Let (X, Y) be a bivariate random vector with joint distribution function FX, Y(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (Xi, Yi), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (Xi, Yi) for which Xi ⩽ Yi. We denote such pairs as (X*i, Yi*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function FX, Y(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ⩽ Y}. This allows also to estimate P{X ⩽ Y} with the truncated observations (X*i, Yi*). The copula of bivariate random vector (X*, Y*) is also derived.
[1]
H. Kile,et al.
Bandwidth Selection in Kernel Density Estimation
,
2010
.
[2]
J. C. Jackson.
The Analysis of Quasar Samples
,
1974
.
[3]
D. Lynden-Bell,et al.
A Method of Allowing for Known Observational Selection in Small Samples Applied to 3CR Quasars
,
1971
.
[4]
Bill Ravens,et al.
An Introduction to Copulas
,
2000,
Technometrics.
[5]
Fabio Spizzichino,et al.
On the relationships between copulas of order statistics and marginal distributions
,
2010
.
[6]
何书元.
ESTIMATING A DISTRIBUTION FUNCTION WITH TRUNCATED DATA
,
1994
.
[7]
S. Sheather.
Density Estimation
,
2004
.
[8]
Berthold Schweizer.
Introduction to Copulas
,
2007
.
[9]
E. Parzen.
On Estimation of a Probability Density Function and Mode
,
1962
.
[10]
Herman Chernoff,et al.
Nonparametric Estimation of the Slope of a Truncated Regression
,
1983
.
[11]
Angel R. Martinez,et al.
Computational Statistics Handbook with MATLAB
,
2001
.