PMLE.Normal {depend.truncation}

Parametric Inference for Bivariate Normal Models with Dependent Truncation

Description

Maximum likelihood estimation (MLE) for dependent truncation data under the bivariate normal distribution. A bivariate normal distribution is assumed for bivariate random variables (L, X). The truncated data (L_j, X_j), subject to L_j<=X_j for all j=1, ..., n, are used to obtain the MLE for the population parameters of (L, X).

Usage

PMLE.Normal(l.trunc, x.trunc, testimator = FALSE,GOF=TRUE)

Arguments

l.trunc

vector of truncation variables satisfying l.trunc<=x.trunc

x.trunc

vector of variables satisfying l.trunc<=x.trunc

testimator

if TRUE, testimator is computed instead of MLE

GOF

if TRUE, goodness-of-fit test is performed

Details

PMLE.Normal performs the maximum likelihood estimation for dependently left-truncated data under the bivariate normal distribution. "PMLE.Normal" implements the methodologies developed in Emura T. & Konno Y. (2012, Statistical Papers 53, 133-149)and can produce the maximum likelihood estimates and their standard errors. Furthermore, "PMLE.Normal" tests the independence assumption between truncation variable and variable of interest via likelihood ratio test. The MLE is obtained by minimizing -logL using "nlm", where L is the log-likelihood.

Value

mu_L

mean of L and its standard error

mu_X

mean of X and its standard error

var_L

variance of L and its standard error

var_X

variance of X and its standard error

cov_LX

covariance between L and X and its standard error

c

inclusion probability, defined by c=Pr(L<=X), and its standard error

test

Likelihood ratio statistic and p-value

C

Cramer-von Mises goodness-of-fit test statistics

K

Kolmogorov-Smirnov goodness-of-fit test statistics

Author(s)

Takeshi EMURA

References

Emura T, Konno Y (2012), Multivariate Normal Distribution Approaches for Dependently Truncated Data. Statistical Papers 53 (No.1), 133-149.

Emura T, Konno Y (2014), Erratum to: Multivariate Normal Distribution Approaches for Dependently Truncated Data, Statistical Papers 55 (No.4): 1233-36

Examples

l.trunc=c(1,2,3,4,5,6,7,8,8)
x.trunc=c(2,4,4,5,5,7,7,9,10)
PMLE.Normal(l.trunc,x.trunc,testimator=FALSE)