# Questions tagged [density-estimation]

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

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### Why is histogram density estimation nonparametric?

My understanding of histogram density estimation: For $k$ predefined equal-width bins $(b_0, b_1], (b_1, b_2], ..., (b_{k-1}, b_k]$ and $n$ observations $x_1,...,x_n \in (b_0,b_k]$, we estimate ...
43 views

### Flexmix maxima are not where they are expected to be

For my dataset I have plotted the density with ggplot. As the data's density is multimodal (a total of 6 destinct modi) I tried to gain insight on the normal distributions associated to each modus. ...
40 views

### Estimating the distribution of a sum of two random variables if the family of one of the variables is known

Assume I have a random variable $Y=X_1+X_2$. I want to estimate the distribution $f$ of $Y$ given a sample $y_1,\ldots,y_N$. If this was all that is known about $Y$ the best way would probably be to ...
36 views

### Density estimation for time series data

Suppose I have a collection of time series for a number of subjects, say $y_{ij}$ are measurements for subject $i$ at time $t_{ij}$. The times are not uniformly sampled and each subject may have a ...
13 views

### Population Estimation And Conditional Probability

Let's say that I have a data set comprised of age data for a large number of individuals, as well as a unique identifier for each individual. For terminology sake let's call this my base population. ...
238 views

55 views

### Storing a probability distribution without saving single values

I saw this question "Storing a probability distribution without saving single values" on stackexchange and thought it deserved a statistical answer. Example Scenario I could see this problem ...
1 vote
104 views

### How to quantitatively compare parametric density fit and kernel density (KDE) fit of a multivariate data?

I am working on modeling the joint distribution of given multivariate data. I can fit some parametric distributions on the data and evaluate the fitted models by LogLiklihood and AIC values. However, ...
30 views

### Meshfree vs grid discretisation of a probability density

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ with density $f$. Suppose I cannot sample from $f$. I am coding an algorithm that involves the unknown density $f$, of course I need ...
1 vote
149 views

### Bias of kernel density estimator of pdf $f$, where $f$ has bounded first derivative $f'$

Let's say the kernel density estimator is given by $$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{X_i-x}{h_n}\right),$$ where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability ...
37 views

### Kernel Density Estimator: Misunderstanding in Taylor Series and the bias of KDE [duplicate]

Let's say the kernel density estimator is given by $\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K(\frac{X_i-x}{h_n})$, where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability distribution ...
1 vote
55 views

### Density Estimation of a Matrix-valued Random Variable?

It seems like the density estimation of a multivariate vector-valued random variable has been well studied, but what if one would like to estimate the probability density of a matrix-valued random ...
1 vote
24 views

### Normalized Density vs Unormalized Density [duplicate]

Edited: I have been researching about density function estimation from a sample of data, and I noticed that there are a lot of researches the estimate the density with the normalizing factor and ...
36 views

### How are probability density functions, that are computed from real-world datasets, stored and represented by computational software?

In probability and statistics, density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. The unobservable density ...
19 views

### Is it correct to say 'estimate of probability density function'?

This question is about terminology: I have a stochastic process from which I get a sample. Ideally I want to know the probability density function (pdf) associated with the process, but from the data ...
1 vote
54 views

### Density Estimation of High-dimensional Data

I would like to estimate the probability density function of a data set with a very large number of samples (50,000+) and a large number of continuous variables (2,048). Compute efficiency is somewhat ...
109 views

### How to fit a copula when zeros abound?

I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
1 vote
350 views

### Fitting a copula vs. directly fitting a multivariate distribution

I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be ...
1 vote
110 views

### How to estimate the conditional probability p(y|x) if y and x are both continuous but y is discrete given x?

For example, $P(Y=f_1(x)|X=x)=g_1(x)$, $P(Y=f_2(x)|X=x)=1-g_1(x)$. (The functions f1,f2 are unknown and need to be learned.) How can I estimate such a conditional probability? I guess that kernel ...
39 views

### How to prove symmetry of a Uniform kernel?

I am trying to prove this kernel is valid, $$K(x) = \frac{1}{2}I(-1 < x < 1)$$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is $\ge$ 0 for ...
450 views

### 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 ...
1 vote
46 views

### 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). (...
1 vote
45 views

### 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 &...
1 vote
119 views

### How to understand the density in machine learning?

We can calculate the conditional density using Eq.1.  p_{\theta, \Lambda}(y \mid \boldsymbol{x})=\frac{\exp \left(f_{\theta, \Lambda}(\boldsymbol{x})[y]\right)}{\sum_{k=1}^{n} \exp \left(f_{\...
1 vote
135 views

### 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 ...
871 views

### 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. ...
109 views

### 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 ...
1 vote
194 views

### 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 ...
1 vote
403 views

### 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?
1 vote
68 views

### 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 ...
1 vote
69 views

### 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 ...
337 views

### 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 ...
14 views

### 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 ...
1 vote
82 views

### 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, ...
1 vote
36 views

### 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 ...
203 views

### 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 ...
1 vote