s11.s22: Variance estimates for test statistics zeta_{n,r}^i, i=1,2...
In RSizeBiased: Hypothesis Testing Based on R-Size Biased Samples

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/s11.s22.R

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

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

`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.

Provided that μ_r, r=1, 2, … is the rth moment of the Weibull or the Gamma distribution, then

σ_{1,r}^2 = μ_r (μ_{2-r}) - 2 μ_1 μ_{1-r} + μ_1^2 μ_{-r}

and

σ_{2,r}^2 = -4μ_r \bigl ( 2μ_{1}^2 - μ_2) - 2) μ_1 μ_{1-r} + (2μ_1^2 - μ_{2})^2 + (8μ_1^2 - 2μ_{2}) μ_{2-r} - 4 μ_1 μ_{3-r} + μ_{4-r} \bigr )

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

Polychronis Economou

R implementation and documentation: Polychronis Economou <peconom@upatras.gr>

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

zeta_plug_in

#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")