Title: | Roy's Bivariate Geometric Distribution |
---|---|
Description: | Implements Roy's bivariate geometric model (Roy (1993) <doi:10.1006/jmva.1993.1065>): joint probability mass function, distribution function, survival function, random generation, parameter estimation, and more. |
Authors: | Alessandro Barbiero |
Maintainer: | Alessandro Barbiero <[email protected]> |
License: | GPL |
Version: | 1.0 |
Built: | 2024-12-01 08:51:13 UTC |
Source: | CRAN |
Implements Roy's bivariate geometric model (Roy (1993) <doi:10.1006/jmva.1993.1065>): joint probability mass function, distribution function, survival function, random generation, parameter estimation, and more.
The DESCRIPTION file:
Package: | bivgeom |
Type: | Package |
Title: | Roy's Bivariate Geometric Distribution |
Version: | 1.0 |
Date: | 2018-10-17 |
Author: | Alessandro Barbiero |
Maintainer: | Alessandro Barbiero <[email protected]> |
Imports: | methods, stats, utils, bbmle, copula |
Description: | Implements Roy's bivariate geometric model (Roy (1993) <doi:10.1006/jmva.1993.1065>): joint probability mass function, distribution function, survival function, random generation, parameter estimation, and more. |
License: | GPL |
NeedsCompilation: | no |
Packaged: | 2018-10-17 16:23:28 UTC; admin |
Repository: | CRAN |
Date/Publication: | 2018-10-26 15:20:06 UTC |
Config/pak/sysreqs: | libgsl0-dev |
Index of help topics:
EyxbivgeomRoy Conditional moment FbivgeomRoy Joint distribution function FyxbivgeomRoy Conditional distribution RelbivgeomRoy Reliability parameter S.n Empirical joint survival function SbivgeomRoy Joint survival function bivgeom-package Roy's Bivariate Geometric Distribution corbivgeomRoy Linear correlation dbivgeomRoy Joint probability mass function estbivgeomRoy Parameter estimation lambda1Roy Bivariate failure rates lambda2Roy Bivariate failure rate loglikgeomRoy Log-likelihood function minuslogRoy Log-likelihood function rbivgeomRoy Pseudo-random generation
Alessandro Barbiero
Maintainer: Alessandro Barbiero ([email protected])
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
Barbiero, A. (2018) Properties and estimation of a bivariate geometric model with locally constant failure rates, submitted
dbivgeomRoy
, rbivgeomRoy
, estbivgeomRoy
, FbivgeomRoy
##################################### #### MONTE CARLO SIMULATION PLAN #### ##################################### # setting the parameters' values theta1 <- 0.3 theta2 <- 0.7 theta3 <- 0.6 N <- 20 # number of Monte Carlo runs n <- 100 # sample size # arranging the array containig the simulation results # N runs, 7 methods, 3 estimates h <- array(0,c(N,7,3)) # setting the seed set.seed(12345) # function for handling missing values # when computing the mean and standard deviation of the estimates: meanrm <- function(x){mean(x,na.rm=TRUE)} sdrm <- function(x){sd(x,na.rm=TRUE)} colnames <- c("ML","MMP","MM1","MM2","MM3","MM4","LS") dimnames(h)[[2]] <- colnames # Monte Carlo simulation: for(i in 1:N) { d <- rbivgeomRoy(n,theta1,theta2,theta3) cat("MC run #",i,"\n") x<-d[,1] y<-d[,2] # implementing all the estimation methods # and saving the point estimates in the array h[i,1,] <- estbivgeomRoy(x, y, "ML") h[i,2,] <- estbivgeomRoy(x, y, "MMP") h[i,3,] <- estbivgeomRoy(x, y, "MM1") h[i,4,] <- estbivgeomRoy(x, y, "MM2") h[i,5,] <- estbivgeomRoy(x, y, "MM3") h[i,6,] <- estbivgeomRoy(x, y, "MM4") h[i,7,] <- estbivgeomRoy(x, y, "LS") } # printing MC expected values and standard errors # for each of the proposed estimation methods cat("hattheta1:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,1],se=apply(h,c(2,3),sdrm)[,1]) cat("hattheta2:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,2],se=apply(h,c(2,3),sdrm)[,2]) cat("hattheta3:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,3],se=apply(h,c(2,3),sdrm)[,3]) # boxplots of MC distribution of the estimators of theta3 boxplot(h[,,3]) abline(h=theta3, lty=3)
##################################### #### MONTE CARLO SIMULATION PLAN #### ##################################### # setting the parameters' values theta1 <- 0.3 theta2 <- 0.7 theta3 <- 0.6 N <- 20 # number of Monte Carlo runs n <- 100 # sample size # arranging the array containig the simulation results # N runs, 7 methods, 3 estimates h <- array(0,c(N,7,3)) # setting the seed set.seed(12345) # function for handling missing values # when computing the mean and standard deviation of the estimates: meanrm <- function(x){mean(x,na.rm=TRUE)} sdrm <- function(x){sd(x,na.rm=TRUE)} colnames <- c("ML","MMP","MM1","MM2","MM3","MM4","LS") dimnames(h)[[2]] <- colnames # Monte Carlo simulation: for(i in 1:N) { d <- rbivgeomRoy(n,theta1,theta2,theta3) cat("MC run #",i,"\n") x<-d[,1] y<-d[,2] # implementing all the estimation methods # and saving the point estimates in the array h[i,1,] <- estbivgeomRoy(x, y, "ML") h[i,2,] <- estbivgeomRoy(x, y, "MMP") h[i,3,] <- estbivgeomRoy(x, y, "MM1") h[i,4,] <- estbivgeomRoy(x, y, "MM2") h[i,5,] <- estbivgeomRoy(x, y, "MM3") h[i,6,] <- estbivgeomRoy(x, y, "MM4") h[i,7,] <- estbivgeomRoy(x, y, "LS") } # printing MC expected values and standard errors # for each of the proposed estimation methods cat("hattheta1:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,1],se=apply(h,c(2,3),sdrm)[,1]) cat("hattheta2:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,2],se=apply(h,c(2,3),sdrm)[,2]) cat("hattheta3:","\n") cbind(mean=apply(h,c(2,3),meanrm)[,3],se=apply(h,c(2,3),sdrm)[,3]) # boxplots of MC distribution of the estimators of theta3 boxplot(h[,,3]) abline(h=theta3, lty=3)
Linear correlation for Roy's bivariate geometric model
corbivgeomRoy(theta1, theta2, theta3)
corbivgeomRoy(theta1, theta2, theta3)
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
the value of Pearson's linear correlation - see Barbiero (2018). The linear correlation for Roy's bivariate geometric distribution is negative (or null, for ) for any feasible choice of its parameters
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
Barbiero, A. (2018) Properties and estimation of a bivariate geometric model with locally constant failure rates, submitted
corbivgeomRoy(0.3,0.7,0.5)
corbivgeomRoy(0.3,0.7,0.5)
Joint probability mass function for Roy's bivariate geometric model
dbivgeomRoy(x, y, theta1, theta2, theta3)
dbivgeomRoy(x, y, theta1, theta2, theta3)
x |
vector of values for the first variable |
y |
vector of values for the second variable |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
Value of the probability .
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
dbivgeomRoy(x=2, y=0, theta1=0.7, theta2=0.2, theta3=0.8) dbivgeomRoy(0:5, y=0, theta1=0.7, theta2=0.2, theta3=0.8) # these are p(0,0), p(1,0), ..., p(5,0) dbivgeomRoy(0:2, 1:3, theta1=0.7, theta2=0.2, theta3=0.8) # these are p(0,1), p(1,2), p(2,3)
dbivgeomRoy(x=2, y=0, theta1=0.7, theta2=0.2, theta3=0.8) dbivgeomRoy(0:5, y=0, theta1=0.7, theta2=0.2, theta3=0.8) # these are p(0,0), p(1,0), ..., p(5,0) dbivgeomRoy(0:2, 1:3, theta1=0.7, theta2=0.2, theta3=0.8) # these are p(0,1), p(1,2), p(2,3)
Parameter estimation for Roy's bivariate geometric model
estbivgeomRoy(x, y, method = "LS")
estbivgeomRoy(x, y, method = "LS")
x |
vector of observations from the first variable |
y |
vector of observations from the first variable |
method |
One of the possible estimation methods: "ML" (maximum likelihood), "LS" (least squares), "MMP" (method of moment and poroportion), "M1", "M2", "M3", and "M4" (several variants of the method of moments) |
a vector of length 3 containing the estimates of ,
, and
Alessandro Barbiero
Barbiero, A. (2018) Properties and estimation of a bivariate geometric model with locally constant failure rates, submitted
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # parameter estimation, using the different proposed methods: hattheta <- estbivgeomRoy(d[,1], d[,2], "ML") hattheta # MLEs estbivgeomRoy(d[,1], d[,2], "LS") estbivgeomRoy(d[,1], d[,2], "MMP")
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # parameter estimation, using the different proposed methods: hattheta <- estbivgeomRoy(d[,1], d[,2], "ML") hattheta # MLEs estbivgeomRoy(d[,1], d[,2], "LS") estbivgeomRoy(d[,1], d[,2], "MMP")
Conditional moment of given
for Roy's bivariate geomtric model
EyxbivgeomRoy(theta1, theta2, theta3, x)
EyxbivgeomRoy(theta1, theta2, theta3, x)
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
x |
value of the conditioning variable |
Value of the conditional moment of given
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 EyxbivgeomRoy(theta1, theta2, theta3, 2)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 EyxbivgeomRoy(theta1, theta2, theta3, 2)
Joint cumulative distribution function for Roy's bivariate geometric model
FbivgeomRoy(x, y, theta1, theta2, theta3)
FbivgeomRoy(x, y, theta1, theta2, theta3)
x |
vector of values for the first variable |
y |
vector of values for the second variable |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
The probability
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that X<=2 and Y<=3: FbivgeomRoy(2, 3, theta1, theta2, theta3)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that X<=2 and Y<=3: FbivgeomRoy(2, 3, theta1, theta2, theta3)
Conditional distribution function of given
FyxbivgeomRoy(y, theta1, theta2, theta3, x)
FyxbivgeomRoy(y, theta1, theta2, theta3, x)
y |
vector of observations from |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
x |
value of the conditioning variable |
The value of the conditional cumulative distribution function in
. Used in
rbivgeomRoy
for conditional sampling
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that Y<=3 given that X=2: FyxbivgeomRoy(3, theta1, theta2, theta3, 2) # the unconditional probability would be pgeom(3, 1-theta2) # i.e. a geometric distribution with parameter 1-theta2
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that Y<=3 given that X=2: FyxbivgeomRoy(3, theta1, theta2, theta3, 2) # the unconditional probability would be pgeom(3, 1-theta2) # i.e. a geometric distribution with parameter 1-theta2
Bivariate failure rate
lambda1Roy(x, y, theta1, theta2, theta3)
lambda1Roy(x, y, theta1, theta2, theta3)
x |
observation from the first variable |
y |
observation from the second variable |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
It is defined as . For this model,
Value of the bivariate failure rate for Roy's bivariate geometric model (Roy, 1993)
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # bivariate failure rate lambda1 # computed in x=1, y=2 x <- 1 y <- 2 lambda1Roy(x,y,theta1,theta2,theta3)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # bivariate failure rate lambda1 # computed in x=1, y=2 x <- 1 y <- 2 lambda1Roy(x,y,theta1,theta2,theta3)
Bivariate failure rate
lambda2Roy(x, y, theta1, theta2, theta3)
lambda2Roy(x, y, theta1, theta2, theta3)
x |
observation from the first variable |
y |
observation from the second variable |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
It is defined as . For this model,
Value of the bivariate failure rate for Roy's bivariate geometric model (Roy, 1993)
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # bivariate failure rate lambda 2 # computed in x=1, y=2 x <- 1 y <- 2 lambda2Roy(x,y,theta1,theta2,theta3)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # bivariate failure rate lambda 2 # computed in x=1, y=2 x <- 1 y <- 2 lambda2Roy(x,y,theta1,theta2,theta3)
Negative log-likelihood function for Roy's bivariate geometric model
loglikgeomRoy(par, x, y)
loglikgeomRoy(par, x, y)
par |
a vector containing the values of the three parameters |
x |
numeric vector of sample |
y |
numeric vector of sample |
Value of the negative log-likelihood function
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # parameter estimation, using the different proposed methods: hattheta <- estbivgeomRoy(d[,1], d[,2], "ML") loglikgeomRoy(hattheta, x=d[,1], y=d[,2]) # negative value of the (maximized) log-likelihood function
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # parameter estimation, using the different proposed methods: hattheta <- estbivgeomRoy(d[,1], d[,2], "ML") loglikgeomRoy(hattheta, x=d[,1], y=d[,2]) # negative value of the (maximized) log-likelihood function
Log-likelihood function (with minus sign) for Roy's bivariate geometric model
minuslogRoy(x, y, theta1 = 0.5, theta2 = 0.5, theta3 = 1)
minuslogRoy(x, y, theta1 = 0.5, theta2 = 0.5, theta3 = 1)
x |
a vector of observed values (non-negative integers) |
y |
a vector of observed values (non-negative integers) of the same length as |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
The value of the log-likelihood function, changed in sign
Just to be used inside the estbivgeomRoy
function
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
Generation of pseudo-random values from Roy's bivariate geometric model
rbivgeomRoy(n, theta1, theta2, theta3)
rbivgeomRoy(n, theta1, theta2, theta3)
n |
a positive integer, corresponding to the sample size |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
A numeric matrix containing the bivariate sample values
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # joint frequency distribution: table(d[,1],d[,2])
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # random sample of size n=1000: set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) # joint frequency distribution: table(d[,1],d[,2])
Stress-strength reliability parameter for Roy's bivariate geometric model
RelbivgeomRoy(theta1, theta2, theta3)
RelbivgeomRoy(theta1, theta2, theta3)
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
The probability for Roy's bivariate geometric model - see Barbiero (2018) for its computation
Alessandro Barbiero
Barbiero, A. (2018) Properties and estimation of a bivariate geometric model with locally constant failure rates, submitted
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 RelbivgeomRoy(theta1, theta2, theta3) # theoretical stress-strength reliability parameter R=P(X<=Y)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 RelbivgeomRoy(theta1, theta2, theta3) # theoretical stress-strength reliability parameter R=P(X<=Y)
Empirical joint survival function
S.n(x, X)
S.n(x, X)
x |
matrix with two columns of non-negative integer values where the empirical joint survival function is computed |
X |
matrix with two columns corresponding to the full observed sample |
value of the empirical joint survival function
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) S.n(cbind(1,1),d) # empirical sf # compare it with the theoretical SbivgeomRoy(1,1,theta1,theta2,theta3)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 set.seed(12345) n <- 1000 d <- rbivgeomRoy(n, theta1, theta2, theta3) S.n(cbind(1,1),d) # empirical sf # compare it with the theoretical SbivgeomRoy(1,1,theta1,theta2,theta3)
Joint survival function for Roy's bivariate geometric model
SbivgeomRoy(x, y, theta1, theta2, theta3)
SbivgeomRoy(x, y, theta1, theta2, theta3)
x |
vector of observations from the first variable |
y |
vector of observations from the second variable |
theta1 |
paramater |
theta2 |
paramater |
theta3 |
paramater |
The probability . For this model it is equal to
Alessandro Barbiero
Roy, D. (1993) Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution, Journal of Multivariate Analysis 46(2), 362-373.
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that X>=2 and Y>=3: SbivgeomRoy(2, 3, theta1, theta2, theta3)
theta1 <- 0.5 theta2 <- 0.7 theta3 <- 0.9 # probability that X>=2 and Y>=3: SbivgeomRoy(2, 3, theta1, theta2, theta3)