Fuzzy Histograms and Density Estimation

The probability density function is a fundamental concept in statistics. Specifying the density function f of a random variable X on Ω gives a natural description of the distribution of X on the universe Ω. When it cannot be specified, an estimate of this density may be performed by using a sample of n observations independent and identically distributed (X1, ..., Xn) of X . Histogram is the oldest and most widely used density estimator for presentation and exploration of observed univariate data. The construction of a histogram consists in partitioning a given reference interval Ω into p bins Ak and in counting the number Acck of observations belonging to each cell Ak. If all the Ak have the same width h, the histogram is said to be uniform or regular. Let 1lAk be the characteristic function of Ak, we have