Size.BiasedMV.Tests {RSizeBiased}

Test statistics.

Description

The function returns the test statistics for testing a null hypothesis for the mean and a null hypothesis for the varaince.

Usage

Size.BiasedMV.Tests(datain_r,r,nullMEAN,nullVAR,start_par,nboot,alpha,prior_sel,distr)

Arguments

datain_r

The available sample points.

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 gamma or theWeibull distribution.

nullMEAN

The null value of the distribution mean.

nullVAR

The null value of the distribution variance.

start_par

Vector with two values, containing the starting values for the MLE for the two parameter distribution (Weibull or gamma) .

nboot

Defines the number of bootstrap replications.

alpha

Significance level.

prior_sel

"normal" for the normal distribution or "gamma" for the gamma.

distr

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

Details

The test statistics implemented are given by the Plug-in and the bootstrap Methods as described in section 3.1 and 3.2 of Economou et al (2021).

Value

An object containing the following components.

par

A vector of the MLE of the distribution parameters.

loglik

A scalar, the maximized log-likelihood.

CovMatrix

The Variance - Covariance matrix of the MLEs.

Zeta_i

A vector of the values of the \zeta_{n,r}^i, i=1,2 test statistics (if defined)

Tivalues

A vector of the values of the T^i_{n,r}, i=1,2 test statistics

T1_bootstrap_quan

A vector of the bootstrap quantiles for the T^1_{n,r} test statistic for each one of the significance levels alpha.

T2_bootstrap_quan

A vector of the bootstrap quantiles for the T^2_{n,r} test statistic for each one of the significance levels alpha.

NullValues

A vector of the null values of the distribution mean and variance.

distribution

Character representing the choice of distribution: "weib" for the Weibull or "gamma" for the gamma distribution.

alpha

A vector of significance levels for the test level.

bootstrap_p_mean

A scalar with the bootstrap p-value for testing the mean.

bootstrap_p_var

A scalar with the bootstrap p-value for testing the variance.

decision

A matrix of 0 and 1 of the decisions taken for each one of the significance levels alpha based on the bootstrap method. The first row corresponds to the null hypothesis for the mean and the second to the null hypothesis for the variance.

asymptotic_p_mean

A scalar with the asymptotic p-value for testing the mean (if \zeta_{n,r}^1 is defined).

asymptotic_p_var

A scalar with the asymptotic p-value for testing the variance (if \zeta_{n,r}^2 is defined).

decisionasympt

A matrix of 0 and 1 of the decisions taken for each one of the significance levels alpha based on the plug-in method and the asymptotic distribution of the test statistics. The first row corresponds to the null hypothesis for the mean and the second to the null hypothesis for the variance.

prior_selection

Character representing the choice of the prior distribution for the bootstrap method: "normal" for the normal distribution or "gamma" for the gamma.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[e-mail]>

References

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

Examples

data(ufc)
datain_r <- ufc[,4]
nullMEAN <- 14 #according to null mean in Sec. 6.3,  Economou et. al. (2021).
nullVAR <- 180 #according to null variance in Sec. 6.3,  Economou et. al. (2021).
Size.BiasedMV.Tests(datain_r, 2, nullMEAN, nullVAR,  c(2,3), 100, 0.05, "normal", "gamma")