5k views

We already know $\gamma$ is bounded between $[-1,1]$ The correlation matrix should be positive semidefinite and hence its principal minors should be nonnegative Thus, \begin{align*} 1(1-\gamma^2)-0.6(...

180k views

> a <- c(179,160,136,227) > sd(a)  38.57892 > sqrt(sum((a-mean(a))^2/(length(a)-1)))  38.57892 `

14k views

Section 2.2.2.1 from lme4 book Because each level of sample occurs with one and only one level of batch we say that sample is nested within batch. Some presentations of mixed-effects models, ...

9k views

$E[(X-E[X])^2] =0 \implies X = E[X]$ Thus $X$ is almost surely constant. A better description for such random variables is that it follows a degenerate distribution.

1k views

EDIT This is explained in the FAQ: I use plot() (npplot()) to plot, say, a density and the resulting plot looks like an inverted density rather than a density This can occur when the ...

3k views

HINT: Quadprog solves the following: \begin{align*} \min_x d^T x + 1/2 x^T D x\\ \text{such that }A^T x \geq x_0 \end{align*} Consider x = \begin{pmatrix} w\\ b \end{pmatrix} \text{and } ... View answer 1 answers 2 votes 570 views Accepted answer 6 votes HINT: \begin{align*} \exp\big(-\frac{\theta}{\kappa_0}-\frac{(\bar{t}-\theta)^2}{2\frac{\sigma^2}{n}}\big) &= \exp\big( -\frac{n}{2\sigma^2} \big((\bar{t}-\theta)^2 + 2\frac{\sigma^2 \theta}{n \... View answer 2 answers 5 votes 633 views 6 votes Statistical Sleuth does not describe the consulting process, but teaches methods using case studies. To quote: The Sleuth was written to train graduate students in disciplines other than Statistics ... View answer 3 answers 9 votes 349 views 5 votes Lyapunov's Inequality (See: Casella and Berger, Statistical Inference 4.7.6): For 1 < r < s < \infty: \mathbb{E}[|X|^r]^\frac{1}{r} \leq \mathbb{E}[|X|^s]^\frac{1}{s} $$Proof: By ... View answer 2 answers 2 votes 3k views Accepted answer 5 votes You can use mixture models to capture the biomodality library(flexmix) set.seed(42) D <- c(rnorm(100,1,1), rnorm(100,5,1)) kde <- density(D) m1 <- FLXMRglm(family = "gaussian") m2 <- ... View answer 1 answers 0 votes 1k views 5 votes Definition 3.5.4 from Casella & Berger: Let f(x) be any pdf. Then for any \sigma >0, the family of pdfs (1/\sigma) f(x/\sigma), indexed by the parameter \sigma, is called the scale ... View answer 1 answers 2 votes 474 views Accepted answer 5 votes Define T=\sum X_i T is a complete sufficient statistic for p. Now, consider indicator I_{X_1=1,X_2=1,X_3=1,X_4=1} which is an unbiased estimator of p^4(As you proved in the first part) Rao-... View answer 1 answers 7 votes 275 views 4 votes Let I_i^c be an indicator variable such that:$$ I_i^c = \begin{cases} 1 & \text{run of length $c$ starts at $i^{th}$ position}, \\ 0 & \text{otherwise} \end{cases} $$To find \mathbb{E}[... View answer 1 answers 0 votes 1k views Accepted answer 4 votes You need to pass it the model and not a string: ts1=arima(TP, order = c(1,1,0), xreg=F) na.kalman(TP, model = ts1model) View answer 1 answers 2 votes 87 views Accepted answer 4 votes$$P(D|S) \quad \underline{deff} \quad ∑_iP(D|M_i)·P(M_i|S) \tag{1}is just a consequence of law of total probability: \begin{align*} P(D|S) &= \sum_{i}P(D,M_i | S)\\ &= \sum_{i} \frac{P(D,... View answer 2 answers 4 votes 55 views 4 votes You might want to try out Gaussian Mixture models for your data. For example, to decompose a mixture of \mathcal{N}(10, 5), \mathcal{N}(22, 3), using flexmix package library(flexmix) set.seed(42) ... View answer 1 answers 1 votes 4k views Accepted answer 4 votes Let \pi_A = \pi_B = \frac{1}{2} represent the probability of selecting coin A and B respectively. Observed data: \{\mathbf{X_1}, \mathbf{X_2}, \mathbf{X_3}, \mathbf{X_4}, \mathbf{X_5} \} ... View answer 1 answers 9 votes 165 views 4 votes Here is an attempt: Consider Z=X-Y such that X \sim \chi^2(\alpha) and Y \sim \chi^2(\beta) with \alpha \geq \beta \mathcal{M}_X(t) = \left(1-2 \, t\right)^{-\alpha/2}  \...

68 views

\begin{align} f(x) = c\ \exp({-x^2-\frac{x}{4}}) &= c\ \exp({-(x+\frac{1}{8})^2+\frac{1}{64}})\\ &= c' \exp(-\frac{y^2}{2})\ \forall -\infty < y < \infty \end{align} where $c'=c\ \... View answer 1 answers 3 votes 441 views Accepted answer 3 votes Using ggvegan and ggrepel library("vegan") library("ggvegan") library("cowplot") library("ggrepel") data(varespec, varechem) vare.cca <- cca(varespec, varechem) obj <- fortify(vare.cca) want &... View answer 1 answers 3 votes 27 views 3 votes$E[E[A|B]]=E[A]$and so the expression is simply the variance of$E[A|B]\begin{align*} \mathrm{Var}(A) &= E[\mathrm{Var}(A|B)] + \mathrm{Var}(E[A|B])\\ \mathrm{Var}(E[A|B]) &= \mathrm{Var}(A)... View answer 1 answers 1 votes 33 views Accepted answer 3 votes It is indeed incorrect.F_{Y_n}(t) = P( \max_{i} X_i \leq t) = P(X_1,X_2, \dots X_n \leq t) $Assuming$X_i$are independent and not simply identically distributed:$P(X_1,X_2, \dots X_n \leq t) =...

2k views

Statistical Inference by Casella et al. is often used as a primary textboook.

161 views

$Var( X_i\epsilon_i)= E[(X_i\epsilon_i)^2]-(E[X_i\epsilon_i])^2 =EX_i^2 \times E\epsilon_i^2-0=(\mu^2+\tau^2)\sigma^2$ Corrected formula: Var(\frac{U}{V}) \approx (\frac{E[U]}{E[V]})^{2}\cdot(\frac{...

191 views

There are examples of using Multiple Linear Regression for similar studies Here is an notebook example of doing this in R.  Genomic ancestry and somatic alterations correlate with age at ...

85 views
Consider $D_i = X_i - Y_i$. Then \begin{align*} D_i &\sim \mathcal{N}\left(\mu_1-\mu_2, 2\sigma^2(1-\rho)\right)\\ \implies \bar{D} &\sim \mathcal{N}\left(\mu_1-\mu_2, \frac{2\sigma^2}{n}(1-\...
Naive way to solve $ii$: Observation 1: $\sum X_i$ is complete and sufficient statistic for $\theta$ Observation 2: $\sum X_i \sim Poisson(n\theta)$ We need to look for an unbiased estimator of $\... View answer 1 answers 6 votes 926 views Accepted answer 2 votes$\chi^2_1(0.95) = 3.841C = \{\mathbf{Y}: (Y_1+3Y_2) \log{\frac{p_0}{\hat{p}}} + (Y_1+3Y_0) \log(\frac{1-p_0}{1-\hat{p}}) \geq \frac{-\chi^2_1(0.95)}{2} \}$When$Y_0 =Y_2$,$\hat{p} = 1/2$Thus, ... View answer 1 answers 3 votes 675 views Accepted answer 2 votes Define$p = P(Z_i > u) = P( \frac{Z-\mathcal{\epsilon}}{\sqrt{\sigma^2+1}} > \frac{u-\mathcal{\epsilon_i}}{\sqrt{\sigma^2+1}}) = \phi(\frac{\mathcal{\epsilon_i}-u}{\sigma{\sigma^2+1}})$The ... View answer 2 answers 2 votes 612 views 2 votes Jaccard Index is often used to calculate similarity of such sample sets. Let's assume there are 4 products$P_1,P_2,P_3,P_4$that can be bought offline or online. So if$S_{off} = [1,0,1,1]$and$S_{...