Type: | Package |
Title: | Count and Continuous Generalized Variability Indexes |
Version: | 1.0.2 |
Date: | 2021-02-25 |
Author: | Aboubacar Y. Touré and Célestin C. Kokonendji |
Maintainer: | Aboubacar Y. Touré <aboubacaryacoubatoure.ussgb@gmail.com> |
Description: | Firstly, both functions of the univariate Poisson dispersion index (DI) for count data and the univariate exponential variation index (VI) for nonnegative continuous data are performed. Next, other functions of univariate indexes such the binomial dispersion index (DIb), the negative binomial dispersion index (DInb) and the inverse Gaussian variation index (VIiG) are given. Finally, we are computed some multivariate versions of these functions such that the generalized dispersion index (GDI) with its marginal one (MDI) and the generalized variation index (GVI) with its marginal one (MVI) too. |
License: | GPL-3 |
Encoding: | UTF-8 |
LazyData: | true |
RoxygenNote: | 7.1.1 |
NeedsCompilation: | no |
Packaged: | 2021-02-25 17:17:37 UTC; PC |
Repository: | CRAN |
Date/Publication: | 2021-02-26 00:00:05 UTC |
Count and continuous generalized variability indexes
Description
Univariate Poisson dispersion index di.fun
, univariate exponential variation index vi.fun
functions are performed. Next, the univariate binomial dispersion index dib.fun
, the univariate negative binomial dispersion index dinb.fun
and the univariate inverse Gaussian variation index viiG.fun
functions are given. Finally, the generalized dispersion index and its marginal one gmdi.fun
, the generalized variation index and its marginal one gmvi.fun
functions are displayed.
Details
- The univariate Poisson dispersion index (DI) and its relative versions with respect to binomial and negative binomial distributions:
The Poisson dispersion phenomenon is well-known and very widely used in practice; see, e.g., Kokonendji (2014) for a review of count (or discrete integer-valued) models. There are many interpretable mechanisms leading to this phenomenon which makes it possible to classify count distributions and make inference; see, e.g., Mizère et al. (2006) and Touré et al. (2020) for approximative statistical tests. Introduced from Fisher (1934), the Poisson dispersion index, also called the Fisher dispersion index, of a count random variable
X
onS=\{0,1,2,\ldots\}=:N_0
can be defined asDI(X)=\frac{VarX}{EX},
the ratio of variance to mean. In fact, the positive quantity
DI(X)
is the ratio of two variances sinceEX
is the expected variance under the Poisson distribution. Hence, one easily deduces the concept of the relative dispersion index (denoted by RDI) by choosing another reference than the Poisson distribution. Indeed, ifX
andY
are two count random variables on the same supportS\subseteq N_0
such thatEX=EY
, thenRDI_Y(X):=\frac{VarX}{Var Y}=\frac{DI(X)}{DI(Y)} >=< 1;
i.e.
X
is over-, equi- and under-dispersed compared toY
ifVarX > VarY
,VarX = VarY
andVarX < VarY
, respectively.
For instance, the binomial dispersion index is defined asRDI_B(X)=\frac{var X}{EX(1-EX/N)},
where
N\in \{1,2,\ldots\}
is the fixed number of trials. Also, the negative binomial dispersion index is defined asRDI_NB(X)=\frac{varX}{EX(1+EX/ \lambda)},
where
\lambda > 0
is the dispersion parameter. See also, Weiss (2018, page 15) and Abid et al. (2021) for more details.
- The univariate variation index (VI) and its relative version with respect to inverse Gaussian distribution:
More recently, Abid et al. (2020) have introduced the exponential variation index for positive continuous random variable
X
on[0,\infty)
asVI(X)=\frac{VarX}{(EX)^2}.
It can be viewed as the squared coefficient of variation. It is used in the framework of reliability to discriminate distribution of increasing/decreasing failure rate on the average (IFRA/DFRA); see, e.g., Barlow and Proschan (1981) in the sense of the coefficient of variation. See also Touré et al. (2020) for more details. Following RDI, the relative variation index (RVI) is defined, for two continuous random variables
X
andY
on the same supportS = [0,\infty)
withEX = EY
, byRVI_Y(X):=\frac{VarX}{VarY}=\frac{VI(X)}{VI(Y)} >=< 1;
i.e.
X
is over-, equi- and under-varied compared toY
ifVarX > VarY
,VarX = VarY
andVarX < VarY
, respectively. For instance, the inverse Gaussian variation index is defined asRVI_IG(X)=\lambda^2\frac{var X}{(EX)^3},
where
\lambda > 0
is the shape parameter.
Next, consider the following notations. Let Y
= (Y_1,\ldots,Y_k)^{\top}
be a nondegenerate count or continuous k
-variate random vector, k\ge 1
. Let also EY
be the mean vector of Y
and covY
= (cov(Y_i,Y_j) )_{i,j\in \{1,\ldots,k\}}
the covariance matrix of Y
.
- The generalized dispersion index (GDI) and marginal dispersion index (MVI):
Kokonendji and Puig (2018) have introduced the generalized dispersion index for count vector
Y
on\{0,1,2,\ldots\}^k
byGDI(Y) =\frac{\sqrt{EY}^{\top} ( covY)\sqrt{EY}}{EY^{\top}EY}.
Note that when
k=1
,GDI(Y)
is just the classical Fisher dispersion index DI.GDI
(Y
) makes it possible to compare the full variability ofY
(in the numerator) with respect to its expected uncorrelated Poissonian variability (in the denominator) which depends only onEY
.GDI(Y)
takes into account the correlation between variables. For only taking into account the dispersion information coming from the margins, the authors defined the "marginal dispersion index":MDI(Y) = \frac{\sqrt{EY}^{\top}( diag varY )\sqrt{EY}}{EY^{\top}EY}=\sum_{j=1}^k\frac{\{E(Y_j)\}^2}{EY^{\top}EY} DI(Y_j).
- The generalized variation index (GVI) and marginal variation index (MVI):
Similarly, Kokonendji et al. (2020) defined the generalized variation index for positive continuous vector
Y
on[0, \infty)^k
byGVI(Y) =\frac{EY^{\top} ( covY) EY}{(EY^{\top}EY)^2}.
Remark that when
k=1
,GVI(Y)
is the univariate variation index VI.GVI(Y)
makes it possible to compare the full variability ofY
(in the numerator) with respect to its expected uncorrelated exponential variability (in the denominator) which depends only onEY
. Also,GVI(Y)
takes into account the correlation between variables. To only take into account the variation information coming from the margins, Kokonendji et al. (2020) defined the "marginal variation index":MVI(Y) = \frac{EY^{\top}( diag varY )EY}{(EY^{\top}EY)^2}=\sum_{j=1}^k\frac{(EY_j)^4}{(EY^{\top}EY)^2} VI(Y_j).
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
Maintainer: Aboubacar Y. Touré <aboubacaryacoubatoure.ussgb@gmail.com>
References
Abid, R., Kokonendji, C.C. and Masmoudi, A. (2020). Geometric Tweedie regression models for continuous and semicontinuous data with variation phenomenon, AStA Advances in Statistical Analysis 104, 33-58.
Abid, R.,Kokonendji, C.C. and Masmoudi, A. (2021). On Poisson-exponential-Tweedie models for ultra-overdispersed count data, AStA Advances in Statistical Analysis 105, 1-23.
Barlow, R.A. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing : Probability Models, Silver Springs, Maryland.
Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies, Annals of Eugenics 6, 13-25.
Kokonendji, C.C., Over- and underdispersion models. In: N. Balakrishnan (Ed.) The Wiley Encyclopedia of Clinical Trials- Methods and Applications of Statistics in Clinical Trials, Vol.2 (Chap.30), pp. 506-526. Wiley, New York (2014).
Kokonendji, C.C. and Puig, P. (2018). Fisher dispersion index for multivariate count distributions : A review and a new proposal, Journal of Multivariate Analysis 165, 180-193.
Kokonendji, C.C., Touré, A.Y. and Sawadogo, A. (2020). Relative variation indexes for multivariate continuous distributions on [0,\infty)^k
and extensions, AStA Advances in Statistical Analysis 104, 285-307.
Mizère, D., Kokonendji, C.C. and Dossou-Gbété, S. (2006). Quelques tests de la loi de Poisson contre des alternatives géenérales basées sur l'indice de dispersion de Fisher, Revue de Statistique Appliquée 54, 61-84.
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Weiss, C.H. (2018). An Introduction to Discrete-Valued Times Series. Wiley, Hoboken NJ.
Function for DI
Description
The function computes the univariate Poisson dispersion index for a count random variable.
Usage
di.fun(X)
Arguments
X |
A count random variable |
Details
di.fun
provides the univariate Poisson dispersion index (Fisher, 1934). We can refer to Touré et al. (2020) for more details on the Poisson dispersion index.
Value
Returns
di |
The Poisson dispersion index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies, Annals of Eugenics 6, 13-25.
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Examples
X<-c(6,7,8,9,8,4,7,6,12,8,0)
di.fun(X)
T<-c(61,72,83,94,85,46,77,68,129,80,10,12,12,3,4,5)
di.fun(T)
Function for DIb
Description
The function computes the binomial dispersion index for a given number of trials N\in \{1,2,\ldots\}
.
Usage
dib.fun(X, N)
Arguments
X |
A count random variable |
N |
The number of trials of binomial distribution |
Details
dib.fun
computes the dispersion index with respect to the binomial distribution. See Touré et al. (2020) and Weiss (2018) for more details.
Value
Returns
dib |
The binomial dispersion index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Weiss, C.H. (2018). An Introduction to Discrete-Valued Times Series. Wiley, Hoboken NJ.
Examples
X<-c(12,9,0,8,5,7,6,5,3,4,9,4)
dib.fun(X,12)
Y<-c(0,0,1,1,0,1,1)
dib.fun(Y,7)
Function for DInb
Description
The function computes the negative binomial dispersion index for a given dispersion parameter l\in (0,\infty)
.
Usage
dinb.fun(X, l)
Arguments
X |
A count random variable |
l |
The dispersion parameter of negative binomial distribution |
Details
dinb.fun
computes the dispersion index with respect to negative binomial distribution. See Touré et al. (2020) and Abid et al. (2021) for more details.
Value
Returns
dinb |
The negative binomial dispersion index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Abid, R.,Kokonendji, C.C. and Masmoudi, A. (2021). On Poisson-exponential-Tweedie models for ultra-overdispersed count data, AStA Advances in Statistical Analysis 105, 1-23.
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Examples
X<-c(12,9,0,8,5,7,6,5,3,4,9,4)
dinb.fun(X,12)
Y<-c(0,6,1,3,4,2,5)
dinb.fun(Y,7)
Function for GDI and MDI
Description
The function computes the GDI and MDI indexes for multivariate count data.
Usage
gmdi.fun(Y)
Arguments
Y |
A matrix of count random variables |
Details
gmdi.fun
computes GDI and MDI indexes introduced by Kokonendji and Puig (2018).
Value
Returns:
gdi |
The generalized dispersion index |
mdi |
The marginal dispersion index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Kokonendji, C.C. and Puig, P. (2018). Fisher dispersion index for multivariate count distributions : A review and a new proposal, Journal of Multivariate Analysis 165, 180-193.
Examples
Y<-cbind(c(1,2,3,4,5,6,7,8),c(1,2,3,4,5,6,7,8))
gmdi.fun(Y)
Z<-cbind(c(1,2,3,4,5,6,7,8),c(1,2,3,4,5,6,7,8),c(1,2,3,4,5,6,7,8),c(1,2,3,4,5,6,7,8))
gmdi.fun(Z)
Function for GVI and MVI
Description
The function computes GVI and MVI indexes for multivariate positive continuous data.
Usage
gmvi.fun(Y)
Arguments
Y |
A matrix of positive continuous random variables |
Details
gmvi.fun
computes the GVI and MVI indexes defined in Kokonendji et al. (2020).
Value
Returns:
gvi |
The generalized variation index |
mvi |
The marginal variation index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Kokonendji, C.C., Touré, A.Y. and Sawadogo, A. (2020). Relative variation indexes for multivariate continuous distributions on [0,\infty)^k
and extensions, AStA Advances in Statistical Analysis 104, 285-307.
Examples
Y<-cbind(c(2.3 ,26.1 ,8.7 ,10.9 ,1.2,1.4),c(9.7 ,7.3,9.3 ,9.4 ,10.5 ,9.8))
gmvi.fun(Y)
Z<-cbind(c(2.3 ,26.1 ,8.7),c(9.7 ,7.3,9.3),c(9.7 ,7.3,9.3),c(9.7 ,7.3,9.3))
gmvi.fun(Z)
Function for VI
Description
The function calculates the univariate exponential variation index for a positive continuous random variable.
Usage
vi.fun(X)
Arguments
X |
A positive continuous random variable |
Details
vi.fun
computes the univariate exponential variation index defined by Abid et al. (2020). See also Touré et al. (2020) for more details on this index.
Value
Returns
vi |
The exponential variation index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Abid, R., Kokonendji, C.C. and Masmoudi, A. (2020). Geometric Tweedie regression models for continuous and semicontinuous data with variation phenomenon, AStA Advances in Statistical Analysis 104, 33-58.
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Examples
X<-c(6.23,7.02,8.94,9.56,8.01,4.34,7.44,6.66,12.72,8.34,0)
vi.fun(X)
T<-c(6.231,7.022,8.943,9.789,8.014,4.423)
vi.fun(T)
Function for VIiG
Description
The function computes the inverse Gaussian variation index with shape parameter l\in (0,\infty)
.
Usage
viiG.fun(X, l)
Arguments
X |
A positive continuous random variable |
l |
The shape parameter of the inverse Gaussian distribution |
Details
viiG.fun
computes the variation index with respect to the inverse Gaussian distribution. See Touré et al. (2020) for more details.
Value
Returns
viiG |
The inverse Gaussian variation index |
Author(s)
Aboubacar Y. Touré and Célestin C. Kokonendji
References
Touré, A.Y., Dossou-Gbété, S. and Kokonendji, C.C. (2020). Asymptotic normality of the test statistics for relative dispersion and relative variation indexes, Journal of Applied Statistics 47, 2479-2491.
Examples
X<-c(0.12,9.11,0.03,8.71,5.02,7.12,6.42,5.73)
viiG.fun(X,0.05)
Y<-c(0.003,6.283,1.001,3.112,4.342,2.890,5.005)
viiG.fun(Y,0.3)