TABLE OF CONTENTS The term "bivariate exponential" usually (but not necessarily) refers to bivariate distributions with both marginals being exponential. There are by now a large number of different kinds of bivariate exponential distributions.
In the following section we shall introduce several types of bivariate exponential distributions and construct estimators of R in the case when (X, Y) is a bivariate exponential random vector. We shall use abbreviation BVED for "bivariate exponential distribution". Unfortunately, the volume of the present book does not allow to exhaust the topic, we shall not be able to cover some available results, such as inference for the Weinmann multivariate exponential distribution studied by Cramer and Kamps (1997) andCramer (2000).
The first attempt to construct a bivariate distribution where X and Y are dependent and have exponential marginals has been undertaken by Gumbel (1960) who introduced BVED with the survival function F (x, y)= P(X>x, Y>y) and pdf given by, respectively,
| (3.166) |  |
| (3.167) |  |
where ? 1, ? 2, ? 12 are positive. It is easy to check that marginal distribution of (3.166) are exponential.
A drawback of the Gumbel's BVED is that it describes no physical reality. To remedy the situation, Freund (1961) introduced another BVED that has been designed, in particular, for the life testing of two-component systems which can function even after one component has failed (for example, two-engine plane, person's kidneys, etc.) The joint pdf of...
