RSizeBiased (version 0.1.0)

s11.s22: Variance estimates for test statistics \(\zeta_{n,r}^i, i=1,2\) specifically for the Weibull and gamma distributions.

Description

Variance estimates for test statistics \(\zeta_{n,r}^i, i=1,2\) specifically for the Weibull and gamma distributions.

Usage

s11.s22(TRpar,r,sgg,dist)

Arguments

TRpar

A vector of length 2, containing the shape and scale parameters of the Weibull distribution.

r

The size (order) of the distribution. The special cases \(r=1,2,3\) correspond to length, area, volume biased samples respectively and are the most frequently encountered in practice. The case \(r=0\) corresponds to random samples from the underlying distribution.

sgg

Character switch ("s11" or "s22"), enables choosing between the s11 and s22 options

dist

Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Value

A scalar with the value of the variance estimate for the test statistic.

Details

Provided that \(\mu_r, r=1, 2, \dots \) is the \(r\)th moment of the Weibull or the Gamma distribution, then

$$ \sigma_{1,r}^2 = \mu_r (\mu_{2-r}) - 2 \mu_1 \mu_{1-r} + \mu_1^2 \mu_{-r}$$

and

$$ \sigma_{2,r}^2 = -4\mu_r \bigl ( 2\mu_{1}^2 - \mu_2) - 2) \mu_1 \mu_{1-r} + (2\mu_1^2 - \mu_{2})^2 + (8\mu_1^2 - 2\mu_{2}) \mu_{2-r} - 4 \mu_1 \mu_{3-r} + \mu_{4-r} \bigr ) $$

References

Economou et. al. (2021). Hypothesis testing for the population mean and variance based on r-size biased samples, under review.

See Also

zeta_plug_in

Examples

# NOT RUN {
#s11 for the Gamma distribution for true parms=(2,3), r=1:
s11.s22(c(2,3),1, "s11", "gamma")
#s22 for for the Weibull distribution for true parms=(2,3), r=1:
s11.s22(c(2,3),1, "s22",  "weib")
# }