Weighted.Desc.Stat (version 1.0)

w.kurtosis: weighted coefficient of kurtosis

Description

Assume that $x=(x_1, x_2, \cdots , x_n)$ is the observed value of a random sample from a fuzzy population. In classical and usual random sample, the degree of belonging $x_i$ into the random sample is equal to 1, for $1 \leq i \leq n$. But considering fuzzy population, we denote the degree of belonging $x_i$ into the fuzzy population (or into the observed value of random sample) by $\mu_i$ which is a real-valued number from [0,1]. Therefore in such situations, it is more appropriate that we show the observed value of the random sample by notation ${ (x_1, \mu_1), (x_2, \mu_2), \cdots , (x_n, \mu_n) }$ which we called it real-valued fuzzy data. The goal of w.kurtosis function is computing the coefficient of kurtosis (or, the weighted coefficient of kurtosis) value of $x_1, \cdots , x_n$ based on real-valued fuzzy data ${ (x_1, \mu_1), \cdots , (x_n, \mu_n) }$ by formula $$k = \frac{\frac{1}{\sum_{i=1}^{n} \mu_i} \sum_{i=1}^{n} \mu_i \left[ x_i - \bar{x} \right]^4}{s^4} - 3.$$

Usage

w.kurtosis(x, mu)

Arguments

x

A vector-valued numeric data which you want to compute its weighted coefficient of kurtosis.

mu

A vector of weights. The length of this vector must be equal to the length of data and each element of it is belongs to interval [0,1].

Value

• The weighted coefficient of kurtosis for the vector x, by considering weights vector mu, is numeric or a vector of length one.

Warning

The length of x and mu must be equal. Also, each element of mu must be in interval [0,1].

Examples

x <- c(1:10)
mu <- c(0.9, 0.7, 0.8, 0.7, 0.6, 0.4, 0.2, 0.3, 0.0, 0.1)
w.kurtosis(x, mu)

## The function is currently defined as
function(x, mu)  (( sum( mu*(x-w.mean(x,mu))^4 ) / sum(mu) ) / w.sd(x,mu)^4)-3