Package 'asymmetry.measures'

Switch between a range of probability density functions.

Description

Returns the user-specified probability density function out of a range of available options evaluated at selected grid points.

Usage

d.sample(s,dist, p1,p2)

Arguments

s	A scalar or vector: the x-axis grid points where the probability density function will be evaluated.
dist	Character string, used as a switch to the user selected distribution function (see details below).
p1	A scalar. Parameter 1 (vector or object) of the selected density.
p2	A scalar. Parameter 2 (vector or object) of the selected density.

Details

Based on user-specified argument dist, the function returns the value of the probability density function at s.

Supported distributions (along with the corresponding dist values) are:

• weib: The weibull distribution is implemented as

  $f(s;p_1,p_2)= \frac{p_1}{p_2} \left (\frac{s}{p_2}\right )^{p_1-1} \exp \left \{- \left (\frac{s}{p_2}\right )^{p_1} \right \}$

  with $s \ge 0$ where $p_1$ is the shape parameter and $p_2$ the scale parameter.

• lognorm: The lognormal distribution is implemented as

  $f(s) = \frac{1}{p_2s\sqrt{2\pi}}e^{-\frac{(log s -p_1)^2}{2p_2^2}}$

  where $p_1$ is the mean and $p_2$ is the standard deviation of the distirbution.

• norm: The normal distribution is implemented as

  $f(s) = \frac{1}{p_2\sqrt{2 \pi}}e^{-\frac{ (s - p_1)^2 }{ 2p_2^2 }}$

  where $p_1$ is the mean and the $p_2$ is the standard deviation of the distirbution.

• uni: The uniform distribution is implemented as

  $f(s) = \frac{1}{p_2-p_1}$

  for $p_1 \le s \le p_2$.

• cauchy: The cauchy distribution is implemented as

  $f(s)=\frac{1}{\pi p_2 \left \{1+( \frac{s-p_1}{p_2})^2\right \} }$

  where $p_1$ is the location parameter and $p_2$ the scale parameter.

• fnorm: The half normal distribution is implemented as

  $2 f(s)-1$

  where

  $f(s) = \frac{1}{sd\sqrt{2 \pi} }e^{-\frac{s^2}{2 sd^2 }},$

  and $sd=\sqrt{\pi/2}/p_1$.

• normmixt:The normal mixture distribution is implemented as

  $f(s)=p_1\frac{1}{p_2[2] \sqrt{2\pi} } e^{- \frac{ (s - p_2[1])^2}{2p_2[2]^2}} +(1-p_1)\frac{1}{p_2[4]\sqrt{2\pi}} e^{-\frac{(s - p_2[3])^2}{2p_2[4]^2 }}$

  where $p1$ is a mixture component(scalar) and $p_2$ a vector of parameters for the mean and variance of the two mixture components $p_2= c(mean1, sd1, mean2, sd2)$.

• skewnorm: The skew normal distribution with parameter $p_1$ is implemented as

  $f(s)=2\phi(s)\Phi(p_1s)$

  .

• fas: The Fernandez and Steel distribution is implemented as

  $f(s; p_1, p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ f_t(s/p_1; p_2) I_{\{s \ge 0\}} + f_t(p_1s; p_2)I_{\{s<0 \}}\right \}$

  where $f_t(x;\nu)$ is the p.d.f. of the $t$ distribution with $\nu = 5$ degrees of freedom. $p_1$ controls the skewness of the distribution with values between $(0, +\infty)$ and $p_2$ denotes the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  $f(s;\mu, p_1, p_2, \tau) = \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}$

  where $r=\sinh(\sinh(z)-(-p_1))$, $c=\cosh(\sinh(z)-(-p_1))$ and $z=((s-\mu)/p2)$. $p_1$ is the vector of skewness, $p_2$ is the scale parameter, $\mu=0$ is the location parameter and $\tau=1$ the kurtosis parameter.

Value

A vector containing the user selected density values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds), Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

r.sample, q.sample, p.sample

Examples

selected.dens <- "weib" #select Weibull as the density
shape <- 2  # specify shape parameter
scale <- 1 # specify scale parameter
xout <- seq(0.1,5,length=50)  #design point where the density is evaluated
d.sample(xout,selected.dens,shape,scale)  # calculate density at xout

---

Empirical cummulative distribution function

Description

Empirical (nonparametric) cummulative distribution function for given a random sample.

Usage

edf(xin, xout)

Arguments

xin	A vector of data points - the available sample.
xout	A vector of design points where the distribution function will be estimated.

Details

The empirical distribution function estimator at $x$ is defined as the number of observations up to $x$, divided by $n$, i.e.

$F_{n}(x) = \frac{\# \{ X_{1},..,X_{n}\} \le x}{n}$

.

Value

A vector with the estimated distribution function at xout.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation:

Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Hollander, M. abd Wolfe, D.A. (1999), Nonparametric Statistical Methods, 2nd edition, Wiley.

Examples

x.in <- rexp(200)
x.out <- seq(0.1,5,length=60)
dist.est <- edf(x.in,x.out)
plot(x.out,dist.est,col="blue",main="Empirical c.d.f.",xlab="x",yla ="probability")

---

Epanechnikov kernel

Description

Implementation of the Epanechnikov kernel.

Usage

Epanechnikov(x)

Arguments

x	A vector of data points between $-\sqrt{5}$ and $\sqrt{5}$ where the kernel will be evaluated.

Details

Implements:

$K(u)= \frac{3}{4\sqrt{5}} \left (1-\frac{x^2}{5} \right )$

for $|x| \le \sqrt{5}$

Value

The value of the kernel at $x$

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Kernel Statistics

See Also

IntEpanechnikov

---

Strong asymmetry measure $eta(X)$.

Description

Returns the strong asymmetry measure $eta(X)$ of Patil, Bagkavos and Wood (2014).

Usage

eta.s(xin, dist, GridLength, p1, p2)

Arguments

xin	A vector of data points - the available sample.
dist	Character string, specifies selected distribution function.
GridLength	A non-negative number, which will be rounded up if fractional.Desired length of the sequence.
p1	A scalar. Parameter 1 (vector or object) of the selected distribution.
p2	A scalar. Parameter 2 (vector or object) of the selected distribution.

Details

Implements

$\eta(X)= -0.5 sign(\rho_1)\max|\rho_p + \rho_p^*|$

with $1/2 \le p \le 1$.

Uses maximum likehood estimates for the unknown functionals in the definition of the measure.

Value

Returns a scalar, the value of the strong asymmetry measure $\eta(X)$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

• Patil P.N., Bagkavos D. and Wood A.T.A., (2014). A measure of asymmetry based on a new necessary and sufficient condition for symmetry, Sankhya A, 76, 123–145.

• Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds), Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve,eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc

Examples

selected.dist <- "norm" #select norm as the distribution
m.use <- mean(GDP.Per.head.dist.2005)
sd.use<- sd(GDP.Per.head.dist.2005)
grid <- 50

s.use<- GDP.Per.head.dist.1995
eta.s(GDP.Per.head.dist.2005,selected.dist,grid,m.use,sd.use)

---

Strong asymmetry measure $eta(X)$.

Description

Returns the strong asymmetry measure $eta(X)$ of Patil, Bagkavos and Wood (2014).

Usage

eta.s.exact(xin, dist, GridLength, p1, p2)

Arguments

xin	A vector of data points - the available sample.
dist	Character string, specifies selected distribution function.
GridLength	A non-negative number, which will be rounded up if fractional.Desired length of the sequence.
p1	A scalar. Parameter 1 (vector or object) of the selected distribution.
p2	A scalar. Parameter 2 (vector or object) of the selected distribution.

Details

Implements

$\eta(X)= -0.5 sign(\rho_1)\max|\rho_p + \rho_p^*|$

with $1/2 \le p \le 1$ This version uses exact p.d.f. and c.d.f. evaluation and not estimates of the unknown functionals.

Value

Returns a scalar, the value of the strong asymmetry measure $\eta(X)$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

• Patil P.N., Bagkavos D. and Wood A.T.A., (2014). A measure of asymmetry based on a new necessary and sufficient condition for symmetry, Sankhya A, 76, 123–145.

• Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., González Manteiga W., Romo J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve,eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc,eta.s

Examples

selected.dist <- "norm" #select norm as the distribution
m.use <- 2
sd.use<- 2
grid <- 50
s.use<- rnorm(100)
eta.s.exact(s.use,selected.dist,grid,m.use,sd.use)  # calculate eta.s at xout

---

Asymmetry coefficient $\breve{\eta}$

Description

Implements the asymmetry coefficient $\breve{\eta}$ of Patil, Patil and Bagkavos (2012).

Usage

eta.w.breve(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2, \dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$, $\breve{\eta}$ is defined by

$\breve{\eta}=-\frac{\sum_{i=1}^n {U_iW_i}-n\bar{U}\bar{W}}{\sqrt{(\sum_{i=1}^n {U_i^2-n\bar{U^2}})(\sum_{i=1}^n{W_i^2-n\bar{W^2}})}}$

where

$U_i = \hat{f}(X_i), \; W_i =F_n(X_i), \; \bar{U}= n^{-1}\sum_{i=1}^n U_i, \; \bar{W}=n^{-1} \sum_{i=1}^{n} W_i.$

Value

Returns a scalar, the estimate of $\breve{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971–985.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc

Examples

eta.w.breve(GDP.Per.head.dist.1995,Epanechnikov)
0.329707 #estimate of etabreve

---

Asymmetry coefficient $\breve{\eta}$ using boundary correction

Description

Implements the asymmetry coefficient $\breve{\eta}$ of Patil, Patil and Bagkavos (2012).

Usage

eta.w.breve.bc(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2,\dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$. $\breve{\eta}$ is defined by

$\breve{\eta}=-\frac{\sum_{i=1}^n {U_iW_i}-n\bar{U}\bar{W}}{\sqrt{(\sum_{i=1}^n {U_i^2-n\bar{U^2}})(\sum_{i=1}^n {W_i^2-n\bar{W^2}})}}$

where

$U_i = \hat{f}(X_i), \; W_i =F_n(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{W}=n^{-1}\sum_{i=1}^n W_i.$

eta.w.breve.bc uses reflection to correct the boundary bias of the kernel density estimate kde

Value

Returns a scalar, the estimate of $\breve{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve, eta.w.tilde,eta.w.tilde.bc

Examples

eta.w.breve.bc(GDP.Per.head.dist.1995,Epanechnikov)
0.329707 #estimate of etabreve

---

Asymmetry coefficient $\hat{\eta}$

Description

Implements the asymmetry coefficient $\hat{\eta}$ of Patil, Patil and Bagkavos (2012).

Usage

eta.w.hat(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2,\dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$, $\hat{\eta}$ is defined by

$\hat{\eta}=-\frac{\sum_{i=1}^n {U_iV_i}-n\bar{U}\bar{V}}{\sqrt{(\sum_{i=1}^n{U_i^2-n\bar{U^2}})(\sum_{i=1}^n{V_i^2-n\bar{V^2}})}}$

where

$U_i = \hat{f}(X_i), \; V_i =\hat{F}(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{V}=n^{-1}\sum_{i=1}^n V_i.$

Value

Returns a scalar, the estimate of $\hat{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat.bc, eta.w.breve, eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc

Examples

eta.w.hat(GDP.Per.head.dist.1995,Epanechnikov)
0.3463025 #estimate of etahat

---

Asymmetry coefficient $\hat{\eta}$ using boundary correction

Description

Implements the asymmetry coefficient $\hat{\eta}$ of Patil, Patil and Bagkavos (2012)

Usage

eta.w.hat.bc(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2, \dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$, $\hat{\eta}$ is defined by

$\hat{\eta}=-\frac{\sum_{i=1}^{n} {U_i V_i}-n\bar{U}\bar{V}}{\sqrt{(\sum_{i=1}^n{U_i^2-n\bar{U^2}})(\sum_{i=1}^n {V_i^2-n\bar{V^2}})}}$

where

$U_i = \hat{f}(X_i), \; V_i =\hat{F}(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{V}=n^{-1}\sum_{i=1}^n V_i.$

eta.w.hat.bc uses reflection to correct the boundary bias issue of the kernel estimate kde.

Value

Returns a scalar, the estimate of $\hat{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat, eta.w.breve, eta.w.breve.bc, eta.w.tilde,eta.w.tilde.bc

Examples

eta.w.hat.bc(GDP.Per.head.dist.1995,Epanechnikov)
0.3463025 #estimate of etahat.bc

---

Asymmetry coefficient $\tilde{\eta}$

Description

Implements the asymmetry coefficient $\tilde{\eta}$ of Patil, Patil and Bagkavos (2012).

Usage

eta.w.tilde(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2,\dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$. $\tilde{\eta}$ is defined by

$\tilde{\eta}=-\frac{\sum_{i=1}^n{U_iV_i}-(n/2)\bar{U}}{\sqrt{(n/12)(\sum_{i=1}^n{U_i^2-n\bar{U^2}})}}$

where

$U_i = \hat{f}(X_i), \; V_i =F(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{V}=n^{-1}\sum_{i=1} V_i.$

Value

Returns a scalar, the estimate of $\tilde{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve.bc, eta.w.breve,eta.w.tilde.bc

Examples

eta.w.tilde(GDP.Per.head.dist.1995,Epanechnikov)
0.3333485 #estimate of etatile

---

Asymmetry coefficient $\tilde{\eta}$ using boundary correction

Description

Implements the asymmetry coefficient $\tilde{\eta}$ of Patil, Patil and Bagkavos (2012).

Usage

eta.w.tilde.bc(xin, kfun)

Arguments

xin	A vector of data points - the available sample.
kfun	The kernel to use in the density estimate.

Details

Given a sample $X_1, X_2,\dots, X_n$ from a continuous density function $f(x)$ and distribution function $F(x)$, $\tilde{\eta}$ is defined by

$\tilde{\eta}=-\frac{\sum_{i=1}^n{U_iV_i}-(n/2)\bar{U}}{\sqrt{(n/12)(\sum^{n}_{i=1}{U_i^2-n\bar{U^2}})}}$

where

$U_i = \hat{f}(X_i), \; V_i =F(X_i), \; \bar{U}=n^{-1}\sum_{i=1}^n U_i, \; \bar{V}=n^{-1}\sum_{i=1}^n V_i.$

eta.w.tilde.bc uses reflection to correct the boundary bias of kde.

Value

Returns a scalar, the estimate of $\tilde{\eta}$.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Patil, P.N., Patil, P.P. and Bagkavos, D., (2012), A measure of asymmetry. Stat. Papers, 53, 971-985.

See Also

eta.w.hat.bc, eta.w.hat, eta.w.breve.bc, eta.w.breve,eta.w.tilde

Examples

eta.w.tilde.bc(GDP.Per.head.dist.1995,Epanechnikov)
0.3333485 #estimate of etatile.bc

---

annual Gross Domestic Product (GDP) per head across 15 European Union (EU) countries

Description

Contains values of the GDP/head distribution of 216 EU regions (the so called NUTS-2 level of the Eurostat categorization of territories within the EU for the year 1995.

Usage

GDP.Per.head.dist.1995

Format

A vector with 184 values of the GDP/head distribution for 1995.

Source

Monfort, P. (2008). Convergence of EU regions measures and evolution. EU short papers on regional research and indicators, Directorate-General for Regional Policy 1/2008.

References

Monfort, P. (2008). Convergence of EU regions measures and evolution. EU short papers on regional research and indicators

See Also

GDP.Per.head.dist.2005

---

annual Gross Domestic Product (GDP) per head across 15 European Union (EU) countries

Description

Contains values of the GDP/head distribution of 216 EU regions (the so called NUTS-2 level of the Eurostat categorization of territories within the EU for the year 2005.

Usage

GDP.Per.head.dist.1995

Format

A vector with 184 values of the GDP/head distribution for 2005.

Source

Monfort, P. (2008). Convergence of EU regions measures and evolution. EU short papers on regional research and indicators, Directorate-General for Regional Policy 1/2008.

References

Monfort, P. (2008). Convergence of EU regions measures and evolution. EU short papers on regional research and indicators

See Also

GDP.Per.head.dist.1995

---

Integrated Epanechnikov function

Description

Implements the Integrated Epanechnikov kernel.

Usage

IntEpanechnikov(x)

Arguments

x	A vector of design points with values from $-\sqrt{5}$ to $\sqrt{5}$.

Details

Implements:

$K(u)= \int_{-\infty}^u \frac{3}{4\sqrt{5}} \left (1-\frac{x^2}{5} \right )\,dx$

for $|x| \le \sqrt{5}$

Value

The value of the integrated kernel function at the user designated points.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Kernel Statistics

See Also

Epanechnikov

---

Integrated Kernel density estimator

Description

Classical univariate integrated kernel density estimator

Usage

IntKde(xin, xout, h, kfun)

Arguments

xin	A vector of data points - the available sample size.
xout	grid points where the distribution function will be estimated.
h	The bandwidth parameter. Defaults to $3.572*\sigma*n^{-1/3}$ according to Bowman et al.(1998).
kfun	The kernel to use in the distribution function estimate.

Details

It implements the classical density integrated kernel estimator.

Let $X_1,X_2,\dots, X_n$ be a univariate independent and identically distributed sample drawn from some unknown distribution function $F$. Its kernel density estimator is

$\hat{F}(x)= n^{-1}\sum_{i=1}^n K\left \{ (x-X_i)h^{-1}\right \}$

where $K$ is an integrated kernel, and $h > 0$ is a smoothing parameter called the bandwidth.

Value

Returns a vector with the estimate of the distribution function at the user specified grid points.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Bowman, A., Hall, P., and Prvan, T., (1998), Bandwidth Selection for the Smoothing of Distribution Functions, Biometrika, 799-808.

See Also

bw.nrd, bw.nrd0, bw.ucv, bw.bcv

Examples

x.in <- rnorm(100)
x.out <- seq(-3.4,3.4,length=60)
kernel <- IntEpanechnikov
dist.est <- IntKde(xin=x.in,xout=x.out,kfun=kernel)
plot(x.out,dist.est, type="l", col="red", main="Kernel c.d.f. estimator")

---

Kernel density estimator.

Description

Classical univariate kernel density estimator.

Usage

kde(xin, xout, h, kfun)

Arguments

xin	A vector of data points. Missing values not allowed.
xout	A vector of grid points at which the estimate will be calculated.
h	A scalar, the bandwidth to use in the estimate, e.g. bw.nrd(xin).
kfun	Kernel function to use.

Details

Implements the classical density kernel estimator based on a sample $X_1,X_2,.., X_n$ of i.i.d observations from a distribution $F$ with density $h$. The estimator is defined by

$\hat{f}(x)= n^{-1}\sum_{i=1}^n K_h(x-X_{i})$

where $h$ is determined by a bandwidth selector such as Silverman's default plug-in rule and $K$, the kernel, is a non-negative probability density function.

Value

A vector with the density estimates at the designated points xout.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.

See Also

bw.nrd, bw.nrd0, bw.ucv, bw.bcv

Examples

x.in <- rnorm(100)
x.out <- seq(-3.4,3.4,length=60)
bandwidth <- bw.nrd(x.in)
kernel <- Epanechnikov
dens.est <- kde(x.in,x.out,bandwidth,kernel)
plot(x.out,dens.est,col="red",main="Kernel density estimator")

---

Switch between a range of available cumulative distribution functions.

Description

Returns the value of the selected cumulative distribution function at user supplied grid points.

Usage

p.sample(s,dist, p1,p2)

Arguments

s	A scalar or vector: the x-axis grid points where the cumulative distribution function is be evaluated.
dist	Character string, used as a switch to the user selected distribution function (see details below).
p1	A scalar. Parameter 1 (vector or object) of the selected distribution.
p2	A scalar. Parameter 2 (vector or object) of the selected distribution.

Details

Based on the user-specified argument dist, the function returns the value of the cumulative distribution function at s.

Supported distributions (along with the corresponding dist values) are:

• weib: The Weibull distribution is implemented as

  $F(s) = 1 - \exp \left \{- \left ( \frac{s}{p_2} \right )^{p_1} \right \}$

  with $s > 0$ where $p_1$ is the shape parameter and $p_2$ the scale parameter.

• lognorm: The lognormal distribution is implemented as

  $F(s)=\Phi \left ( \frac{\ln s-p_1 }{p_2} \right )$

  where $p_1$ is the mean,$p_2$ is the standard deviation and $\Phi$ is the cumulative distribution function of the standard normal distribution.

• norm: The normal distribution is implemented as

  $\Phi(s)={\frac {1}{\sqrt {2\pi}p_2 }}\int_{-\infty }^s e^{-\frac{(t-p_1)^2}{2p_2^2}}\,dt$

  where $p_1$ is the mean and the $p_2$ is the standard deviation.

• uni: The uniform distribution is implemented as

  $F(s)=\frac{s-p_1}{p_2-p_1}$

  for $p_1 \le s \le p_2.$

• cauchy: The cauchy distribution is implemented as

  $F(s;p_1,p_2)=\frac{1}{\pi}\arctan \left ( \frac{s-p_1}{p_2} \right ) + \frac{1}{2}$

  where $p_1$ is the location parameter and $p_2$ the scale parameter.

• fnorm: The half normal distribution is implemented as

  $F_S(s;\sigma)=\int_0^s \frac{\sqrt{2/\pi}}{\sigma} \exp \left \{ -\frac{x^2}{2\sigma^2} \right \} \,dx$

  where $mean=0$ and $sd=\sqrt{\pi/2}/p_1$.

• normmixt: The normal mixture distribution is implemented as

  $F(s)=p_1\frac{1}{p_2[2]\sqrt{2\pi}}\int_{-\infty }^{s}e^{-\frac{(t - p_2[1])^2}{2p_2[2]^2}}\,dt + (1-p_1) \frac{1}{p_2[4]\sqrt{2\pi}} \int_{-\infty }^s e^{-\frac{(t - p2[3])^2}{2p_2[4]^2}}\,dt$

  where $p_1$ is a mixture component(scalar) and $p_2$ a vector of parameters for the mean and variance of the two mixture components $p_2=c(mean1,sd1,mean2,sd2)$.

• skewnorm: The skew normal distribution is implemented as

  $F(y; p_1) = \Phi \left ( \frac{y-\xi}{\omega} \right )-2 T \left ( \frac{y-\xi}{\omega},p_1 \right )$

  where $location=\xi=0$, $scale=\omega=1$, $parameter=p_1$ and $T(h, a)$ is the Owens T function, defined by

  $T(h,a) = \frac{1}{2\pi}\int_{0}^{a} \exp \left \{ \frac{- 0.5 h^2 (1+x^2) }{1+x^2} \right \} \,dx, -\infty \le h, a \le \infty$

• fas: The Fernandez and Steel distribution is implemented as

  $F(s;p_1,p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ \int_{-\infty}^s f_t(x/p_1; p_2)I_{\{x \ge 0\}} \,dx + \int_{-\infty}^s f_t(p_1 x; p_2)I_{\{x<0\}}\, dx \right \}$

  where $f_t(x; \nu)$ is the p.d.f. of the t distribution with $\nu = 5$ degrees of freedom.$p_1$ controls the skewness of the distribution with values between $(0, +\infty)$ and $p_2$ is the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  $F(s;\mu, p_2, p_1, \tau) =\int_{-\infty}^s \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}\,dz$

  where $r=\sinh(\sinh(z)- p_1)$, $c=\cosh(\sinh(z)- p_1)$ and $z=(s-\mu)/p_2$. $p_1$ is the vector of skewness, $p_2$ is the scale parameter, $\mu=0$ is the location parameter and $\tau=1$ the kurtosis parameter.

Value

A vector containing the cumulative distribution function values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> , Lucia Gamez Gallardo <[email protected]>

References

Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

r.sample, q.sample, d.sample

Examples

selected.d <- "weib" #select Weibull as the CDF
shape <- 2  # specify shape parameter
scale <- 1 # specify scale parameter
xout <- seq(0.1,5,length=50)  #design point where the CDF is evaluated
p.sample(xout,selected.d,shape,scale)  # calculate CDF at xout

---

Calculate $f^2(x)$

Description

Calculates the square of a density.

Usage

pdfsq(s,dist, p1,p2)

Arguments

s	A scalar or vector: the x-axis grid points where the probability density function will be evaluated.
dist	Character string, used as a switch to the user selected distribution function (see details below).
p1	A scalar. Parameter 1 (vector or object) of the selected density.
p2	A scalar. Parameter 2 (vector or object) of the selected density.

Details

Based on user-specified argument dist, the function returns the value of $f^2(x)dx$, used in the definitions of $\rho_p^*$, $\rho_p$ and their exact versions.

Supported distributions (along with the corresponding dist values) are:

• weib: The weibull distribution is implemented as

  $f(s;p_1,p_2)= \frac{p_1}{p_2} \left (\frac{s}{p_2}\right )^{p_1-1} \exp \left \{- \left (\frac{s}{p_2}\right )^{p_1} \right \}$

  with $s \ge 0$ where $p_1$ is the shape parameter and $p_2$ the scale parameter.

• lognorm: The lognormal distribution is implemented as

  $f(s) = \frac{1}{p_2s\sqrt{2\pi}}e^{-\frac{(log s -p_1)^2}{2p_2^2}}$

  where $p_1$ is the mean and $p_2$ is the standard deviation of the distirbution.

• norm: The normal distribution is implemented as

  $f(s) = \frac{1}{p_2\sqrt{2 \pi}}e^{-\frac{ (s - p_1)^2 }{ 2p_2^2 }}$

  where $p_1$ is the mean and the $p_2$ is the standard deviation of the distirbution.

• uni: The uniform distribution is implemented as

  $f(s) = \frac{1}{p_2-p_1}$

  for $p_1 \le s \le p_2$.

• cauchy: The cauchy distribution is implemented as

  $f(s)=\frac{1}{\pi p_2 \left \{1+( \frac{s-p_1}{p_2})^2\right \} }$

  where $p_1$ is the location parameter and $p_2$ the scale parameter.

• fnorm: The half normal distribution is implemented as

  $2 f(s)-1$

  where

  $f(s) = \frac{1}{sd\sqrt{2 \pi} }e^{-\frac{s^2}{2 sd^2 }},$

  and $sd=\sqrt{\pi/2}/p_1$.

• normmixt:The normal mixture distribution is implemented as

  $f(s)=p_1\frac{1}{p_2[2] \sqrt{2\pi} } e^{- \frac{ (s - p_2[1])^2}{2p_2[2]^2}} +(1-p_1)\frac{1}{p_2[4]\sqrt{2\pi}} e^{-\frac{(s - p_2[3])^2}{2p_2[4]^2 }}$

  where $p1$ is a mixture component(scalar) and $p_2$ a vector of parameters for the mean and variance of the two mixture components $p_2= c(mean1, sd1, mean2, sd2)$.

• skewnorm: The skew normal distribution with parameter $p_1$ is implemented as

  $f(s)=2\phi(s)\Phi(p_1s)$

  .

• fas: The Fernandez and Steel distribution is implemented as

  $f(s; p_1, p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ f_t(s/p_1; p_2) I_{\{s \ge 0\}} + f_t(p_1s; p_2)I_{\{s<0 \}}\right \}$

  where $f_t(x;\nu)$ is the p.d.f. of the $t$ distribution with $\nu = 5$ degrees of freedom. $p_1$ controls the skewness of the distribution with values between $(0, +\infty)$ and $p_2$ denotes the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  $f(s;\mu, p_1, p_2, \tau) = \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}$

  where $r=\sinh(\sinh(z)-(-p_1))$, $c=\cosh(\sinh(z)-(-p_1))$ and $z=((s-\mu)/p2)$. $p_1$ is the vector of skewness, $p_2$ is the scale parameter, $\mu=0$ is the location parameter and $\tau=1$ the kurtosis parameter.

Value

A vector containing the user selected density values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

r.sample, q.sample, p.sample

Examples

selected.dens <- "weib" #select Weibull
  shape <- 2  # specify shape parameter
  scale <- 1 # specify scale parameter
  xout <- seq(0.1,5,length=50)  #design point
  pdfsq(xout,selected.dens,shape,scale)  # calculate the square density at xout

---

Calculate $f^2(x)F(x)$

Description

Return the product $f^2(x)F(x)$

Usage

pdfsqcdf(s,dist, p1,p2)

Arguments

s	A scalar or vector: the x-axis grid points where the probability density function will be evaluated.
dist	Character string, used as a switch to the user selected distribution function (see details below).
p1	A scalar. Parameter 1 (vector or object) of the selected density.
p2	A scalar. Parameter 2 (vector or object) of the selected density.

Details

Based on user-specified argument dist, the function returns the value of $f^2(x)F(x)dx$, used in the definitions of $\rho_p^*$, $\rho_p$ and their exact versions.

Supported distributions (along with the corresponding dist values) are:

• weib: The weibull distribution is implemented as

  $f(s;p_1,p_2)= \frac{p_1}{p_2} \left (\frac{s}{p_2}\right )^{p_1-1} \exp \left \{- \left (\frac{s}{p_2}\right )^{p_1} \right \}$

  with $s \ge 0$ where $p_1$ is the shape parameter and $p_2$ the scale parameter.

• lognorm: The lognormal distribution is implemented as

  $f(s) = \frac{1}{p_2s\sqrt{2\pi}}e^{-\frac{(log s -p_1)^2}{2p_2^2}}$

  where $p_1$ is the mean and $p_2$ is the standard deviation of the distirbution.

• norm: The normal distribution is implemented as

  $f(s) = \frac{1}{p_2\sqrt{2 \pi}}e^{-\frac{ (s - p_1)^2 }{ 2p_2^2 }}$

  where $p_1$ is the mean and the $p_2$ is the standard deviation of the distirbution.

• uni: The uniform distribution is implemented as

  $f(s) = \frac{1}{p_2-p_1}$

  for $p_1 \le s \le p_2$.

• cauchy: The cauchy distribution is implemented as

  $f(s)=\frac{1}{\pi p_2 \left \{1+( \frac{s-p_1}{p_2})^2\right \} }$

  where $p_1$ is the location parameter and $p_2$ the scale parameter.

• fnorm: The half normal distribution is implemented as

  $2 f(s)-1$

  where

  $f(s) = \frac{1}{sd\sqrt{2 \pi} }e^{-\frac{s^2}{2 sd^2 }},$

  and $sd=\sqrt{\pi/2}/p_1$.

• normmixt:The normal mixture distribution is implemented as

  $f(s)=p_1\frac{1}{p_2[2] \sqrt{2\pi} } e^{- \frac{ (s - p_2[1])^2}{2p_2[2]^2}} +(1-p_1)\frac{1}{p_2[4]\sqrt{2\pi}} e^{-\frac{(s - p_2[3])^2}{2p_2[4]^2 }}$

  where $p1$ is a mixture component(scalar) and $p_2$ a vector of parameters for the mean and variance of the two mixture components $p_2= c(mean1, sd1, mean2, sd2)$.

• skewnorm: The skew normal distribution with parameter $p_1$ is implemented as

  $f(s)=2\phi(s)\Phi(p_1s)$

  .

• fas: The Fernandez and Steel distribution is implemented as

  $f(s; p_1, p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ f_t(s/p_1; p_2) I_{\{s \ge 0\}} + f_t(p_1s; p_2)I_{\{s<0 \}}\right \}$

  where $f_t(x;\nu)$ is the p.d.f. of the $t$ distribution with $\nu = 5$ degrees of freedom. $p_1$ controls the skewness of the distribution with values between $(0, +\infty)$ and $p_2$ denotes the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  $f(s;\mu, p_1, p_2, \tau) = \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}$

  where $r=\sinh(\sinh(z)-(-p_1))$, $c=\cosh(\sinh(z)-(-p_1))$ and $z=((s-\mu)/p2)$. $p_1$ is the vector of skewness, $p_2$ is the scale parameter, $\mu=0$ is the location parameter and $\tau=1$ the kurtosis parameter.

Value

A vector containing the user selected density values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]> ,Lucia Gamez Gallardo <[email protected]>

References

Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

r.sample, q.sample, p.sample

Examples

selected.dens <- "weib" #select Weibull
  shape <- 2  # specify shape parameter
  scale <- 1 # specify scale parameter
  xout <- seq(0.1,5,length=50)  #design point
  pdfsqcdf(xout,selected.dens,shape,scale)  # calculate pdfsqcdf function at xout

---

Calculate $f^2(x)(1-F(x))$.

Description

Return the product $f^2(x)(1-F(x))$.

Usage

pdfsqcdfstar(s,dist, p1,p2)

Arguments

s	A scalar or vector: the x-axis grid points where the probability density function will be evaluated.
dist	Character string, used as a switch to the user selected distribution function (see details below).
p1	A scalar. Parameter 1 (vector or object) of the selected density.
p2	A scalar. Parameter 2 (vector or object) of the selected density.

Details

Based on user-specified argument dist, the function returns the value of $f^2(x)(1-F(x))dx$, used in the definitions of $\rho_p^*$, $\rho_p$ and their exact versions.

Supported distributions (along with the corresponding dist values) are:

• weib: The weibull distribution is implemented as

  $f(s;p_1,p_2)= \frac{p_1}{p_2} \left (\frac{s}{p_2}\right )^{p_1-1} \exp \left \{- \left (\frac{s}{p_2}\right )^{p_1} \right \}$

  with $s \ge 0$ where $p_1$ is the shape parameter and $p_2$ the scale parameter.

• lognorm: The lognormal distribution is implemented as

  $f(s) = \frac{1}{p_2s\sqrt{2\pi}}e^{-\frac{(log s -p_1)^2}{2p_2^2}}$

  where $p_1$ is the mean and $p_2$ is the standard deviation of the distirbution.

• norm: The normal distribution is implemented as

  $f(s) = \frac{1}{p_2\sqrt{2 \pi}}e^{-\frac{ (s - p_1)^2 }{ 2p_2^2 }}$

  where $p_1$ is the mean and the $p_2$ is the standard deviation of the distirbution.

• uni: The uniform distribution is implemented as

  $f(s) = \frac{1}{p_2-p_1}$

  for $p_1 \le s \le p_2$.

• cauchy: The cauchy distribution is implemented as

  $f(s)=\frac{1}{\pi p_2 \left \{1+( \frac{s-p_1}{p_2})^2\right \} }$

  where $p_1$ is the location parameter and $p_2$ the scale parameter.

• fnorm: The half normal distribution is implemented as

  $2 f(s)-1$

  where

  $f(s) = \frac{1}{sd\sqrt{2 \pi} }e^{-\frac{s^2}{2 sd^2 }},$

  and $sd=\sqrt{\pi/2}/p_1$.

• normmixt:The normal mixture distribution is implemented as

  $f(s)=p_1\frac{1}{p_2[2] \sqrt{2\pi} } e^{- \frac{ (s - p_2[1])^2}{2p_2[2]^2}} +(1-p_1)\frac{1}{p_2[4]\sqrt{2\pi}} e^{-\frac{(s - p_2[3])^2}{2p_2[4]^2 }}$

  where $p1$ is a mixture component(scalar) and $p_2$ a vector of parameters for the mean and variance of the two mixture components $p_2= c(mean1, sd1, mean2, sd2)$.

• skewnorm: The skew normal distribution with parameter $p_1$ is implemented as

  $f(s)=2\phi(s)\Phi(p_1s)$

  .

• fas: The Fernandez and Steel distribution is implemented as

  $f(s; p_1, p_2) = \frac{2}{p_1+\frac{1}{p_1}} \left \{ f_t(s/p_1; p_2) I_{\{s \ge 0\}} + f_t(p_1s; p_2)I_{\{s<0 \}}\right \}$

  where $f_t(x;\nu)$ is the p.d.f. of the $t$ distribution with $\nu = 5$ degrees of freedom. $p_1$ controls the skewness of the distribution with values between $(0, +\infty)$ and $p_2$ denotes the degrees of freedom.

• shash: The Sinh-Arcsinh distribution is implemented as

  $f(s;\mu, p_1, p_2, \tau) = \frac{ce^{-r^2/2}}{\sqrt{2\pi }} \frac{1}{p_2} \frac{1}{2} \sqrt{1+z^2}$

  where $r=\sinh(\sinh(z)-(-p_1))$, $c=\cosh(\sinh(z)-(-p_1))$ and $z=((s-\mu)/p2)$. $p_1$ is the vector of skewness, $p_2$ is the scale parameter, $\mu=0$ is the location parameter and $\tau=1$ the kurtosis parameter.

Value

A vector containing the user selected density values at the user specified points s.

Author(s)

Dimitrios Bagkavos and Lucia Gamez Gallardo

R implementation and documentation: Dimitrios Bagkavos <[email protected]>, Lucia Gamez Gallardo <[email protected]>

References

Bagkavos D., Patil P.N., Wood A.T.A. (2016), A Numerical Study of the Power Function of a New Symmetry Test. In: Cao R., Gonzalez Manteiga W., Romo J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics and Statistics, vol 175, Springer.

See Also

r.sample, q.sample, p.sample

Examples

selected.dens <- "weib" #select Weibull
  shape <- 2  # specify shape parameter
  scale <- 1 # specify scale parameter
  xout <- seq(0.1,5,length=50)  #design point
  pdfsqcdfstar(xout,selected.dens,shape,scale)  #return f^2(xout)F(xout)

---