All Questions
Tagged with density or density-function
448 questions with no upvoted or accepted answers
3
votes
0
answers
48
views
Error bounds when approximating densities
I was curious whether it is possible to obtain approximation error bounds on continuous densities from approximation error bounds of random variables.
To make my question more precise: We consider ...
- 475
3
votes
0
answers
157
views
How to calculate CDF of g(X)
Let
$X$ a random variable with distribution $F_X(x)$
$$Y=g(X) = \left\{
\begin{array}{lr}
X-c & : X > c\\
0 & : -c < X \le c \\
X+c & : X \le -c
\end{array}
\right\}$$
...
- 589
3
votes
0
answers
475
views
Orthogonal series density estimation
I am going through this paper Orthogonal series density estimation. I have a doubt in following
Assume that the random variable X is supported on [0, 1], that is, P(X ∈ [0, 1]) = 1, and that the ...
- 592
3
votes
0
answers
77
views
Shouldn't a function of data from a PDF repeated over and over on new data eventually yield a Gaussian PDF?
I got into an interesting discussion with a co-worker today and we are not sure what the answer is:
We have $N=1000$ samples from a Rayleigh PDF. We take those $N$ samples, and compute their (...
- 1,685
3
votes
0
answers
120
views
I want to prove that these definitions of expected value hold
Let $(\Omega,\mathcal B,P)$ be a probability space. I have two (related) questions. Assuming that $g:\mathbb{R}\to\mathbb{R}$ is Borel measurable, and understanding that
$$E(g(X)) = \int_{\Omega}g(X(...
- 403
3
votes
0
answers
2k
views
How to calculate confidence intervals using subsampling after a nonparametric estimator about the empirical distribution function?
I have a problem where I think subsampling is more appropriate than the bootstrap. (Reason in another post.)
However, I found no quick reference on subsampling CIs, and my naive inversion of the ...
- 987
3
votes
0
answers
281
views
Is excess mass estimation smooth enough to bootstrap? At what rate might a bunching estimator converge?
The recent public finance literature often estimates relative excess mass around specific points of the earnings distribution ("kink points" or "notches" of tax schedules, say), and then bootstraps to ...
- 987
3
votes
0
answers
956
views
What is the relationship between two points on probability density function?
The Wikipedia entry for Probability Density Function states that the PDF "describes the relative likelihood for this random variable to take on a given value." Two questions:
Does that mean that the ...
- 159
3
votes
0
answers
329
views
Goodness-of-fit test without analytical PDF and CDF
I have closed form moment-generating function and characteristic function of a distribution, which describes waiting time of a continuous univariate random process. However, I cannot analytically ...
2
votes
0
answers
37
views
To what extent can likelihood methods be used for functional responses?
Let's suppose that we are working with a functional data set, $Y_i(t)$, $Y_i\in L^2[0,1]$, $1\le i\le n$. If we were working with univariate or even multivariate data set, likelihood methods would ...
- 1,413
2
votes
0
answers
38
views
Distribution supported on $(0,\infty)$ for which moments of its truncated distribution are elementary functions of the truncation point and power
I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\...
- 525
2
votes
0
answers
100
views
Find PDF from approximated MGF
I have an array of values of MGF (it is evaluated at some points).
The plot of it is shown (blue curve): .
Is it possible to find PDF knowing MGF in such form?
I tried to fit MGF with some curve (you ...
- 173
2
votes
1
answer
217
views
Questions about the conditional Radon-Nikodym derivative
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ ...
- 863
2
votes
0
answers
98
views
PDF of the sine of a wrapped Normal distribution
I have a random variable which is an angle $\Theta$ that follows a wrapped Normal distribution. The angle $\Theta$ has a relatively small variance, so despite having a range from $(-\pi,\pi)$, ...
- 131
2
votes
1
answer
103
views
Are the terms in the diffusion model equation random variables or probability density functions?
Are all terms in the first line(71) of the equation random variables or probability density functions? If they are probability density functions, is there a possibility of obtaining a value that is ...
- 21
2
votes
0
answers
36
views
The Wikipedia example of a Statistical model and its PDF
The example section of Wikipedia's article on Statistical model says:
Suppose that we have a population of children, with the ages of the children distributed uniformly, in the population. The height ...
- 121
2
votes
0
answers
55
views
Package for Multidimensional Density Estimation
I may be missing something obvious, but is there a python package that can reliably do density estimation of a PDF in high dimensions (e.g. 512)? I know of scipy's ...
- 21
2
votes
0
answers
26
views
What is the proper representation of the density of a transformed variable-- 3 examples with different plotting axes
I am having trouble understanding when PDFs of transformed variables need to be modified (i.e., take derivative) for proper plotting. I've looked at Intuitive explanation for density of transformed ...
- 133
2
votes
0
answers
43
views
Fast measure of "clusteredness" of points?
I have a cloud of points in a bounded volume in 2D (lets say 2d for now, though it'd be nice to generalize to any dimension):
$<p_n \in \mathbb [0, 1]^2: n \in [1..N]>$
I'm looking for some ...
- 614
2
votes
0
answers
407
views
Confused about inverse function (quantile function)
I read a post that says:
"Math definition is that the quantile function is the inverse of the distribution function at α. It specifies the value of the random variable such that the probability ...
- 1,680
2
votes
0
answers
290
views
Distribution of Sample Variances for Half Normal Distributions
If $X$ is distributed according to a normal distribution with zero-mean $\mathcal{N}(0, \sigma_N^2)$, $Y:=\vert X\vert$ is said to be distributed according to a half-normal distribution, cf. 2.
I am ...
- 71
2
votes
0
answers
23
views
PDF of estimated Bernoulli parameter
This is my first question on this part of the stack exchange, so please bear with me and correct me if I am missing something obvious. Statistics is not my main field of expertise.
Background
I wish ...
- 21
2
votes
0
answers
64
views
Intuition behind the inverse of the copula density $\frac{1}{c(u,v)}$
If the inverse of a probability $\frac{1}{p(x)}$ represents the unpredictability or surprisal of a sample from random variable $X$,
then what is the intuition behind the point-wise inverse of the ...
- 4,049
2
votes
0
answers
683
views
Deriving the PDF of an exponentially modified Gaussian RV
For a random variable $Z = X + Y$, where $X$ is an exponential RV with $λ = 1$ and $Y$ is a Gaussian (Normal) random variable with mean $μ$ and standard deviation $σ$, how could we derive the ...
- 21
2
votes
0
answers
321
views
If a zero entropy distribution implies high information a priori, what does it mean ex posteriori?
The following counteracts the statements made for the maximum entropy principle case in order to posit a pseudo "minimum entropy principle" case that is simply the polar opposite of the ...
- 4,049
2
votes
0
answers
92
views
Estimation of the conditional density functions
Suppose we have a random vector $x=[x_1, x_2, ..., x_{10}]\in\mathbb{R}^{10}$. Obviously, $x_1$, $x_2$, ..., $x_{10}$ are random variables theirselves. I have 500 observed samples of $x$ which I am ...
- 121
2
votes
0
answers
36
views
The monotonicity of entropy operator
Define the entropy operator of a distribution as $\mathbb{H}(p) = -\int p \log p$, how does the entropy change for distributions that are proportional to the powers of $p$?
For example, define $\...
- 581
2
votes
0
answers
555
views
Inverse transform method with piecewise pdf
I am having trouble using the inverse transform method with the generalized inverse
$$F^{-1}(u) = \inf \{x : F(x) \geq u\}$$
In this case, I have a piecewise pdf
$$f(x) = \begin{cases}x, & 0 \...
- 253
2
votes
0
answers
33
views
How are probability functions derived? E.g. normal, Poisson, t-distribution
The idea behind PMFs is simple - how likely is a given discrete event to happen? For example, it is intuitive to see why the PMF of a binomial distribution is $
Pr(X=k)=
\left( {
\begin{array}{}
...
- 484
2
votes
0
answers
103
views
Predictive density via LOOCV
I am looking for a way to generate a density prediction (in contrast to a point prediction or a prediction interval) in a multiple regression setting without relying on stringent parametric ...
- 69.5k
2
votes
0
answers
106
views
How to fully estimate a probability density from only a sample of minimum values?
We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$.
By means of ...
- 1,733
2
votes
0
answers
173
views
Characterizing a distribution
I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year.
If I have some data in the form ...
- 123
2
votes
0
answers
63
views
Determining a probability distribution from constraints on where its mass is
Let $X$ be a random variable over the real line.
Suppose that we know that $X$ is a Pearson distribution. Furthermore, suppose we know how the mass of $X$ is distributed into 6 intervals, so that if $...
- 21
2
votes
0
answers
1k
views
fitting non-normal multivariate distributions in R
I have many (n=317,823) observations on two variables. I want to fit a bivariate distribution to my observations, in order to identify descriptive features of the distribution (quantiles). However, my ...
- 51
2
votes
0
answers
333
views
Finding probability of a point using bivariate copula density
I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
- 21
2
votes
0
answers
57
views
Joint distribution of multivariate normal
Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$.
\begin{align*}
P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\
&...
- 681
2
votes
0
answers
190
views
In the choice of bandwidth for kernel density estimator. Why usually minimize MISE instead of minimizing ISE?
Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Let $\xi$ be a random variable with density function $f$ ...
- 491
2
votes
0
answers
49
views
Assess whether the generated data follows the distribution
Using the Inverse function method I managed to draw a sample of 500 random data from a Cumulative distribution function. $f(x)$,$F(x)$ and $F_X^{-1}(u)$ are as follows:
$$f_X=\frac{x}{5}exp\left({\...
- 171
2
votes
0
answers
668
views
Differential histogram bin calculation
I want to be able to minimize a difference between two distributions $P(x|\theta)$ and $Q$. I can choose Q (e.g. to be a Gaussian normal), but $P(x|\theta)$ is an unknown distribution, so I am ...
- 21
2
votes
0
answers
859
views
Fisher's Derviation of t Distribution
I was looking for a geometrical explanation of degrees of freedom and how they are determined by "the rank of a certain quadratic form", which is the definition my Professor gives but did not prove.
...
- 61
2
votes
0
answers
59
views
Goodness of Fit test with PDF, not data
I have a sample probability distribution and want to know how well a Gaussian would fit to it. However, I just have a probability distribution to test, and not an actual data sample. (Nor do I want ...
- 21
2
votes
0
answers
110
views
Algebra on random variables
I have the feeling this should be doable, or at least have an approximation, but I'm failing to find one.
Let's consider a random variable $C$, that belongs to a Truncated Exponential distribution ...
- 787
2
votes
0
answers
68
views
Density Estimation of multivariate random variable
I have a dataset ($300 \times 14$ matrix). This means it has 14 features and 300 observations.
$n=14$
$$
\begin{pmatrix}
a_{11} & 0 & \ldots & a_{1n}\\
0 & a_{22} & \ldots &...
- 143
2
votes
0
answers
170
views
Quantiles based on raw data or density
I've created violin plots of my data and included quantile lines. These were created with the geom_violin function from ggplot2 ...
- 257
2
votes
0
answers
806
views
min/max and probability distributions
We have two identically distributed, independent, uniform variables on interval $[0,1]$ : $X_{1}$, $X_{2}$. And $Y_{1}=\max(X_{1},X_{2})$, $Y_{2}=\min(X_{1},X_{2})$. I want to find distribution $f(y_{...
- 273
2
votes
0
answers
320
views
Density Estimators: Is average test set log-likelihood adequate to assess performance? Why isn't noise data used for low probability testing?
reading NADE papers, I noticed that the average (negative) log-likelihood is often used to assess the accuracy of the density estimation model.
At the beginning this made sense to me, but then I ...
- 938
2
votes
0
answers
55
views
Identify subset of population to match given moments
I have a Gaussian probability density function $f(x;\mu,\sigma)$. The mean and the standard deviation of this pdf is known.
I have an unknown probability density function $g(x)$. I only know $g(x)$ ...
2
votes
0
answers
145
views
Sum of Correlated Empirical pdf via Gaussian Copula
I'm new to R.
My goal is to calculate and plot the probability density function of the sum of 3 correlated empirical random variables (X1+X2+X3), given the correlation matrix.
I want to aggregate the ...
2
votes
0
answers
217
views
pdf of sum of squares error
If $Y_i$ (i=1,...,n) are iid N($\mu$,$\sigma^2$), how would I calculate the marginal pdf of the SSE?
On Wikipedia, I saw that $\sum$($Y_i$-$\bar{Y}$)$^2$ ~ $\sigma^2$$\chi$$^2$(n-1).
Any help would ...
- 21
2
votes
0
answers
3k
views
Comparing Two Kernel Density Estimates
I developing a kernel density estimate in Java for a control and test sample population given a certain treatment of the data. I am wondering the best way to test the similarity of the distributions ...
- 21