution function. It can be evaluated in closed form. Upon identifying the terms varying with $n,$ write your function in a simpler form as $$F_{U_i}(y)=\frac{2(R^{\alpha}y/\theta)^k}{\Gamma(k)}\sum_{n=0}^\infty \frac {(-R^\alpha y / \theta)^n}{n!(k+n)((k+n)\alpha+2)} = \frac{2x^k}{\Gamma(k)}\sum_{n=0}^\infty \frac{(-x)^n}{n!...


1

The old rule of thumb was that the sample size is too small when the expected count of one of the cells of the contingency table is lower than five. Recall that the expected count of a cell is the ratio $$\frac{row\_count \times column\_count}{total}.$$ In http://www.biostathandbook.com/small.html and http://www.biostathandbook.com/fishers.html, owing ...


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Utilizing your reproducible example, I have worked up what appears to be the idea behind VIM using nls. library(matrixStats) set.seed(101) df <- data.frame( "ID" = paste("ID", seq(1,100,1), sep = ""), "x1" = sample(90:220, size = 100, replace = T), "x2" = sample(90:220, size = 100, replace = T), "x3"...


1

Adding (or subtracting) a small number to (or from) a not-small number results in a number that is roughly the same as the not-small number, hence in many situations can be skipped, meaning the small number can be ignored. As the small number gets even smaller, the approximation obtained by ignoring it and just using the not-small number gets better and ...


1

Want to punish all criminals? Throw everyone in jail. Never want to punish an innocent defendant? Don’t ever prosecute anyone. As you require a higher standard of proof ($\alpha$ or specificity), it’s harder to convict. If you need fingerprints, DNA, and two eyewitness, that’s a lot of evidence compared to just relying on one eyewitness and some prints. You ...


1

If $\sigma$ is known, you use Gaussian distribution. When it is unknown, to account for the uncertainty over possible $\sigma$'s, you use student-t distribution. So, the issue is the unknown variance. When the student-t distribution is well approximated by Gaussian with large samples, the problem with uncertainty over $\sigma$ decreases and the two ...


0

Generate different 10% validation 90% training splits. Run model generation on each training set. Determine model performance on accompanying testing set. Determine confidence intervals from resulting vector of metrics. Formore indepth look see https://www.sisostds.org/DesktopModules/Bring2mind/DMX/API/Entries/Download?Command=Core_Download&EntryId=36208&...


4

In this general setting, what would be the best approach to calculate corr with null values? Input or discard? I don't know the size of these datasets, but they are generally "big". I'm assuming that you have only 2 variables and you want to compute the correlation between them. Imputing the missing values, if you have no knowledge of why they are missing,...


1

Penalized regression does not do an initial regression and then downweight points "outlying" from this initial regression. $L_1$ and $L_2$-penalized regressions do not weight observations at all (de facto). In fact, $L_1$-penalized regression (LASSO) is more prone to bias d/t outlier because it tends to select regressors based on large coefficient values. ...


2

I actually just wrote it out. So here goes: Let $$ \frac{\partial}{\partial \mu} E(X^{n}) = \int \frac{\partial}{\partial \mu} \frac{1}{\sqrt{2\pi}} x^n e^{-(x-\mu)^2/2} dx $$, then if I expand the square (in the exponent) to get $\mu^2 - 2\mu x - x^2$, I can factor out only the terms the involve $\mu$ and use those in the derivative. This gives, $$ \...


1

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. I can't really see how... This $$ \mathbb{E}_{p_{\theta}(\mathbf{z})}\left[f\left(\mathbf{z}^{(i)}\right)\right]= \mathbb{E}_{p(\epsilon)}\left[f\left(g_{\...


0

That $n_1\delta_1+n_2\delta_2$ has finite second moment has been shown in the other answer. To prove the linear combination is UMVUE, I would use this necessary-sufficient condition which states that an unbiased estimator (with finite second moment) is UMVUE if and only if it is uncorrelated with every unbiased estimator of zero. Let $\mathcal U_0$ be the ...


1

One possible way is to compute the CDF and hen differentiate with respect to $y$. Here's a start, try to complete it. Let $y \in (0,1)$, \begin{align}Pr(Y \le y)&= Pr(-4X+4\sqrt{X}-y \le 0)\\ &=Pr\left(\sqrt{X} \le \frac{1-\sqrt{1-y}}{2}\right) + Pr\left(\sqrt{X} \ge \frac{1+\sqrt{1-y}}{2}\right)\\ &=\left( \frac{1-\sqrt{1-y}}{2}\right)^2 +1- \...


2

The cited paper by David Pollard indeed analyses an example due to Kagan and Shepp [The American Statistician 59 (2005) 54–56]. That example gives a statistic which is not sufficient, but still the Fisher information based on the insufficient statistic is equal to the Fisher information based on the complete data. I will not state that example here, but ...


1

What tests generally do is that they tell you whether what you have observed is unlikely under the null hypothesis, which is a probability model for random data. Note that this does not mean that you can only use them if the data are indeed random. If you apply a statistical test to nonrandom data and you get a significant result, it means that your data ...


0

I had a similar problem. R code demonstration. t.test(c(1.1,2),conf.level = .9)$co t.test(c(2.1,2),conf.level = .9)$co t.test(c(3.1,2),conf.level = .9)$co It is caused by the standard deviation shrinking. I offset the problem by using 75th percentile of standard deviations of all items linearly combined with actual sd. However my goals were a bit different ...


19

One precise formulation of Bayes' Theorem is the following, taken verbatim from Schervish's Theory of Statistics (1995). The conditional distribution of $\Theta$ given $X=x$ is called the posterior distribution of $\Theta$. The next theorem shows us how to calculate the posterior distribution of a parameter in the case in which there is a measure $\nu$ ...


0

Rather than giving a full response I would like add a factor in the distinction between the two. Let's make the example of a neural network used for classification, most of the times when people get the results they wanted they don't know exactly why they are getting those results. While statistics is more rigorous and always comes with a measure of the ...


1

Just for the sake of argument, I am putting my two cents here. As I find the answers above/below so far are pretty explanatory. David DN rounded your question up nicely, I think. This subject is very new and therefore, take what you get and run with it. I worked with stats and I worked in research. I also worked on predictive research. Even the big ...


3

Personally, I find it very hard to draw a line between the two, as there is clearly some overlapping. Machine Learning is a field that is based on classical statistics and USES statistic models heavily. Also, the mathematics behind Machine Learning can get extremely complicated, so I really would not use the mathematical argument as a discriminant. One ...


2

This reminds me of a recent conversation with my dad who was surprised to hear that I still run models taking days which he did decades ago: Are you still doing computations that take longer than a day? Shouldn't everything run fast now? What are you people (physicists) improving on modeling nowadays, when there have already been satisfying models in the ...


7

In my view, MCMC/bootstrapping/permutation methods all fall under the category of computational techniques. They aren't tied down to a specific approach or way of thinking about a problem but rather an algorithmic approach to a class of problems. Techniques that involve resampling and iteration don't arise from a machine learning framework, they come out of ...


2

The Weibull survival function with shape parameter $k$ and scale parameter $\lambda$ (both positive) has the form $$S(x; \lambda, k) = \exp\left(-(x/\lambda)^k\right)$$ for $x \gt 0.$ A finite mixture of $n$ such distributions is determined by positive mixture weights $p_i$ (necessarily summing to unity) and corresponding parameters and has survival ...


1

Yes, they are equivalent because $\{\omega | X(\omega)+2=x+2\}$ is equal to $\{\omega | X(\omega)=x\}$. To see this, suppose we have for a particular $\omega$ that $X(\omega)+2=x+2$, we can subtract two from both sides. Conversely,we can add $2$ to both sides. $P(X+2=x+2)$ is actually short hand for $P(\{\omega | X(\omega)+2=x+2\})$. As for your ...


0

First, get predictions for the desired inputs. The predicted class for each input is the class with the largest predicted probability.


0

The question has been answered in depth here: Math Stack Exchange The idea is writing the joint density and switching to polar coordinates, where it can be seen that the transformation of the random variables (which are the ones you ask about) gives a joint density which can be simply factorized, which implies that the two newfound variables are ...


1

You can define the following process: $$ Y_t=c+Y_{t-1}+u_t $$ Assuming an initial condition $Y_0=0$ for simplicity, recursive substitution yields $$ Y_t=c\cdot t+\sum_{s=1}^tu_s, $$ a so-called random walk with drift.


0

it depends on the signification of the parameter if for example 'a' is a parameter depends on the mean so you should find a rough estimation for the mean.


1

There is a lot to unpack here, so I'll just answer a few of these. Confounding occurs when a predictor and the outcome share a common cause. Usually, the presence of unadjusted confounding yields a biased estimate for the relationship between the predictor and the outcome. If you are building a predictive model, you don't need to think about confounding ...


0

Just for your reference, there isn't much difference between left and right truncation in principle. If we define the time to some event as a random variable $T$ then the left-truncated version of this random variable is simply $$T=t\,|\,T>u$$ where $u$ is some truncation point. If we were to define some model and try to fit data to it using, for ...


3

You have asked three different questions, perhaps without realizing that you have: 1) Are the two probes getting different results? 2) Are they equally accurate? 3) Are they getting the same results? Assuming that the data is paired (which is implied by "measuring the same area") then for 1) you could use a paired t-test (or a nonparametric version). ...


1

I Googled "left censoring survival analysis" and got 570,000 hits. A better route might be Google Scholar with the same search terms. That has 120,000 hits. I'd suggest starting with a good book (or more than one) on survival analysis, maybe one general one and one specific to R. Assuming that you have a good basic understanding of survival analysis (...


0

Hint: If $Y+1 \sim \text{Ga}(2,1)$ then the random variable $Y$ has the density function: $$g(y) = \text{Ga}(y+1|2,1) = \frac{1}{e} \cdot (1+y) e^{-y} \quad \quad \quad \text{for all } -1 < y < \infty.$$ This density function bears a striking resemblance to the target distribution. Have a think about how you might use this result, in conjunction ...


0

T1 will be more efficient if it has a small variance compered to T2


0

As pointed out in comments, with a certain positive (maybe small, but always positive) probability, the expression $Y_1/(Y_1+Y_2)$ becomes $0/0$, so the random variable $Y_1/(Y_1+Y_2)$ is not unambiguously defined. You must clarify your question, how do you define your random variable in that case?


1

First of all, it is SSR, and using regression coefficient estimates, we can calculate estimated target values: $\hat y_i = \hat \beta x_i+\hat \beta_0$, then substitute into the formulation of SSR: $$\begin{align}\text{SSR}&=\sum_{i=1}^n(\hat y_i-\bar y)^2=\sum_{i=1}^n \hat y_i^2 -2 \bar y\sum_{i=1}^n \hat y_i+\underbrace{\sum_{i=1}^n \bar y^2}_{\text{...


0

Does this image clear it up? Source: https://www.datasciencecentral.com/profiles/blogs/machine-learning-vs-statistics-in-one-picture


4

Think of the set of questions one can ask about data as living on a simplex, where the vertices represent Confirmatory questions, Exploratory Questions, and Predictive questions about the data. Here is a visual aid I've taken from a course my supervisor has taught. Included are some questions that could be asked about data concerning how many people are in ...


1

Statistics is a mathematical science that studies the collection, analysis, interpretation, and presentation of data. Statistical/Machine Learning is the application of statistical methods (mostly regression) to make predictions about unseen data. Statistical Learning and Machine Learning are broadly the same thing. The main distinction between them is in ...


3

I have not studied this but as far as I can tell statistical learning and machine learning are the same thing. One can make inferences and predictions in both statistical learning and inferential statistics, but the goal in statistical learning tends to be prediction over inference, whereas the reverse is true of inferential statistics. The main ...


0

If we assume that the random variables $Z_i=(Y_i, X_i)$ are i.i.d., $i = 1, \dots, |\mathcal{D}|$, and that $p(x_i \mid \mathbf{w}) = p(x_i), \forall i$ (that is, $x_i$ does not depend on the parameters) and considering that $P(X, Y \mid Z) = P(Y \mid X, Z) P(X \mid Z)$, then we have \begin{align} \operatorname{Pr}(\mathcal{D} \mid \mathbf{w}) &= \...


4

Since Chernoff (1952) already uses the letter $M$ to denote the moment generating function, it is preferable to use a different symbol for the underlying sample space. Thus, I will suppose that the problem is located in the probability space $(\mathcal{S}, \mathscr{S}, P)$. When dealing with random variables that are generated as functions of other random ...


0

First of all, permit me to elucidate a couple of terms to find the common ground: $\delta(x)$ - decision rule $\gamma(\theta, \delta(x))$ - the loss function $R(\theta, \delta) = \mathbb{E}_{x} [R(\theta, \delta(x))]$ - the risk function $r(\delta) = \mathbb{E}_{\theta}[R(\theta, \delta(x))]=\mathbb{E}_{\theta}(\mathbb{E}_{x} [R(\theta, \delta(x))])$ - the ...


0

Your question deals with the problem of training a generative model Eq.(2) versus training a discriminative model Eq.(3). As highlighted in this article, you cannot go from Eq. (3) to Eq. (2), there are two definitions of a quantity to optimize; both being often misleadingly called likelihood. From the article: While this is a valid way of obtaining a ...


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