Questions tagged [empirical-cumulative-distr-fn]

Empirical cumulative distribution function: a step function increasing by $1/n$ at each unique $X$-value that occurred in the sample.

Filter by
Sorted by
Tagged with
0 votes
0 answers
9 views

Reference request: Almost sure weak convergence

(Reposting from Math StackExchange) I've encountered the term "almost sure weak convergence" of empirical measures in several places but haven't been able to find a textbook reference. Could ...
  • 121
5 votes
1 answer
230 views

Is there an equivalent to an ECDF with a "<" sign?

The empirical cumulative distribution function is defined as $$ F(x)=\frac{1}{n} \sum_{i=1}^{n} \mathbb I_{x_i \leq x} $$ Is there an equivalent interpretation of this function as: $$ F(x)=\frac{1}{n} ...
0 votes
0 answers
64 views

Estimating density function at specific percentile using empirical cdf

Summary I'm running an experiment where I'm using the empirical CDF of a known random variable to approximate the density of the random variable at a specific percentile. In general I'm interested in ...
  • 1
0 votes
0 answers
15 views

Simplifying empirical distribution functions

I have an empirical distribution function, that is I have say 2000 points $x_i$ each with a probability $w_i>0$ and $\sum w_i=1$ . Now, I want to reduce this empirical probability distribution by ...
0 votes
0 answers
29 views

Convergence of standarized empirical distribution functions to distribution of standarized sample mean

Suppose we have a family of $i.i.d.$ random variables $\{X_n\}_{n=1}^\infty$, with distribution function $F(\cdot;\mu,\sigma^2)$, with mean $\mu\in\mathbb{R}$ and variance $\sigma^2>0$. By virtue ...
2 votes
1 answer
46 views

Covariance of the empirical probability mass function

Suppose a discrete random variable $Y$ takes $k$ levels of different values $y_1,y_2,...,y_k$. Let $P(Y=y_k):=p_k$. Suppose we have $n$ i.i.d. samples of $Y$, my question is: How can we compute the ...
2 votes
0 answers
19 views

Simulating New Observations from a Empirical Distribution Function [duplicate]

Recently, I found out that it is possible to simulate new observations from an Empirical Distribution Function (EDF) . Suppose I collected some data (e.g. heights of baksetball players): $x_1, x_2,\...
  • 6,028
0 votes
0 answers
35 views

Measure for similarity of two distribution ECDF with sensitivity to the tails

Currently, I am looking for a measure to quantify the overall dissimilarity or similarity of two sample distributions (possibly of different size). I would like to compare observed data with model ...
  • 11
0 votes
0 answers
13 views

Is the empirical distribution the only unbiased distribution estimator?

Given $n$ samples, if $\hat{p}$ is the empirical distribution of $p \in \Delta_{\mathcal{X}}$ where $\mathcal{X}$ is a finite domain, we know that $\mathbb{E} \hat{p} - p = 0$. Is the empirical ...
1 vote
0 answers
28 views

Empirical distribution for feature binning

In paper "A simple yet effective baseline for non-attributed graph classification" (https://arxiv.org/pdf/1811.03508.pdf) authors use empirical distribution for feature binning. Precisely, ...
  • 229
2 votes
1 answer
76 views

symmetrization in glivenko-cantelli proof

In this proof of the Glivenko-Cantelli theorem, page 2 of these notes, two types of symmetrization are used. The first transforms the sup of the centered empirical cdf $$P(\sup_{z\in\mathbb{R}}|(1/n)\...
1 vote
1 answer
28 views

rank to quantile estimate?

R> x=c(92, 3, 1, 4, 15, 4) R> rank(x) [1] 6.0 2.0 1.0 3.5 5.0 3.5 Given the rank results of an input vector of sampled data, I can estimate the quantiles ...
0 votes
0 answers
39 views

Does empirical cumulative distribution function (ECDF) has its Akaike information criterion (AIC)?

Working on multivariate distribution fitting, and right now I have marginal univariate transform models and a copula model. Was thinking if I pick ECDF for marginals, do I still have meaningful AIC? ...
1 vote
1 answer
61 views

How to average several posteriors distributions from a Monte Carlo Simulation

Say you produce several posteriors distributions from different runs of the same model under different seeds. That is to say you have something like the following: ...
  • 239
0 votes
0 answers
15 views

Doubt about distribution function of continuous distribution with (Generalized) Pareto tails

In this paper: https://www.sciencedirect.com/science/article/pii/S0167947315003163 They proposed an estimation method for the parameters of the Generalized Pareto Distribution. Defining: $$ F_n(x) \...
  • 69
0 votes
0 answers
13 views

Expressions for entropy of an observed set of points (continuous variables) [duplicate]

The entropy of a discrete variable can be defined as $$H(x) = - \sum_{i=1}^n p(x) \log p(x) $$ or for continuous distributions and a density $f(x)$ we could compute $$h[f] = \int_{\mathbb{X}} f(x) \...
2 votes
1 answer
223 views

Distance between two categorical distributions

I want to test whether two (empirical) categorical distributions taking on $K$ possible values (e.g. 5, with no innate underlying ordering) with associated (empirical) probabilities $p_k$ are the same....
  • 67
0 votes
0 answers
41 views

Probabilities from ecdf and normalised data - is this acceptable difference? (using R)

I have a dataset of 500 observations, definitely not normally distributed: The minimum is around 50 000 in the data. If I use ecdf to predict e.g. 100 000, I get: ...
1 vote
1 answer
156 views

What is a "bootstrap from the Empirical CDF"?

I am given the following set of questions. In all the questions the basic background is a sample X_1, ... , X_n. If n=3 and you observe the numbers 1, 2, and 4, what is the sample median. If you ...
  • 531
0 votes
1 answer
134 views

Determine a product's rating given a known 3-sigma tolerance

Consider a 100Ω resistor with a 10% tolerance. We can assume this is the 3-sigma value since this is typical in manufacturing. Thus, we can expect ≈99.73% of such resistors to range from 90Ω to 110Ω. ...
  • 103
1 vote
1 answer
48 views

Difference in means

For a df that looks something like the following ...
0 votes
1 answer
111 views

Smoothed CDF to calculate asymptotic normality

If we have the following estimator: $\hat{F_Z}(z)=\frac{1}{N}\sum_{i=1}^N1\{Z_i\leq z\}$. The CDF of $Z$ is defined as $F_Z(z)=Pr(Z\leq z)$. $Z_1, ..., Z_N$ is i.i.d. data. What would be the steps to ...
2 votes
0 answers
101 views

Plug-in principle with kernel density estimate

The plug-in principle says that to estimate a statistical functional of the form $$ T(\mu) = \int f(x)\ d\mu(x) $$ we can replace $\mu$ with the empirical distribution $\mu_n$ depending on data $X_1,\...
  • 186
7 votes
1 answer
207 views

What are the simplest examples of nonlinear statistical functionals?

I am reading Wasserman's book "All of Statistics" in which he defines a statistical functional as any function $T(F)$ of the cumulative distribution function $F(x)$ that outputs a real ...
  • 603
2 votes
1 answer
47 views

Picking a specific estimated CDF from a set of CDFs provided by an ECDF

Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...
  • 23
2 votes
2 answers
104 views

Notation for ECDF

I'm reviewing statistical functionals and U-statistics, trying to make notes, and I am tripping on notation. From my understanding, $X$ is used to denote a random variable and $x$ is used to denote an ...
2 votes
1 answer
234 views

Density estimation from ECDF - numerical derivatives and scaled domains

Suppose we want to get a density estimate of some data X. One way is to compute the empirical CDF, ...
  • 43
1 vote
0 answers
28 views

Nonparameteric Empirical Estimator For Stochastic Process

Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...
  • 544
0 votes
0 answers
114 views

How to deal with a PDF that takes CDF data as input? (Metalog distributions)

I am trying to utilise the Metalog distribution in a Machine Learning project. For this project, I need to be able to obtain likelihoods using the PDF of the distribution. https://en.wikipedia.org/...
1 vote
0 answers
87 views

Is it possible to convert a pdf obtained with density() to an ecdf?

Consider the following code that gives us (an estimate of) the pdf of a random variable $X$: X = rnorm(100,10,1) XDensity = density(X) I want to obtain the ecdf of ...
  • 860
2 votes
0 answers
152 views

Fitting a function to ECDF [closed]

I have some ECDFs. I would like to summaries the ECDFs with functional approximations. I was thinking that a polynomial, spline, or other line fitting procedure would generate a nice parsimonious ...
  • 983
0 votes
0 answers
26 views

Computing the most representative sample of a random variable

Let $X$ be a real-valued random variable and $n > 0$. Using numerical methods, how can we find the vector $\vec v$ of $n$ real numbers that is most characteristic of $X$, in the sense that the ...
  • 19.3k
3 votes
1 answer
154 views

Median from two unmergable datasets

I am trying to calculate the "overall" median of a variable that is spread across two datasets. I have access to the raw data in each dataset but can't bring their raw data together. What ...
2 votes
0 answers
578 views

Why are people using KS statistic to choose cutoff for binary predictions?

I've seen that people are sometimes using Kolmogorov-Smirnov statistic to determine a cutoff in a binary classification models, such as logistic regression. However, i do not fully understand why and ...
  • 411
3 votes
1 answer
129 views

Obtaining an expression for empirical mean from empirical CDF definition

This is my first post so I will try to be as clear and concise as possible. I am doing a course in statistics and we define the true mean of a random gaussian variable to be as follows: $\mu$ = $\int_{...
1 vote
1 answer
691 views

Confidence Interval of p-Quantile from Empirical CDF

I am trying to provide an interval estimate for the 0.8-quantile of some numeric data, which is assumed to be an IID sample from some unknown, continuous distribution. I constructed an Empirical CDF ...
0 votes
0 answers
145 views

Empirical estimation of conditional distribution $Y|X$ at the boundaries of X

I want to estimate conditional distributions of Y | X. Where X contains several continuous covariates. I'm coding in R. I tried several methods so far, but none gives me entirely satisfactory results ...
  • 157
1 vote
1 answer
75 views

Does the Glivenko-Cantelli theorem work back and forth?

If we have a sample $X_1, X_2, \ldots, X_n \sim F$ then $\hspace{1mm}sup_x|F_n(x) -F(x)|\xrightarrow{a.s./p}0$. Now, if I can come up with a theoretical cdf $F$ such that $\hspace{1mm}sup_x|F_n(x) -F(...
  • 43
0 votes
0 answers
322 views

Is it accurate to take the maximum distance between CDF and ECDF only at the edges? (Kolmogorov-Smirnov Test)

I have two samples, one obtained empirically and the other is the result of a simulation. I want to tune the simulation so that the result resembles the reality, for that I will minimize the KS ...
0 votes
1 answer
884 views

Example of cumulative distribution function and the empirical distribution function [closed]

A random of 100 rolls of the die. The outcomes 1, 2, 3, 4, 5, 6, occurred 13, 19, 10, 17, 14, 27 times, respectively. Calculate the cumulative distribution function and the empirical distribution ...
0 votes
1 answer
775 views

How can i find empirical survival function using survival function in R?

The survival function is given by: S(y; α, λ) = (α/α−1)* (1 − α^(−e^(−λy ))), if α is not equal 1 = e^(−λy) if α =1 y = 1 4 4 7 11 13 15 15 17 18 19 19 20 20 ...
  • 25
37 votes
4 answers
5k views

Intuitive explanation of Kolmogorov Smirnov Test

What is the cleanest, easiest way to explain someone the concept of Kolmogorov Smirnov Test? What does it intuitively mean? It's a concept that I have difficulty in articulating - especially when ...
  • 2,461
3 votes
0 answers
154 views

Comparing empirical distributions with an interaction effect

My experiment includes two subject groups: group1 and group2. Both groups undergo a behavioral test, the result of which is called “movement celerity” (this is my dependent variable). Both groups are ...
1 vote
1 answer
159 views

Fit cdf and pdf from points on empirical cdf

I have cumulative counts with respect to a variable x, which looks like: ...
  • 1,112
1 vote
1 answer
123 views

Can we apply the Probability Integral Transform to Dependent Random Variables?

Let's suppose we deal with a non-homogeneous Poisson process having intensity function λ(t), t ≥ 0. The event times X1, X2, … of ...
  • 263
0 votes
1 answer
52 views

How to perform van Zyl's (2018) ecf-based normality test in R [closed]

I have been reading quite a bit on empirical characteristic function (ecf) based normality tests, but cannot find any functions to perform such a test in R and lack the mathematical ability to figure ...
  • 45
2 votes
0 answers
44 views

What are “absolute ECDFs” called (if anybody uses them)?

By absolute ECDF, I am referring to an ECDF that shows the absolute number of samples above some value as opposed to the fraction of such samples. For example, the bottom plot below displays such an ...
  • 2,257
1 vote
0 answers
24 views

Quantile-calculation example: Why the term $c$?

I need help to interpret a solution to the following example: Based on historical claim amounts $x_1,...,x_{47}$, which are assumed to be outcomes from an unknown claim distribution, you are ...
0 votes
2 answers
945 views

Simulate data from ECDF in R [closed]

Consider the following 11 continuous non-negative observations: 0, 0.3, 0.31, 0.33, 0.37, 0.49, 0.51, 0.53, 0.59, 0.6 Obtain the empirical cumulative distribution function for these observations. ...
2 votes
1 answer
175 views

How to quantify distribution concentration in cdf? [closed]

For example, let's say I have a list of users, each user has its revenue. I can plot the cdf of both user and revenue, to see if there is some concentration, for example, may be 40% user contribute ...
  • 1,175