The normalp Package. October 2, 2007

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1 Version Date The normal Package October 2, 2007 Title Package for exonential ower distributions (EPD) Author Maintainer Deends R (>= 1.5.0) Descrition A collection of utilities refered to exonential ower distributions, also known as General Error Distribution License GPL-2 URL htt:// htt://dssm.unia.it/elio R toics documented: dnorm estimate grahn kurtosis lm aram lot.lm lot.simul.lm lot.simul.m norm qnorm qqnorm rnorm simul.lm simul.m summary.lm summary.simul.lm Index 20 1

2 2 dnorm dnorm Density function of an exonential ower distribution Descrition Density function for the exonential ower distribution with location arameter mu, scale arameter sigma and shae arameter. dnorm(x, mu=0, sigma=1, =2, log=false) x mu sigma log Vector of quantiles. Vector of location arameters. Vector of scale arameters. Shae arameter. Logical; if TRUE, the density is given as log(density). Details If mu, sigma or are not secified they assume the default values 0, 1 and 2, resectively. The exonential ower distribution has density function f(x) = 1 e x µ σ 2 (1/) Γ(1 + 1/)σ where µ is the location arameter, σ the scale arameter and the shae arameter. When = 2 the exonential ower distribution becomes the Normal Distribution, when = 1 the exonential ower distribution becomes the Lalace Distribution, when the exonential ower distribution becomes the Uniform Distribution. dnorm gives the density function of an exonential ower distribution. See Also Normal for the Normal distribution, Uniform for the Uniform distribution, and Secial for the Gamma function.

3 estimate 3 Examles ## Comute the density for a vector x with mu=0, sigma=1 and =1.5 ## At the end we have the grah of the exonential ower distribution ## density function with =1.5 x <- c(-1, 1) f <- dnorm(x, =1.5) rint(f) lot(function(x) dnorm(x, =1.5), -4, 4, main = "Exonential ower distribution density function (=1.5)", ylab="f(x)") estimate Estimation of Descrition The estimate function estimates the shae arameter from a vector of observations. estimate(x, mu, =2, method=c("inverse","direct")) x mu method Vector of observations. An estimate of the location arameter. Starting value of the shae arameter. Method used to estimate from a samle. Details The used algorithm is based on a method roosed by A.M. Mineo (1994), which uses a articular index of kurtosis, called V I Γ(1/)Γ(3/) V I =. Γ(2/) With method the user can choice between an inverse interolation (faster) or a direct solution of the equation An estimate of from a samle of observations. Vˆ Γ(1/)Γ(3/) I =. Γ(2/)

4 4 grahn References Mineo, A.M. (1994) Un nuovo metodo di stima di er una corretta valutazione dei arametri di intensità e di scala di una curva normale di ordine, Atti della XXXVII Riunione Scientifica della Società Italiana di Statistica, San Remo, Vol. 2, Examles x<-rnorm(300,mu=1,sigma=2,=4) <-estimate(x,mu=1,=2) grahn Plot of exonential ower distributions Descrition The function grahn returns on the same device, marked with different colours, from one to five exonential ower distributions. grahn(=c(1,2,3,4,5), mu=0, sigma=1, title="exonential Power Distributions") mu sigma title A vector of values. His length must be maximum five. of the location arameter. of the scale arameter. The title of the lot. Details If one or more values of are greater than or equal to 50, grahn will lot the density function of an uniform distribution. A grahic device with till five different curves. The curves have different colours and the device is comleted by a legend.

5 kurtosis 5 Examles ## Plot four different curves with =1,2,3,4 ## and 50 (it will lot an uniform distribution) grahn(c(1:4,50)) kurtosis Indices of kurtosis Descrition This function comutes the theoretical and emirical values of three indices of kurtosis. kurtosis(x = NULL,, value = c("estimate", "arameter")) x value A samle of observations. the shae arameter. If is set to estimate, evaluate the indices using an estimate of. Otherwise, if is set to arameter it uses the value secified in. It returns the vector of the three indices of kurtosis V I, β 2 and β. Giving a vector as argument, it returns the estimates of the three indices, comuted on the samle. On the other hand, giving the value of the shae arameter, it returns the theoretical indices. References Mineo, A.M. (1996) La migliore combinazione delle osservazioni: curve normali di ordine e stimatori di norma L. PhD thesis. Examles kurtosis(=2) x<-rnorm(50,mu=0,sigma=2,=1.5) kurtosis(x,=2)

6 6 lm lm Fitted linear model with exonential ower distribution errors Descrition The function lm is used to fit linear model. It can be used when the errors are distributed as an exonential ower distribution. lm(formula, data, ) formula data A symbolic descrition of the model to be fitted. An otional data frame containing the variables in the model. By default the variables are taken from the environment. The shae arameter. If secified, this function estimates the arameter by using the L norm method. Details To evaluate the coefficients of the linear model, lm uses the maximum likelihood estimators. This function can give some roblems if the number of regressors is too high. The function lm returns an object of class "lm" and "lm". The function summary rint a summary of the results. The generic accessor functions coefficients, effects, fitted.values and residuals extract various useful features of the value returned by lm. An object of class "lm" is a list containing at least the following comonents: coefficients A named vector of coefficients. residuals The residuals, that is resonses minus fitted values. fitted.values The fitted values. rank df.residual call terms kn model iter The numeric rank of the fitted linear model. The residual degrees of freedom comuted as in lm. The matched call. The terms object used. Estimate of the shae arameter comuted on residuals. A logical arameter used by summary. The model frame used. If its value is 1 we have had a difficult convergence.

7 aram 7 References Mineo, A.M. (1995) Stima dei arametri di regressione lineare semlice quando gli errori seguono una distribuzione normale di ordine ( incognito). Annali della Facoltà di Economia dell Università di Palermo (Area Statistico-Matematica), Examles e<-rnorm(n=100,mu=0,sigma=4,=3,method="d") x<-runif(100) y<-0.5+2*x+e lm(y~x) aram Estimation of location and scale arameters Descrition The function aram returns a list with five elements: arithmetic mean, M, standard deviation, S, and shae arameter, estimated on a samle. aram(x, ) x A vector of observations. If secified, the algorithm uses this value for, i.e. the algorithm does not use an estimate of.... Further arguments assed to or from other methods. The estimation of µ and is based on an iterative method. To show differences between the least squares method and the maximum likelihood method, it rints out also the mean and the standard deviation. Mean M Sd S iter Arithmetic mean. The estimated value of the location arameter. Standard deviation. The estimated value of the scale arameter. The estimated value of the shae arameter. If its value is 1, we have had roblems on convergence.

8 8 lot.lm References Mineo, A.M. (1996) La migliore combinazione delle osservazioni: curve normali di ordine e stimatori di norma L. PhD thesis. Examles x<-rnorm(1000,2,3,4.2) aram(x) lot.lm Diagnostic lots for a lm object Descrition This function roduces four lots: a lot of residuals against fitted value, a Normal Q-Q lot, an Exonential Power Distribution Q-Q lot, a Scale-Location lot, with a -root of the standardized residuals against the fitted values. ## S3 method for class 'lm': lot(x,...) x A lm object, tyically result of lm.... Further arguments assed to or from other methods. Details The standardized residuals in the Normal Q-Q lot are those of an object lm; in the Exonential Power distribution Q-Q lot and in the scale location lot the standardized residuals are comuted as (e i m )/(s ).

9 lot.simul.lm 9 Examles x<-1:20 z<-runif(20) e<-rnorm(20,mu=0,sigma=1,=3) y<-0.5+x+z+e lm.res<-lm(y~x+z) lot(lm.res) lot.simul.lm Plots of the results of a simulation lan on a linear regression model Descrition It returns the histograms of the estimates of the regression coefficients, of the scale arameter σ and of the shae arameter. ## S3 method for class 'simul.lm': lot(x,...) x A simul.lm object, tyically result of simul.lm... Further arguments assed to or from other methods The histograms of all the coefficients of the linear regression model and of the estimates of the scale arameter σ and of the structure arameter. Examles sim<-simul.lm(n=10,m=50,q=1,data=1.5,int=0,sigma=1,=3.5) lot(sim)

10 10 norm lot.simul.m Plots of the results of a simulation lan on the arameters of an exonential ower distribution Descrition It returns the histograms of the vector of means, estimates of µ, standard deviations, estimates of σ and estimates of. ## S3 method for class 'simul.m': lot(x,...) x A simul.m object, tyically result of simul.m... Further arguments assed to or from other methods The histograms of the estimates of the arameters of an exonential ower distribution. Examles ## The histograms of all the comuted estimates a<-simul.m(100,50,mu=0,sigma=1,=3) lot(a) norm Probability function of an exonential ower distribution Descrition Probability function for the exonential ower distribution with location arameter mu, scale arameter sigma and shae arameter. norm(q, mu=0, sigma=1, =2, lower.tail=true, log.r=false)

11 norm 11 q mu sigma lower.tail log.r Vector of quantiles. Vector of location arameters. Vector of scale arameters. Shae arameter. Logical; if TRUE (default), robabilities are P [X x], otherwise, P [X > x]. Logical; if TRUE, robabilities r are given as log(r). Details If mu, sigma or are not secified they assume the default values 0, 1 and 2, resectively. The exonential ower distribution has density function f(x) = 1 e x µ σ 2 (1/) Γ(1 + 1/)σ where µ is the location arameter, σ the scale arameter and the shae arameter. When = 2 the exonential ower distribution becomes the Normal Distribution, when = 1 the exonential ower distribution becomes the Lalace Distribution, when the exonential ower distribution becomes the Uniform Distribution. norm gives the robability of an exonential ower distribution. See Also Normal for the Normal distribution, Uniform for the Uniform distribution, and Secial for the Gamma function. Examles ## Comute the distribution function for a vector x with mu=0, sigma=1 and =1.5 ## At the end we have the grah of the exonential ower distribution function with =1.5. x <- c(-1, 1) r <- norm(x, =1.5) rint(r) lot(function(x) norm(x, =1.5), -4, 4, main = "Exonential Power Distribution Function (=1.5)", ylab="f(x)")

12 12 qnorm qnorm Quantiles of an exonential ower distribution Descrition Quantiles for the exonential ower distribution with location arameter mu, scale arameter sigma and shae arameter. qnorm(r, mu=0, sigma=1, =2, lower.tail=true, log.r=false) r mu sigma lower.tail log.r Vector of robabilities. Vector of location arameters. Vector of scale arameters. Shae arameter. Logical; if TRUE (default), robabilities are P [X x], otherwise, P [X > x]. Logical; if TRUE, robabilities r are given as log(r). Details If mu, sigma or are not secified they assume the default values 0, 1 and 2, resectively. The exonential ower distribution has density function f(x) = 1 e x µ σ 2 (1/) Γ(1 + 1/)σ where µ is the location arameter, σ the scale arameter and the shae arameter. When = 2 the exonential ower distribution becomes the Normal Distribution, when = 1 the exonential ower distribution becomes the Lalace Distribution, when the exonential ower distribution becomes the Uniform Distribution. qnorm gives the quantiles of an exonential ower distribution. See Also Normal for the Normal distribution, Uniform for the Uniform distribution, and Secial for the Gamma function.

13 qqnorm 13 Examles ## Comute the quantiles for a vector of robabilities x ## with mu=1, sigma=2 and =1.5 x <- 0.3 q <- qnorm(x, 1, 2, 1.5) q qqnorm Quantile-Quantile lot for an exonential ower distribution Descrition The function qqnorm roduces an exonential ower distribution Q-Q lot of the values in y. The function qqline adds a line to an exonential ower distribution Q-Q lot going through the first and the third quartile. qqnorm(y, ylim,, main, xlab, ylab,...) qqline(y, =2,...) y Vector of observations. The shae arameter. main,xlab,ylab Plot labels. ylim,... Grahical arameters Examles ## Exonential ower distribution Q-Q lot for a samle of 100 observations. e<-rnorm(100,mu=0,sigma=1,=3) qqnorm(e,=3) qqline(e,=3)

14 14 rnorm rnorm Pseudo-random numbers from an exonential ower distribution Descrition Generation of seudo-random numbers from an exonential ower distribution with location arameter mu, scale arameter sigma and shae arameter. rnorm(n, mu = 0, sigma = 1, = 2, method = c("def", "chiodi")) n mu sigma method Number of observations. Vector of location arameters. Vector of scale arameters. Shae arameter. If is set to the default method "def", it uses the method based on the transformation of a Gamma random variable. If set to "chiodi", it uses an algorithm based on a generalization of the Marsaglia formula to generate seudo-random numbers from a normal distribution. The default method "def" is faster than the "chiodi" one (this one is introduced only for "historical" uroses). Details If mu, sigma or are not secified they assume the default values 0, 1 and 2, resectively. The exonential ower distribution has density function f(x) = 1 e x µ σ 2 (1/) Γ(1 + 1/)σ where µ is the location arameter, σ the scale arameter and the shae arameter. When = 2 the exonential ower distribution becomes the Normal Distribution, when = 1 the exonential ower distribution becomes the Lalace Distribution, when the exonential ower distribution becomes the Uniform Distribution. rnorm gives a vector of n seudo-random numbers from an exonential ower distribution.

15 simul.lm 15 References Chiodi, M. (1986) Procedures for generating seudo-random numbers from a normal distribution of order (>1), Statistica Alicata, 1, Marsaglia, G. and Bray, T.A. (1964) A convenient method for generating normal variables, SIAM rev., 6, See Also Normal for the Normal distribution, Uniform for the Uniform distribution, Secial for the Gamma function and.random.seed for the random number generation. Examles ## Generate a random samle x from an exonential ower distribution ## At the end we have the histogram of x x <- rnorm(1000, 1, 2, 1.5) hist(x, main="histogram of the random samle") simul.lm Simulation lanning for a linear regression model with errors distributed as an exonential ower distribution Descrition This function erforms a Monte Carlo simulation to comare least squares estimators and Maximum Likelihood estimators for a linear regression model with errors distributed as an exonential ower distribution. The regressors are drawn from an Uniform distribution. simul.lm(n, m, q, data, int=0, sigma=1, =2, l=false) n m q data int sigma l Samle size. Number of samles. Number of regressors. A vector of coefficients. of the intercet. The scale arameter. The shae arameter. Logical. If TRUE, it evaluates the coefficients with known.

16 16 simul.m The function simul.lm returns an object of class "simul.lm". A comonent of this object is a table of means and variances of the m estimates of the regression coefficients and of the scale aramenter σ. The summary shows this table and the arguments of the simulation lan. The function lot returns the histograms of the comuted estimates. References Mineo, A.M. (1995) Stima dei arametri di regressione lineare semlice quando gli errori seguono una distribuzione normale di ordine ( incognito). Annali della Facoltà di Economia dell Università di Palermo (Area Statistico-Matematica), Examles ## Simulation of 50 samles of size 10 for a linear regression model with 1 regressor. simul.lm(10,50,1,data=1.5,int=1,sigma=1,=3,l=false) simul.m Simulation lanning for the arameters of an exonential ower distribution Descrition This function erforms a Monte Carlo simulation to comare least square estimators and Maximum Likelihood estimators for the arameters of an exonential ower distribution. For each samle, it calls the function aram, returning the arithmetic means, the max-likelihood estimates of the location arameter, the standard deviations, the max-likelihood estimates of the scale arameter and the estimates of the shae arameter. simul.m(n, m, mu=0, sigma=1, =2) n m mu sigma Samle size. Number of samles. of the location arameter. of the scale arameter. the shae arameter.

17 summary.lm 17 This function is useful to comare several kinds of estimators. It returns an object of class "simul.m", a list containing the following comonents: dat table A matrix m 5 containing the results of aram for each samle. A matrix reorting the means and the variances of the values of the five estimators. References Mineo, A.M. (1995) Stima dei arametri di intensità e di scala di una curva normale di ordine ( incognito). Annali della Facoltà di Economia dell Università di Palermo (Area Statistico- Matematica), Examles ## Simulation lan for 100 samles of size 20, with mu=0, sigma=1, =3. simul.m(20,100,mu=0,sigma=1,=3) summary.lm Summarize linear model fits with exonential ower distribution errors Descrition This function is the summary method for class "lm". This function roduces a set of results for a linear regression model. By assuming that in a linear regression model the errors are distributed as an exonential ower distribution, we can use the function lm. ## S3 method for class 'lm': summary(object,...) ## S3 method for class 'summary.lm': rint(x,...) object An object of class "lm", a result of a call to lm. x An object of class "summary.lm".... Further arguments assed to or from other methods.

18 18 summary.simul.lm The function summary returns a list of summary statistics of the fitted linear model given in lm, using the comonents (list elements) call and terms from its argument, lus Call The matched call. Residuals A summary of the vector of residuals e i. Coefficients Vector of coefficients. Estimate of An estimate of the shae arameter. Power deviation of order The ower deviation of order given by Examles x<-runif(30) e<-rnorm(30,0,3,1.25) y<-0.5+x+e L<-lm(y~x) summary(l) S = [ e ] 1 i n q where q is either the number of the estimated regression coefficients if is known, either the number of the estimated regression coefficients lus 1 if is estimated. summary.simul.lm Summarize simulation results on linear regression model Descrition This function is the summary method for class "simul.lm". This function roduces a set of results for a simulation lan for a linear regression model with errors distributed as an exonential ower distribution. ## S3 method for class 'simul.lm': summary(object,...) ## S3 method for class 'summary.simul.lm': rint(x,...)

19 summary.simul.lm 19 object x An object of class "simul.lm", a result of a call to simul.lm. An object of class "summary.simul.lm", usually a result of a call to summary.simul.lm.... Further arguments assed to or from other methods. This function returns this information: Results Table containing the simulation results. Coefficients The true values of coefficients used on the simulation model. Formula The used linear regression model. Number of samles Number of samles generated. of of the shae arameter used to draw the samles. Number of samles with roblems on convergence If is estimated, we have information on the number of samles with roblems on convergence. Examles ris<-simul.lm(100,20,2,data=c(3,2),int=0,sigma=1,=3) summary(ris)

20 Index Toic alot grahn, 4 Toic distribution dnorm, 1 norm, 10 qnorm, 12 rnorm, 14 Toic hlot lot.lm, 8 lot.simul.lm, 9 lot.simul.m, 10 qqnorm, 13 Toic regression lm, 6 simul.lm, 15 summary.lm, 17 summary.simul.lm, 18 Toic univar estimate, 3 kurtosis, 5 aram, 7 simul.m, 16.Random.seed, 15 call, 18 class, 6 lot.lm, 8 lot.simul.lm, 9 lot.simul.m, 10 norm, 10 rint.aram (aram), 7 rint.simul.m (simul.m), 16 rint.summary.lm (summary.lm), 17 rint.summary.simul.lm (summary.simul.lm), 18 qnorm, 12 qqline (qqnorm), 13 qqnorm, 13 rnorm, 14 simul.lm, 9, 15, 19 simul.m, 10, 16 Secial, 2, 11, 12, 15 summary.lm, 17 summary.simul.lm, 18, 19 terms, 6, 18 Uniform, 2, 11, 12, 15 dnorm, 1 estimate, 3 grahn, 4 kurtosis, 5 lm, 8 lm, 6, 8, 17, 18 Normal, 2, 11, 12, 15 aram, 7, 16, 17 20