Questions tagged [density-estimation]

Estimation of probability density functions, whether by kernel density estimation, log-spline estimation or other methods.

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Density estimation on labelled data

I'm looking for a nonparametric density estimation for a particular classification. Let A={2,3,5,7,10,13} And Dens be the density of A Dens={x: x lies within the density region of A} , based on a ...
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Sufficient Statistic and UMVUE [closed]

I have this example and I'm struggling to solve it Say that I have this Independent and identically distributed sample with density $ f(x,\theta)= \frac{3x^2}{\theta^3} , 0 \le x \le \theta,$ $\...
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Estimate parameters of an unknown negative binomial distribution based on known distribution

The PDF of a known NBD given in Equation (1). The parameter a and r are function of $μ$ = sample mean, and $s^2$ = sample variance, as given in Equation (2) and (3) respectively. $r$ = number of ...
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Universal Approximation Capabilities of Mixture of Weibulls

Can a mixture of $N$ Weibull distributions approximate any continuous density with non-negative support, if $N$ is sufficiently large? (If so, a reference to the proof would be greatly appreciated). (...
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Kernel Density: How do the terms 'global' and 'pilot' translate?

I nearly most of the articles on kernel smoothing or concepts that use kernel density estimations, authors speak of 'pilot' and 'global'. https://link.springer.com/article/10.1023/A:1008925425102 &...
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How to understand the density in machine learning?

We can calculate the conditional density using Eq.1[3]. $$ p_{\theta, \Lambda}(y \mid \boldsymbol{x})=\frac{\exp \left(f_{\theta, \Lambda}(\boldsymbol{x})[y]\right)}{\sum_{k=1}^{n} \exp \left(f_{\...
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Kernel Density Estimation using a Two-Boundary-Kernel à la Jones

I'm trying to understand how to perform a KDE on a bounded support, i.e. with lower and upper boundary, when using a kernel that is specifically designed to ensure consistency/$h^2$-bias at the ...
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Kernel Density Estimate for Cauchy

As far as I understand, kernel density estimation does not make any assumptions on the moments of the underlying density, and just requires smoothness. The Cauchy density function is quite smooth. ...
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Approximation of a polynomial via histogram

Note: I originally tried to pose this question generally, without discussing the specific type of stochastic process. I hope that this can still be an interesting question generally. Assume that we ...
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Exponentially distributed bins for viewing and analysing the tails of a histogram

Is there a rule for visualising and analysing density estimates of heavy tailed observations on a log-log scale? For example, equispaced bins and exponentially distributed bins: ...
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How to sample from a distribution approximated by a Neural Network?

There are a few models already that approximate distributions with a neural network i.e.: energy models define a density function $f(x)= e^{S(x,w)}/Z$ where $S$ is a neural network and $Z$ is a ...
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Bandwidth Selection for Kernel Density Estimation

Are there any heuristics for selecting the bandwidth for kernel density estimation? In other words, is a spiky curve better or a smooth one?
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I have difficulties to understand the input normalization - density estimation in the same context in ML or DL

I know (by experimenting with different ML and DL algorithms) that input normalization helps to improve the performance of the model. When we do normalization in training, with the same mean and ...
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Understanding the methodology of evaluating density forecats

I have difficulties with under understanding the idea that Diebold came up with in 1998 in his essay about the evaluation of the accuracy of forecasting density p(y), he used the probability integral ...
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Optimal rate of convergence of nonparametric density estimators

Suppose that $X_1, X_2, \dots, X_n$ forms an independent and identically distributed sample from some $d$-dimensional probability distribution with unknown probability density function $f$. Let $x$ be ...
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Is possible to compare two density distributions 'trends'?

I have responses from two groups (A and B) on a confidence rating about a distance judgment task. The participants saw pairs of stimuli and after they were asked to rate their confidence about the ...
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Is there a model particularly suited to predicting a discrete density function?

I would like to forecast a demand density function over the 24 hours of a day. For example, consider customers at a bank with a number of tellers. I would like to forecast how the demand for tellers ...
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Density estimation for 50 input variables - individually or together

I am a beginner to statistics and was wondering if it is possible to estimate the density where there are 50 input variables. Do I perform density estimation for each variable individually then ...
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Expectation of kernel density estimators using sharpened data

My question regards the proof of the bias of the kernel density estimator obtained using "sharpened" data. The method comes from the paper by Choi and Hall (1999). Specifically, assume $X_1, ...
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Estimating conditional density, each observation being conditional on belonging to some range of values

I have iid observations $(Y_1, X_1), (Y_2, X_2), ... , (Y_n, X_n)$, where the conditional density of $Y_i|X_i$ is known to have form $$f(y|x) = \begin{cases} \frac{D(y)}{\int_{||t-x|| \leq 1} D(t)...
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Driver based forecasting using past distributions

I have reduced my original forecasting problem (Short context : I need to forecast hotel bookings and checkins for the next 3 months. I already have a reasonable forecast for bookings and need to ...
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Determine traffic speed distribution while driving

I assume that the vehicle speed of cars on a highway is normally distributed around the posted speed limit. I could verify this by sitting by the road with a radar gun and measure vehicle speed for a ...
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How is Dowd's (2007) resampling procedure supposed to mitigate the problem of autocorrelated multiple-step-ahead forecasts?

Dowd "Validating multiple‐period density‐forecasting models" (2007) considers evaluation of multiple-step-ahead density forecasts. There is a know problem of dependence between forecast ...
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Practical Uses of Kernel Density Estimators

Perhaps this question is too broad, but I would like to know - how does one use a kernel density estimate in practice? I know of course that one can use it to draw pretty pictures on top of histograms,...
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Gain of power by a smart choice of goodness of fit test

Suppose one would like to test that a sample of observations comes from Uniform(0,1) distribution. Instead of applying the Kolmogorov-Smirnov test on the sample, one may first apply the inverse CDF (...
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Bayesian Parameter estimation (Pattern Classification by Duda, et al

I have been trying to solve question 17 of chapter 3 (Maximum Likelihood and bayesian estimation) of the book "Pattern Classification" by Duda, et al. The question goes as follows: Now the ...
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Interpretation of test set negative log likelihood in neural density estimation applications

I have seen people splitting a dataset into a train and test sets and learning the parameters of a mixture density network using the negative log likelihood cost function on the train set and ...
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Measure for evaluating a density estimation procedure

Given an implementation of a multivariate density estimation scheme, what would be a suitable measure to evaluate the accuracy of the procedure? I am currently evaluating the procedure using three ...
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Can a neural net approximate any conditional density asymptotically?

Assume that the conditional density of $ y \vert x $ is a Beta distribution for all values of x. Can a Beta distribution with parameters computed by a neural net, i.e. Beta($\hat{\alpha}$, $\hat{\beta}...
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cross validation for distribution

Assume we have a linear regression model. Then, assume we have response values $y_{1}, \dots, y_{n}$ and covariates vectors $\textbf{x}_{1}, \dots, \textbf{x}_{n}$. To assess the prediction ...
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Methods for Predicting Destination Locations based on Start Locations in 2D Space

I have location data $(x_1, x_2)$ and the associated labels $(y_1, y_2)$, which can be interpreted as origin and destination points in physical space. Now I would like to predict destination locations ...
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what is the number of bits assigned by a generative model to a K-class dataset, in theory

Consider a dataset, something like MNIST or CIFAR-10, that consists of (for example) 50000 images taken from 10 classes and balanced with an equal number in each class. For a discriminitive classifier,...
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Kernel Density Estimation for a Discrete Variable

I was tying to estimate the distribution for a discrete variable. However, suddenly I thought that "Is a simple histogram sufficient? because I have observations for every evaluation point" ...
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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,\...
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Probability Density Estimation vs Function Approximation [closed]

I have a function $f: \mathbb{R} \to \mathbb{R}_+$ and I would like to estimate it. The data pairs $\{(x_i, f(x_i))\}$ arrive at different times $t$. I have two questions: In this case, since the ...
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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, ...
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histogram vs. kernel in density estimation

Assume we have a problem of estimation of a density $f(x)$ over an interval $[0, 1]$. Can a regular histogram (i.e. with equal-sized bins) be viewed as some kind of a kernel?
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Geometric distribution and entropy

According to wikipedia, among all discrete probability distributions supported on $\{1, 2, 3, ... \}$ with given expected value $\mu$, the geometric distribution X with parameter $p = \frac{1}{ \mu} $ ...
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Find x-axis value given a tail probability of an custom PDF estimated using approxfun(density()) in r

I know qnorm() function returns the z-statistic of the applicable normal distribution that corresponds to a given probability (see example below). ...
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Interpreting Kernel density Plot

Below I am showing the kernel density with the size of the informal economy, and would appreciate support on interpreting this. For instance, what does the of the Kdensity line around .017 represent ...
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Why is bw.nrd0() function and bw="nrd0" showing different results of density plot?

According to density function documentation in R (source), they use "nrd0" as default setting. However, when I tried <...
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Extraction of modes from a multi-modal density function

I am trying to extract modes from a multi-modal density function and not just peaks. For example, in the two density functions below (images), I would like to extract the curves contained in the black ...
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Density function estimation from histogram with infinite bin

Given a histogram such as the following Histogram 1 Bin Count −3.5 to −2.51 9 −2.5 to −1.51 32 −1.5 to −0.51 109 −0.5 to 0.49 180 0.5 to 1.49 132 1.5 to Inf 38 What would be the best approach ...
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About computation of Brier score

Assume that we have some count data $x_{1}, \dots, x_{n}$, generated by probability mass function $\textbf{p} = \{p_{1}, \dots, p_{s} \}$. Let $\hat{\theta}$ be some estimator of $\textbf{p}$. In ...
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Peaks of estimated probability distribution are always lower than those of true distribution - why?

I have some code (shown below) for sampling from and then estimating a normal mixture distribution in one dimension. When I plot the estimate (blue) against the true distribution (black) I get images ...
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Hypothesis Testing Probability Density Estimates [closed]

Is there a good way to test an probability density estimate against observed data? I know there exist various approaches to probability density estimation and testing individual parameters of density ...
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6 votes
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Calculating the area under two overlapping distribution

I have two overlapping frequency distribution, one of the buyers' demand or willingness to pay and the other one is seller's reservation price frequency distribution. The two distributions overlap and ...
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can I estimate density function of 2d/3d data with kernel smoothing (e.g. ks package R), or are there better estimation methods

I have a 2d matrix of positive values (non integer), where the values can be thought of intensity at an x,y coordinate indexed by the row and column. I want to estimate a density function across this ...
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What determines the functional form of maximum entropy constraints?

I'm familiar with the maximum entropy (ME) principle in statistical mechanics, where, for example, the Boltzmann distribution $p(\epsilon_i|\beta)$ is identified as the ME distribution constrained by ...
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Convex hull version of density estimation (or lines of constant density)

Background: So I had a thought, tried it out, and liked what it did. I'm sure someone else has done this. It feels very convenient. It also gives an interesting take on robust nonparametric density ...
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