Optimality

332 R. M. Mnatsakanov and F. H. Ruymgaart
)
x
(
S
0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1.
2
0 5 10 15 20
Fig 2.
Figure 2.
h = n −1/5 , respectively. To construct the graphs for the moment-type estimator
ˆSα defined by (5.4) and the kernel-type estimator Sh defined in a similar way as
the one given by (6.1) let us generate n = 400 copies of r.v.’s Y1, . . . , Yn with pdf g
from (5.1) with W = 4 and
x −
1−F(x) = e 2 + x x
e− 2 , x≥0 .
2
We generated Y1, . . . , Yn as a mixture of two gamma G(1,2) and G(2, 2) distributions
with equal proportions. In the Figure 2 the solid line represents the graph of
S = 1−F while the dashed and dotted lines correspond to ˆ Sα and Sh, respectively.
Here again we have α = n 2/5 and h = n −1/5 .
References
[1] Bhattacharyya, B. B., Kazempour, M. K. and Richardson, G. D.
(1991). Length biased density estimation of fibres. J. Nonparametr. Statist. 1,
127–141.
[2] Cox, D.R. (1969). Some sampling problems in technology. In New Developments
in Survey Sampling (Johnson, N.L. and Smith, H. Jr., eds.). Wiley, New
York, 506–527.
[3] Feller, W. (1966). An Introduction to Probability Theory and Its Applications,
Vol. II. Wiley, New York.
[4] Jones, M. C. (1991). Kernel density estimation for length biased data. Biometrika
78, 511–519.
[5] Mnatsakanov, R. and Ruymgaart, F. H. (2003). Some properties of
moment-empirical cdf’s with application to some inverse estimation problems.
Math. Meth. Stat. 12, 478–495.
[6] Mnatsakanov, R. and Ruymgaart, F. H. (2004). Some properties of
moment-density estimators. Math. Meth. Statist., to appear.