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)

Arguments

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 <peconom@upatras.gr>

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")

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

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.

Author(s)

Polychronis Economou

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

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")

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) 

Arguments

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 <peconom@upatras.gr>

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)

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) 

Arguments

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 <peconom@upatras.gr>

References

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

See Also

p_rsize_Weibull, r_rsize_Weibull

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))

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)

Arguments

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 <peconom@upatras.gr>

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")

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) 

Arguments

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 <peconom@upatras.gr>

References

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

See Also

d_rsize_Weibull, r_rsize_Weibull

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)

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)

Arguments

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 <peconom@upatras.gr>

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")

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) 

Arguments

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 <peconom@upatras.gr>

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.

See Also

d_rsize_Weibull, p_rsize_Weibull

Examples

#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)

Arguments

Details

Provided that \mu_r, r=1, 2, \dots is the rth 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 <peconom@upatras.gr>

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

#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")

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 

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)
  

\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)

Arguments

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 <peconom@upatras.gr>

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