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.

Details

The DESCRIPTION file:

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

Author(s)

Alessandro Barbiero

Maintainer: Alessandro Barbiero (alessandro.barbiero@unimi.it)

References

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

See Also

dbivgeomRoy, rbivgeomRoy, estbivgeomRoy, FbivgeomRoy

Examples

#####################################
#### 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)

Conditional moment

Description

Conditional moment of Y given X=x for Roy's bivariate geomtric model

Usage

EyxbivgeomRoy(theta1, theta2, theta3, x)

Arguments

Value

Value of the conditional moment of Y given X=x

Author(s)

Alessandro Barbiero

References

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.

See Also

FyxbivgeomRoy

Examples

theta1 <- 0.5
theta2 <- 0.7
theta3 <- 0.9
EyxbivgeomRoy(theta1, theta2, theta3, 2)

Joint distribution function

Description

Joint cumulative distribution function for Roy's bivariate geometric model

Usage

FbivgeomRoy(x, y, theta1, theta2, theta3)

Arguments

Value

The probability F(x,y):=P(X\leq x,Y\leq x)

Author(s)

Alessandro Barbiero

References

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.

See Also

dbivgeomRoy, SbivgeomRoy

Examples

theta1 <- 0.5
theta2 <- 0.7
theta3 <- 0.9
# probability that X<=2 and Y<=3:
FbivgeomRoy(2, 3, theta1, theta2, theta3)

Conditional distribution

Description

Conditional distribution function of Y given X=x

Usage

FyxbivgeomRoy(y, theta1, theta2, theta3, x)

Arguments

Value

The value of the conditional cumulative distribution function F_{Y|x} in y. Used in rbivgeomRoy for conditional sampling

Author(s)

Alessandro Barbiero

References

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.

See Also

EyxbivgeomRoy, rbivgeomRoy

Examples

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

Reliability parameter

Description

Stress-strength reliability parameter R for Roy's bivariate geometric model

Usage

RelbivgeomRoy(theta1, theta2, theta3)

Arguments

Value

The probability R:=P(X\leq Y) for Roy's bivariate geometric model - see Barbiero (2018) for its computation

Author(s)

Alessandro Barbiero

References

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.

See Also

dbivgeomRoy, FbivgeomRoy

Examples

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

Description

Empirical joint survival function

Usage

S.n(x, X)

Arguments

Value

value of the empirical joint survival function \hat{S}_X(x)

Author(s)

Alessandro Barbiero

References

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.

See Also

estbivgeomRoy

Examples

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

Description

Joint survival function for Roy's bivariate geometric model

Usage

SbivgeomRoy(x, y, theta1, theta2, theta3)

Arguments

Value

The probability P(X\geq x, Y\geq y. For this model it is equal to S(x,y)=\theta_1^x\theta_2^y\theta_3^{xy}

Author(s)

Alessandro Barbiero

References

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.

See Also

FbivgeomRoy

Examples

theta1 <- 0.5
theta2 <- 0.7
theta3 <- 0.9
# probability that X>=2 and Y>=3:
SbivgeomRoy(2, 3, theta1, theta2, theta3)

Linear correlation

Description

Linear correlation for Roy's bivariate geometric model

Usage

corbivgeomRoy(theta1, theta2, theta3)

Arguments

Value

the value of Pearson's linear correlation - see Barbiero (2018). The linear correlation for Roy's bivariate geometric distribution is negative (or null, for \theta_3=1) for any feasible choice of its parameters

Author(s)

Alessandro Barbiero

References

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

See Also

dbivgeomRoy

Examples

corbivgeomRoy(0.3,0.7,0.5)

Joint probability mass function

Description

Joint probability mass function for Roy's bivariate geometric model

Usage

dbivgeomRoy(x, y, theta1, theta2, theta3)

Arguments

Value

Value of the probability p(x,y):=P(X=x,Y=y).

Author(s)

Alessandro Barbiero

References

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.

See Also

FbivgeomRoy

Examples

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

Description

Parameter estimation for Roy's bivariate geometric model

Usage

estbivgeomRoy(x, y, method = "LS")

Arguments

Value

a vector of length 3 containing the estimates of theta_1, theta_2, and theta_3

Author(s)

Alessandro Barbiero

References

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.

See Also

dbivgeomRoy, minuslogRoy

Examples

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")

Bivariate failure rates

Description

Bivariate failure rate \lambda_1

Usage

lambda1Roy(x, y, theta1, theta2, theta3)

Arguments

Details

It is defined as P(X=x,Y\geq y)/P(X\geq x,Y\geq y). For this model, \lambda_1(x,y)=1-\theta_1\theta_3^y

Value

Value of the bivariate failure rate \lambda_1 for Roy's bivariate geometric model (Roy, 1993)

Author(s)

Alessandro Barbiero

References

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.

See Also

lambda2Roy

Examples

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

Description

Bivariate failure rate \lambda_2

Usage

lambda2Roy(x, y, theta1, theta2, theta3)

Arguments

Details

It is defined as P(X\geq x,Y=y)/P(X\geq x,Y\geq y). For this model, \lambda_2(x,y)=1-\theta_2\theta_3^x

Value

Value of the bivariate failure rate \lambda_2 for Roy's bivariate geometric model (Roy, 1993)

Author(s)

Alessandro Barbiero

References

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.

See Also

lambda1Roy

Examples

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)

Log-likelihood function

Description

Negative log-likelihood function for Roy's bivariate geometric model

Usage

loglikgeomRoy(par, x, y)

Arguments

Value

Value of the negative log-likelihood function

Author(s)

Alessandro Barbiero

References

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.

See Also

dbivgeomRoy

Examples

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

Description

Log-likelihood function (with minus sign) for Roy's bivariate geometric model

Usage

minuslogRoy(x, y, theta1 = 0.5, theta2 = 0.5, theta3 = 1)

Arguments

Value

The value of the log-likelihood function, changed in sign

Note

Just to be used inside the estbivgeomRoy function

Author(s)

Alessandro Barbiero

References

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.

See Also

estbivgeomRoy

Pseudo-random generation

Description

Generation of pseudo-random values from Roy's bivariate geometric model

Usage

rbivgeomRoy(n, theta1, theta2, theta3)

Arguments

Value

A n\times 2 numeric matrix containing the bivariate sample values

Author(s)

Alessandro Barbiero

References

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.

See Also

dbivgeomRoy, FbivgeomRoy

Examples

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])