gaussratiovegind (version 3.0.0)

pnormratio: Cumulative Distribution of a Normal Ratio Distribution

Description

Cumulative distribution of the ratio of two Gaussian distributions.

Usage

pnormratio(z, bet, rho, delta, r = 0)

Value

Numeric: the value of density.

Arguments

z

length \(p\) vector of quantiles.

bet, rho, delta

numeric values. The parameters \((\beta, \rho, \delta_y)\) of the distribution, see Details.

r

numeric. The correlation coefficient. Default r=0 (the two distributions are considered independent).

Author

Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel

Details

Let two random variables \(X \sim N(\mu_x, \sigma_x)\) and \(Y \sim N(\mu_y, \sigma_y)\) with correlation coefficient \(r\).

If we denote \(\displaystyle{ f_Z(z; \beta, \rho, \delta_y, r)}\) the probability distribution function of the ratio \(\displaystyle{Z = \frac{X}{Y}}\), with \(\beta = \frac{\mu_x}{\mu_y}\), \(\displaystyle{\rho = \frac{\sigma_y}{\sigma_x}}\), \(\displaystyle{\delta_y = \frac{\sigma_y}{\mu_y}}\) and \(\displaystyle{r = Cor(X, Y) = \frac{Cov(X, Y)}{\sigma_x \sigma_y}}\) (see dnormratio(), Details section).

The cumulative distribution for \(Z\) is given by: $$\displaystyle{F(z; \beta, \rho, \delta_y) = \int_{-\infty}^z{f_Z(z; \beta, \rho, \delta_y, r) dz}}$$

This integral is computed using numerical integration.

References

El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). tools:::Rd_expr_doi("10.3390/rs15020528")

Marsaglia, G. 2006. Ratios of Normal Variables. Journal of Statistical Software 16. tools:::Rd_expr_doi("10.18637/jss.v016.i04")

Díaz-Francés, E., Rubio, F.J., On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. Stat Papers 54, 309–323 (2013). tools:::Rd_expr_doi("10.1007/s00362-012-0429-2")

See Also

dnormratio(): density function.

rnormratio(): sample simulation.

estparnormratio(): parameter estimation.

Examples

# First example: ratio of independent variables
beta1 <- 0.15
rho1 <- 5.75
delta1 <- 0.22
pnormratio(0, bet = beta1, rho = rho1, delta = delta1)
pnormratio(0.5, bet = beta1, rho = rho1, delta = delta1)
curve(pnormratio(x, bet = beta1, rho = rho1, delta = delta1), from = -0.1, to = 0.7)

# Second example: ratio of correlated variables
beta2 <- 2
rho2 <- 2
delta2 <- 2
r2 <- 0.8
pnormratio(0, bet = beta2, rho = rho2, delta = delta2, r = r2)
pnormratio(1, bet = beta2, rho = rho2, delta = delta2, r = r2)
curve(pnormratio(x, bet = beta2, rho = rho2, delta = delta2, r = r2),
      from = -1.5, to = 2.5)