The tag has no usage guidance.

learn more… | top users | synonyms

3
votes
1answer
42 views

determine constant for biweight (bisquare) function

In robust statistics a biweight (bisquare) function is defined as follows $$\rho \left( x \right) = \gamma\left( {1 - {{\left( {1 - {{\left( {\frac{x}{c}} \right)}^2}} \right)}^3}} ...
0
votes
0answers
24 views

How to Compute the Integral of the Auto-correlation Function for a Discrete Time Series

Using the covariance $$ c(u) = \frac{1}{N}\sum^{N-u}_{t=1}(x_t - \bar{x})(x_{t+u}-\bar{x}), $$ I've computed the auto-correlation function $$ r(u) = \frac{c(u)}{c(0)}, $$ where $x$ is a time ...
2
votes
1answer
115 views

Overlap between two normal pdfs [duplicate]

I have two normally distributed random variables (estimated from two different sets of samples), and I'd like to know how "similar" those variables are (in order to compare the sets). I had the idea ...
0
votes
0answers
42 views

Minimizing KL Divergence With An Unintegrable Function in Expectation Propagation

I am trying to match the mean and variance of my posterior by minimizing the KL divergence as per the EP algorithm. However, my likelihood function is of the form:$\exp(\exp(-||\theta - x||))$ where ...
0
votes
2answers
77 views

How to deal with 'cut-off' selection bias/sampling bias? (truncated distribution)

In short When measuring an outcome with a normal distribution, but whos mean is below the detection threshold, can you still make statements about differences between populations? Example Say I ...
1
vote
0answers
43 views

How to calculate E(1/Y) when Y is Inverse Gaussian distributed?

The Inverse Gaussian Distribution density is : $$\frac{\phi^{\frac{1}{2}}}{\sqrt{2\pi y^3}} exp[\frac{-\phi(y - \mu)^2}{2\mu^2y}]$$ Got to this integral: $$\int_0^\infty \frac{1}{y} ...
0
votes
0answers
21 views

Application of line integral or surface integral to machine learning?

I am exploring the kernel methods in machine learning, and found an interesting post on this. In my point of view, kernel method is a way of reducing dimensions. I have an intuitive understanding that ...
3
votes
1answer
55 views

Distribution of stochastic integral

I would like to find the distributions of the following random variables: $Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process. What is the ...
0
votes
0answers
23 views

Uncertainty on ratio of integrals (correlated)?

I have a histogram, and a quantity defined as the integral of a subrange of that histogram (int_sub) divided by the integral of the full range (...
0
votes
0answers
23 views

What regression to use for integral and scale data?

I am hoping to compare reaction times with ratings from a questionnaire. The ratings on the questionnaire could range from -9 to +9 although in the data as expected they are only positive results ...
0
votes
1answer
52 views

Given a sample of size n from a normal distribution, estimate the probability of picking a value larger than X from this distribution

Say I picked 10 random samples from a normal distribution with unknown parameters: 1.7, 2.6, 3.0, 4.4, 1.6, 2.1, 2.4, 2.7, 5.2, 3.3 What is the probability that I will pick a value larger than ...
4
votes
1answer
116 views

Gibbs Sampler transition kernel

Let $\pi$ be the target distribution on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R^d}))$ which is absolutely continuously wrt to the $d$-dimensional Lebesgue measure, i.e : $\pi$ admits a density ...
0
votes
0answers
22 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
0
votes
1answer
46 views

indicator variable - dirac delta or step function

I am trying to solve the following equation, \begin{equation} = \int_{-\infty}^{\infty} \frac{1}{\sqrt{ (2\pi)^{k_{Y}} | \Sigma |}} \cdot \mathrm{exp} \{ -\frac{1}{2} (Y - Xm)^{T} \Sigma^{-1} (Y ...
13
votes
3answers
2k views

“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
5
votes
1answer
180 views

Conditional independence iff joint factorizes

I have proven that: $X⊥Y|Z\ {\rm iff}\ p(x,y|z)=p(x|z)p(y|z)$ for all $x,y,z$ such that $p(z)>0$. The next question is to prove an alternative definition: $X⊥Y|Z$ iff there exist functions $g$ ...
7
votes
1answer
118 views

Truncated Von Mises-Fisher distribution

I am putting a von Mises-Fisher prior on my data. The data does lie on a unit sphere, but the only problem is that my data is always positive. So I feel like I am wasting my prior on unnecessary ...
1
vote
0answers
30 views

Monte carlo integrations with metropolis hastings step

Consider the following problem: Suppose we want to compute the following integral $$ f(y_1|y_2) = \int_{\theta} \int_{x_1} \int_{x_2} f(y_1| x_1,x_2,\theta, y_2) f(x_1,x_2|\theta,y_2) f(\theta | ...
2
votes
0answers
93 views

A Integral question

I am having problem with integral: $$\int_{-\infty}^\infty\frac{1}{2\pi} e^{-itx} \left( 0.5 \left( e^{-4t^2} + 1 +\cos(t) + \cos(2t) \right)\right) \,dt$$ @DilipSarwate I have proved lim sup(cf)=1 ...
4
votes
2answers
118 views

Expectation of von Mises Fisher Distribution

The von Mises- Fisher distribution is defined as $$ \frac{\kappa^{p/2-1}}{2\pi I_{p/2-1}(\kappa)}\exp(\kappa \mu^Tx) $$ It is defined over the unit sphere i.e. $||x||_2^2=1$. My question is what is ...
3
votes
2answers
74 views

conditional expectations value

I need to calculate the following integral $$\int_{\mu+c}^{\infty} y\cdot \frac{1}{\sigma\sqrt{2\pi}}e^{(y-\mu-w)^2/2\sigma^2}dy$$ So essentially $y\sim N (\mu+w, \sigma^2)$ and im trying to ...
0
votes
1answer
34 views

Compute the cumulative hazard of a time interval

I'm a little confused in calculating the cumulative hazard within a time interval. We know that $H(t)=\int^{t}_{0}h(u)du$, if I have $\Delta t=t_1-t_0$ (1) $H(\Delta t)=\int^{\Delta t}_{0}h(u)du$ ...
1
vote
1answer
71 views

Explanation of density rewriting?

Can somebody please explain the math behind this statement to me? I am not sure how they represent the left hand side by that integral and finally how it is proportional to that. \begin{align} ...
1
vote
1answer
65 views

[Revised]Proving the expected \bold{density} of being the Nth order statistics is decreasing in sample size

(Sorry that I've previously formulated the question in a wrong way, which confused everyone including myself. This is a better version of the question. Thanks!) Here's another order statistics ...
3
votes
2answers
165 views

Proving some properties of expected first order statistics with respect to sample size

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)= ...
1
vote
0answers
198 views

Integrate first derivation - area under curve of first derivation

I want to calculate the area under the first derivate. Can I do that with splinefun()? ...
2
votes
1answer
66 views

Moment generating function if the PDF is $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$

Let $z$ a random variable with PDF : $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$, where $0\leq z \leq d$, $F$ is the Hypergeometric function, $k$ is a positive integer, $-a+k+1 ...
7
votes
1answer
323 views

Integrating an empirical CDF

I have an empirical distribution $G(x)$. I calculate it as follows x <- seq(0, 1000, 0.1) g <- ecdf(var1) G <- g(x) I denote $h(x) = dG/dx$, ...
1
vote
3answers
178 views

Why is the marginal liklihood term(the denominator) in Bayes' updation not a function of theta(parameters)?

The snippet from the book is pasted above. I don't understand how the denominator is not a function of theta as it is integrated over theta. Thank you.
1
vote
0answers
96 views

Moment of random variable on a integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
1
vote
0answers
131 views

Monte Carlo integration

I am calculating a simple integral $\int^1_{-2} \exp{x^2}(x+1)dx $ with Monte Carlo method using a linear density function $p_\xi (x) = \frac{4}{9} + \frac{2}{9} x $. Let say I have a a sample which ...
0
votes
1answer
32 views

Approximation of Pr(X > a), with X a multivariate normal rv

Let $X = (X_1, ..., X_p)$ a random variable with a $N(\mu, \Sigma)$ distribution. $$ $$ $$ \Pr(X_1 > a_1, ..., X_p > a_p) \\ =\int_{a_1}^\infty ... \int_{a_p}^\infty (2\pi)^{-p/2} ...
0
votes
1answer
93 views

Integration problem in Bayes factor calculation for multinomial model

This is one integration problem I encountered during the calculation of Bayes factor between two models given data $D$ One of the model, $M_0$ assumes the data accords to multinomial distribution, ...
1
vote
2answers
85 views

Ordered gamma variables led to an ugly integral

Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f. $$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$ and let $Y_1,...,Y_n$ be the order statistics for these variables. a) Find the conditional p. ...
1
vote
1answer
72 views

Integral of a function with uniform kernel

I am trying to understand question 9-1 on p.334 in Cameron & Trivedi (link) where I have to calculate the bias of a kernel density estimate at $x=1$ and $n=100$, where we assume that the ...
4
votes
1answer
100 views

How to arrive at a specific formulation of the relative median deviation?

I am an economist currently working with this book: Frank Cowell - Measuring Inequality On page 25 a formulation of the relative mean deviation is given as follows: $$ M = 2 \left[ ...
1
vote
2answers
60 views

What is the expected partial value function really called?

If f is a pdf, the integral of x*f(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, x*f(x) from negative infinity up to t, ...
0
votes
1answer
57 views

Calculating joint probability problem

This is the example from Casella and Berger 2nd page 146. you have $$f(x,y)=e^{-y}, 0<x<y< \infty $$ and you are interested in finding $$P(X+Y\ge1)$$ They solved this problem the following ...
1
vote
0answers
28 views

Integrability of a sequence of iid random variables

I'd really appreciate some hints on the first part of the following question: Let $f_n, n\in \mathbb{N}$ be a sequence of iid random variables over $(\Omega, A,P)$. That is, $P(\{f_1 \in ...
1
vote
0answers
37 views

Sufficient and necessary condition for the integrability of a random variable

Could anyone give me a hint on the following problem? Thanks! Let $f$ be a random variable over a probability space $(\Omega,A,P)$. Show that $f$ is integrable $\iff $ $\sum\limits_{k=1}^\infty ...
2
votes
1answer
62 views

$E(x^k)$ under truncated $\mathcal{N}(\mu,1)$

There is a similar question in $E(x^k)$ under a Gaussian. However, it doesn't seem to be trivial when $\mu\ne0$. As mentioned in the previous question $k$ is not an integer. The integral that I need ...
0
votes
1answer
249 views

Integral in Stata when upper limit is infinity [closed]

How can we calculate the integral of the integrand with the lower limit 0 and upper limit infinity in Stata? I am aware of the integ command, but I am not sure ...
4
votes
0answers
172 views

Integrating a Gaussian Distribution multiplied by 1/x^(2/3), any ideas?

I am trying to integrate the following function over $x$ and $g$: $f(z)= \int_{a/z}^{d} \int_{a}^{zg} \frac{e^{-(g-\mu)^2/2\sigma^2}}{x^{2/3+1}} dx~dg$ When I integrate over $x$ I get: $f(z)= 3 ...
3
votes
2answers
397 views

Improved Monte-Carlo method vs. hit-and-miss method

I do not understand Which is more accurate, the hit-and-miss method or the improved Monte-Carlo method? Here it is written that that the hit-and-miss has a higher variance but they showed ...
2
votes
1answer
77 views

Find the pdf of Y

$$ f(x)=\frac{1}{2}e^{-|x|} , -\infty < x < \infty ; Y=|X|^{3} \ $$ I understand that I have to divide it in two parts and write it in cdf form $$ F_{x}(y^\frac{1}{3}) - F_{x}(-y^\frac{1}{3}) ...
3
votes
0answers
173 views

Integration step in Bayesian linear regression

I want to obtain the integral: $$\int_{{\mathbb R}^p} \frac{1}{(2\pi)^{\frac{n}{2}}\vert\Sigma\vert^{\frac{1}{2}}}\exp\left[-\frac{1}{2}({\bf y} - {\bf X}{\beta})^{\top} \Sigma^{-1}({\bf y} - {\bf ...
1
vote
4answers
135 views

For what broad types of statistics are integral calculus useful? [closed]

I understand how differential calculus is useful for basic Maximum Likelihood estimation techniques. However, my question is: what broad types of statistics require an understanding of integral ...
2
votes
1answer
132 views

Sampling from marginal using integrated conditional

I would like to sample from: $$ p(\theta_2|x)=\int p(\theta_2|\theta_1,x) . p(\theta_1|x) . d\theta_1 $$ knowing that I can easily sample from $p(\theta_1|x)$ and (less easily) from ...
3
votes
2answers
284 views

why isn't the the marginal distribution needed when using a conjugate prior?

What is a good explanation as to why you wouldn't have to integrate to find the posterior when you use a conjugate prior. Most examples (like for instance: http://www.youtube.com/watch?v=0XD6C_MQXXE) ...
1
vote
0answers
92 views

Can the proof of the central limit theorem be expressed as the limit of a convolution with an operator

The central limit theorem, as I understand it (engineer, non-statistician), says that the distribution comprised of means of some other reasonably behaved distributions converges to a normal ...