Package 'RSizeBiased' reference manual

Title:	Hypothesis Testing Based on R-Size Biased Samples
Description:	Provides functions and examples for testing hypothesis about the population mean and variance on samples drawn by r-size biased sampling schemes.
Authors:	Dimitrios Bagkavos [aut, cre], Polychronis Economou [aut], Apostolos Batsidis [aut], Gorgios Tzavelas [aut]
Maintainer:	Dimitrios Bagkavos <[email protected]>
License:	GPL (>= 2)
Version:	0.1.0
Built:	2024-11-28 06:33:37 UTC
Source:	CRAN

Kullback-Leibler divergence between the (parametrized with respect to shape and mean or variance) of the Weibull or gamma distribution and its (assumed) maximum likelihood estimates.

Description

The function returns the Kullback-Leibler divergence (minus a constant) between the (parametrized with respect to shape and mean or variance) underlying Weibull or gamma distribution and its (assumed) maximum likelihood estimates.

Usage

Cond.KL.Weib.Gamma(par,nullvalue,hata,hatb,type,dist)Cond.KL.Weib.Gamma(par,nullvalue,hata,hatb,type,dist)

Arguments

`par`	The (actual) shape parameter $\alpha$ of the distribution.
`nullvalue`	The (actual) distribution mean or variance.
`hata`	Maximum likelihood estimate of the shape parameter of the distribution.
`hatb`	Maximum likelihood estimate of the scale parameter of the distribution.
`type`	Numeric switch, enables the choice of mean or variance: type: 1 for mean, 2 (or any other value != 1) for variance.
`dist`	Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Details

The Kullback-Leibler divergence between the Weibull( $\alpha, \beta)$ or the gamma( $\alpha, \beta)$ and its maximum likelihood estimate Gamma( $\hat \alpha, \hat \beta)$ is given by

$D_{KL} = (\hat \alpha -1)\Psi(\hat \alpha) - \log\hat \beta - \hat \alpha - \log \Gamma(\hat \alpha) + \log\Gamma( \alpha) + \alpha \log \beta - (\alpha -1)(\Psi(\hat \alpha) + \log \hat \beta) + \frac{ \hat \beta \hat \alpha}{\lambda}.$

Since $D_{KL}$ is used to determine the closest distribution - given its mean or variance - to the estimated gamma p.d.f., the first four terms are omitted from the function outcome, i.e. the function returns the result of the following quantity:

$\log\Gamma( \alpha) + \alpha \log \beta - (\alpha -1)(\Psi(\hat \alpha) + \log \hat \beta) + \frac{ \hat \beta \hat \alpha}{\lambda}.$

For the Weibull distribution the corresponding formulas are

$D_{KL} = \log \frac{\hat \alpha}{{\hat \beta}^{\hat \alpha}} - \log \frac{\alpha}{{\beta}^{\alpha}} + (\hat \alpha - \alpha) \left ( \log \hat \beta - \frac{\gamma}{\hat \alpha} \right ) + \left (\frac{\hat \beta}{\beta} \right )^\alpha \Gamma\left ( \frac{\alpha}{\hat \alpha} +1 \right ) -1$

and since $D_{KL}$ is used to determine the closest distribution - given its mean or variance - to the estimated gamma p.d.f., the first term is omitted from the function outcome, i.e. the function returns the result of the following quantity:

$- \log \frac{\alpha}{{\beta}^{\alpha}} + (\hat \alpha - \alpha) \left ( \log \hat \beta - \frac{\gamma}{\hat \alpha} \right ) + \left (\frac{\hat \beta}{\beta} \right )^\alpha \Gamma\left ( \frac{\alpha}{\hat \alpha} +1 \right ) -1$

Value

A scalar, the value of the Kullback-Leibler divergence (minus a constant).

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

#K-L divergence for the Gamma distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "gamma")
#K-L divergence for the Weibull distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "weib")
#K-L divergence for the Gamma distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "gamma")
#K-L divergence for the Weibull distribution for shape=2
#and variance=3 and their assumed MLE=(1,1):
 Cond.KL.Weib.Gamma(2,3,1,1,2, "weib")

Weibull size biased distribution of order $r$ .

Description

Calculates the density of the $r-$ size biased Weibull distribution.

Usage

d_rsize_Weibull(x,TRpar,r) d_rsize_Weibull(x,TRpar,r)

Arguments

`x`	Grid points where the functional is being calculated.
`TRpar`	A vector of length 2, containing the shape and scale parameters of the 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 Weibull distribution.

Details

The $r-$ size density of the observed biased sample $X_1, \dots, X_n$ is defined by

$f_r(x; \theta)=\frac{x^r f(x; \theta)}{E(X^r)}$

where $f(x; \theta)$ is the density of the Weibull distribution and $\theta$ the vector of the shape and scale parameters of the distribution.

Value

A vector of length equal to the length of $x$ .

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

# example of r-size Weibull distribution, r=0,1,2
x<- seq(0, 10, length=50)
dens.0.size<-d_rsize_Weibull(x,c(2,3),0)
dens.1.size<-d_rsize_Weibull(x,c(2,3),1)
dens.2.size<-d_rsize_Weibull(x,c(2,3),2)
plot(x, dens.0.size, type="l", ylab="r-denisty")
lines(x, dens.1.size, col=2)
lines(x, dens.2.size, col=3)
legend("topright",legend=c("r= 0","r= 1","r= 2"),
       col=c("black","red","green"),lty=c(1,1,1))
# example of r-size Weibull distribution, r=0,1,2
x<- seq(0, 10, length=50)
dens.0.size<-d_rsize_Weibull(x,c(2,3),0)
dens.1.size<-d_rsize_Weibull(x,c(2,3),1)
dens.2.size<-d_rsize_Weibull(x,c(2,3),2)
plot(x, dens.0.size, type="l", ylab="r-denisty")
lines(x, dens.1.size, col=2)
lines(x, dens.2.size, col=3)
legend("topright",legend=c("r= 0","r= 1","r= 2"),
       col=c("black","red","green"),lty=c(1,1,1))

Log likelihood function for the weighted gamma or Weibull distributions.

Description

Calculates the log-likelihood function of the weighted gamma or Weibull (depends on user input) distribution.

Usage

log_Lik_Weib_gamma_weighted(TRpar,datain,r,dist)log_Lik_Weib_gamma_weighted(TRpar,datain,r,dist)

Arguments

`TRpar`	A vector of length 2, containing the shape and scale parameters of the distribution.
`datain`	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 distribution.
`dist`	Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Details

The log likelihood function of the weighted gamma distribution is defined by

$\log L = \sum_{i=1}^n log f_r(X_i; \theta)$

where $f_r(x; \theta)$ is the density of the $r-$ size biased gamma distribution. Setting $r=0$ corresponds to the log likelihood of the Gamma distribution.

In the case of Weibull, the log likelihood is defined by

$\log L = \sum_{i=1}^n log f_r(X_i; \theta)$

where $f_r(x; \theta)$ is the density of the $r-$ size biased Weibull distribution. Setting $r=0$ corresponds to the log likelihood of the Weibull distribution.

Value

A scalar, the result of the log likelihood calculation.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples


#Log-likelihood for the gamma distribution for true parms=(2,3), r=0:
log_Lik_Weib_gamma_weighted(c(2,3), rgamma(100, shape=2, scale=3), 0, "gamma")
#Log-likelihood for the Weibull distribution for true parms=(2,3), r=0:
log_Lik_Weib_gamma_weighted(c(2,3), rweibull(100, shape=2, scale=3), 0, "weib")
#Log-likelihood for the gamma distribution for true parms=(2,3), r=0:
log_Lik_Weib_gamma_weighted(c(2,3), rgamma(100, shape=2, scale=3), 0, "gamma")
#Log-likelihood for the Weibull distribution for true parms=(2,3), r=0:
log_Lik_Weib_gamma_weighted(c(2,3), rweibull(100, shape=2, scale=3), 0, "weib")

Weibull size biased c.d.f. of order $r$ .

Description

Calculates the cumulative distribution of the $r-$ size biased Weibull distribution.

Usage

p_rsize_Weibull(q,TRpar,r) p_rsize_Weibull(q,TRpar,r)

Arguments

`q`	Points where the functional is being calculated.
`TRpar`	A vector of length 2, containing the shape and scale parameters of the 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 Weibull distribution.

Details

The $r-$ size c.d.f. of the Weibull density is defined by

$F_r(y; \theta)=\int_{0}^{y} \frac{x^r f(x; \theta)}{E(X^r)} \,dx$

where $\theta$ is a bivariate vector with the the shape and scale of the Weibull distribution.

Value

A vector of length equal to the lemgth of $x$ .

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

# c.d.f of the r-size Weibull distribution, r=0,1,2 evalutated at a specific point x.
x<- 2
dist.0.size<-p_rsize_Weibull(x,c(2,3),0)
dist.1.size<-p_rsize_Weibull(x,c(2,3),1)
dist.2.size<-p_rsize_Weibull(x,c(2,3),2)
# c.d.f of the r-size Weibull distribution, r=0,1,2 evalutated at a specific point x.
x<- 2
dist.0.size<-p_rsize_Weibull(x,c(2,3),0)
dist.1.size<-p_rsize_Weibull(x,c(2,3),1)
dist.2.size<-p_rsize_Weibull(x,c(2,3),2)

$r-$ th moment of the gamma or the Weibull distribution.

Description

Calculates the $r-$ th moment of the gamma or Weibull distribution.

Usage

r_moment_gamma_Weib(TRpar,r,dist)r_moment_gamma_Weib(TRpar,r,dist)

Arguments

`TRpar`	A vector of length 2, containing the shape and scale parameters of the 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 Gamma distribution.
`dist`	Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Details

In the case of the $\Gamma(\alpha, \beta)$ distribution the $r$ -th moment is given by

$\mu_r = \int_0^{\infty} x^r f(x;\alpha, \beta)\,dx =\beta^r \frac{\Gamma(\alpha+r)}{\Gamma(\alpha)}, \alpha> -r$

while for the $W(\alpha, \beta)$ distribution the $r$ -th moment is given by

$\mu_r = \int_0^{\infty} x^r f(x;\alpha, \beta)\,dx = \beta^r \Gamma\left(1+\frac{\alpha}{r}\right), \alpha> -r$

Value

A scalar, the value of the moment.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

#r-moment for the Gamma distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "gamma")
#r-moment for for the Weibull distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "weib")
#r-moment for the Gamma distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "gamma")
#r-moment for for the Weibull distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "weib")

Weibull size biased random number generation of order $r$ (modified).

Description

Provides a random sample of size $n$ from the $r-$ size biased Weibull distribution (modified).

Usage

r_rsize_Weibull(n,TRpar,r) r_rsize_Weibull(n,TRpar,r)

Arguments

`n`	Number of th sample data points to be provided.
`TRpar`	A vector of length 2, containing the shape and scale parameters of the 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 Weibull distribution.

Details

The $r-$ size random number generator from the Weibull distribution is implemented based on a change-of-variable technique, to the standard gamma distribution as described by Gove and Patil (1998).

Value

A vector of length $n$ with the random sample.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

Gove J.H. and Patil G.P. (1998). Modeling the Basal Area-size Distribution of Forest Stands: A Compatible Approach. Forest Science, 44(2), 285-297.

Examples

#Random number geenration for the r-size Weibull distribution.
r_rsize_Weibull(100,c(2,3),1)
#Random number geenration for the r-size Weibull distribution.
r_rsize_Weibull(100,c(2,3),1)

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)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.

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

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

Value

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

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

#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")
#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")

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)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 <[email protected]>

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")
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")

Test statistic $T_{n,r}^1$ or $T_{n,r}^2$ depending on user input.

Description

The test statistics $T_{n,r}^1$ and $T_{n,r}^2$ are consistent estimators of the mean value $\mathrm{E}(X)$ and variance $\mathrm{Var}(X)$ respectively given an $r-$ size biased sample.

Usage

T1T2.Mean.Var(datain,r, type) T1T2.Mean.Var(datain,r, type)

Arguments

`datain`	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 underlying distribution.
`type`	Numeric switch: type =1 corresponds to the T1 statistic while any other numeric value will cause calculation of T2.

Details

The test statistic $T_{n,r}^1$ is defined by

$T_{n,r}^{1}=\frac{\sum_{i=1}^n X_i^{1-r}}{\sum_{i=1}^n X_i^{-r}}.$

The test statistic $T_{n,r}^2$ is defined by

$T_{n,r}^{2}= \frac{\sum_{i=1}^n X_i^{2-r}}{\sum_{i=1}^nX_i^{-r}}-{\left(\frac{\sum_{i=1}^n X_i^{1-r}}{\sum_{i=1}^n X_i^{-r}}\right)^2}.$

Value

A scalar, the value of the test statistic for the given sample.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

References

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

Examples

#e.g.:
T1T2.Mean.Var(rgamma(100, 2,3),0, 1)
#e.g.:
T1T2.Mean.Var(rgamma(100, 2,3),0, 1)

Upper Flat Creek forest cruise tree data

Description

Forest measurement data from the Upper Flat Creek unit of the University of Idaho Experimental Forest, measured in 1991.

Usage

  ufc ufc

Format

A data frame with 336 observations on the following 5 variables; plot (plot label), tree (tree label), species (species kbd with levels DF, GF, WC, WL), dbh.cm (tree diameter at 1.37 m. from the ground, measured in centimetres.), height.m (tree height measured in metres).

Details

The inventory was based on variable radius plots with 6.43 sq. m. per ha. BAF (Basal Area Factor). The forest stand was 121.5 ha. This version of the data omits errors, trees with missing heights, and uncommon species. The four species are Douglas-fir, grand fir, western red cedar, and western larch.

Source

Harold Osborne and Ross Appelgren of the University of Idaho Experimental Forest.

References

Robinson, A.P., and J.D. Hamann. 2010. Forest Analytics with R: an Introduction. Springer.

Examples

  data(ufc)
  data(ufc)

$\zeta_{n,r}^i, i=1,2$ test statistic for the Weibull or the gamma distribution (depending on user input.

Description

Studentized version of the $T^i_{n,r}, i=1,2$ test statistic for the Weibull/gamma distribution.

Usage

zeta_plug_in(null_value, datain,r,EST_par,type, dist)zeta_plug_in(null_value, datain,r,EST_par,type, dist)

Arguments

`null_value`	The parameter value in the hypothesis test under the null
`datain`	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 underlying distribution.
`EST_par`	A vector of length 2, containing the shape and scale parameters of the Weibull distribution.
`type`	Numeric switch: type =1 returns the $\zeta_{n,r}^1$ test statistic, any other value returns $\zeta_{n,r}^2$
`dist`	Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Details

When type=1 the function returns

$\sqrt{n} \frac{T_{n,r^1} - \mu^0}{ \sigma_{1,r}(\hat \theta_n)} \rightarrow N(0,1)$

after using the fact that under the null we have $\mu_1=\mu^0$ . Any other value for type returns

$\sqrt{n} \frac{T_{n,r^2} - \sigma_0^2}{ \sigma_{2,r}(\hat \theta_n)} \rightarrow N(0,1)$

in which case the fact that var(X) $=\sigma_0^2$ under the null has been used.

Value

A scalar with the value of the test statistic.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[email protected]>

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
# ml estimates = c(2.6555,8.0376),  taken from section 6.2 in Economou et. al. (2021).
zeta_plug_in(nullMEAN, datain_r, 2, c(2.6555,8.0376),1, "gamma") #corresponds to mean

nullVar <- 180
zeta_plug_in(nullVar, datain_r, 2, c(2.6555,8.0376),2, "gamma") #corresponds to var

data(ufc)
datain_r <- ufc[,4]
nullMEAN <- 14
# ml estimates = c(2.6555,8.0376),  taken from section 6.2 in Economou et. al. (2021).
zeta_plug_in(nullMEAN, datain_r, 2, c(2.6555,8.0376),1, "gamma") #corresponds to mean

nullVar <- 180
zeta_plug_in(nullVar, datain_r, 2, c(2.6555,8.0376),2, "gamma") #corresponds to var

Package 'RSizeBiased'

Help Index

Kullback-Leibler divergence between the (parametrized with respect to shape and mean or variance) of the Weibull or gamma distribution and its (assumed) maximum likelihood estimates.

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Weibull size biased distribution of order rrr.

Description

Usage

Arguments

Details

Value

Author(s)

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See Also

Examples

Log likelihood function for the weighted gamma or Weibull distributions.

Description

Usage

Arguments

Details

Value

Author(s)

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Examples

Weibull size biased c.d.f. of order rrr.

Description

Usage

Arguments

Details

Value

Author(s)

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See Also

Examples

r−r-r−th moment of the gamma or the Weibull distribution.

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Weibull size biased random number generation of order rrr (modified).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

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

Description

Usage

Arguments

Details

Value

Author(s)

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See Also

Examples

Test statistics.

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Test statistic Tn,r1T_{n,r}^1Tn,r1​ or Tn,r2T_{n,r}^2Tn,r2​ depending on user input.

Description

Weibull size biased distribution of order $r$ .

Weibull size biased c.d.f. of order $r$ .

$r-$ th moment of the gamma or the Weibull distribution.

Weibull size biased random number generation of order $r$ (modified).

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

Test statistic $T_{n,r}^1$ or $T_{n,r}^2$ depending on user input.

$\zeta_{n,r}^i, i=1,2$ test statistic for the Weibull or the gamma distribution (depending on user input.