# WeightedData

WeightedData[{x1,x2,},{w1,w2,}]

represents observations xi with weights wi.

WeightedData[{x1,x2,},fn]

represents observations xi with weighting function fn.

# Details • WeightedData augments data with weights for each data point.
• The data {x1,x2,} and weights {w1,w2,} should be lists of equal length.
• The weight function fn is applied to the list {x1,x2,} and should return an explicit list of weights {w1,w2,}.
• WeightedData can be used in statistics functions including:
•  Mean,Variance,… descriptive statistics functions EmpiricalDistribution,… nonparametric distribution estimation EstimatedDistribution,… parametric distribution estimation
• WeightedData[{x1,x2,}] gives data with equal weights.
• Properties of WeightedData can be obtained by specifying WeightedData[]["property"].
• A list of available properties can be obtained using WeightedData[]["Properties"].
• WeightedData has the following properties:
•  "EmpiricalPDF" data values and estimated weights "InputData" unweighted input data values "MetaInformation" a list containing meta-information rules "Weights" a list containing the data weights

# Examples

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## Basic Examples(1)

Create data with weights:

Compute a weighted Mean and StandardDeviation:

## Scope(10)

Create weighted univariate data:

Some weighted descriptive statistics:

Add weights to a set of multivariate values:

A set of weighted multivariate descriptive statistics:

Use a pure function to create weighted data values:

Weighted means and variances:

Visualize the impact of the various weighting schemes:

Fit nonparametric distributions to weighted data:

Fit parametric distributions to weighted data:

Compare the estimated and empirical distributions:

Extract the input data from a WeightedData object:

Compare the distributions of the unweighted and weighted data:

Obtain the weights from a WeightedData object:

Visually inspect the effect of the weights on the data values:

Compute a weighted mean from the empirical PDF:

The weighted mean can be computed directly using Mean:

Find the weighted average of an irregularly sampled TimeSeries:

Compare with the average of values:

Create weighted data involving quantities:

Some weighted descriptive statistics:

## Applications(2)

Create a weighted histogram:

Estimate confidence interval for maximum likelihood estimates of distribution parameters:

Apply fractional random weight bootstrap to estimate confidence interval, by repeating weighted estimation with weights sampled from a DirichletDistribution with unit parameters:

Generate one thousand estimates of the distribution parameters:

Visualize bootstrap estimates:

Fit joint Gaussian distribution to the bootstrapped parameters:

## Properties & Relations(2)

Descriptive statistics are based on the underlying EmpiricalDistribution:

Sample estimates are given when they differ from population estimates:

WeightedData works for TimeSeries objects:

For , the weights are proportional to :

Compare with binsampled time average:

Compute integration limits: