## Qx z Sl S2 S3 S4 S2xdsi S2 S3 S4 I z9

5 The distribution of random vector (L(X ),R(X ),L(Z ),R(Z )).

For the right truncated density function f (x) we shall use the following notation fa (x)= J fU) du ,a](x).

Now we suppose that for fixed z and fixed value of t = t, random variable X is taken from the truncated distribution with density f\$(x). Here 3 = z(t,z) = L(z). It follows from (9) that in that case the distribution Pz of random vector (L(X),R(X),L(z),R(z)) has density (with respect to the measure v) q(si, S2, S3, S4 | z), q(si, S2,u,vI z) = J qx,z(si, S2,u,v) fu(x) dx, and (see (8))

q(si,s2,u,v I z) = J fu(x) dx x d*(si,s2,S3,S4 | z), si where for s = (si, S2, S3, S4)

' d3(si, S2, S3)!(s2, s3](z), if si < S2 = S3 < S4 d*(s|z) = { d4(Si, S2, S3, S4)H(S3, s4](z), if si < S2 < S3 < S4

0, else

Therefore the distribution Pz is absolutely continuous with respect to the measure v*, which is defined for continuous nonnegative functions ^(s) by the relation

4>(Si,S2,S2,S4) dsids2ds4 + / si,S2,S3,S4) dsids2ds3ds4,

Now suppose that Z is a random variable with density g, which is independent from the random covering •&(• ). For fixed values Z = z and t = t, random variable X is taken from the truncated distribution with density fz(x), Z = l(t,z) = L(z). Denote by P* the distribution of random vector (L(X),R(X),L(Z),R(Z)). It is clear that the distribution P* has density q(s) with respect to the measure v*, q(si, S2,u, S4) =

Now consider the random vector W = (L(X), R(X), L(Z)). Let j be the measure on R3, defined for continuous nonnegative functions ^ by

4>(si,s2,s2) ds1ds2 + J JJ ^(s1,s2,s3) ds1ds2 ds3,

It is clear that the distribution Pw of random vector W is absolutely continuous with respect to the measure z and p(y) = p(vi,v2,v3) = dz = J q(yl,y2,y3,u)du, ■

Therefore, where p(u,v,z) = j fz(x) dx x r(u,v,z), u r(u,v,z)= J d(u,v,z,x) dx.

6 Maximum likelihood estimators.

Let W,W1,..., Wn be i.i.d. random vectors, W = (L(X), R(X), L(Z)), with unknown density v f f(x) dx p(u, v, z) = r(u, v, z) x f f (x) dx x<z

We assume that the baseline density r and density f belong to given sets G and F correspondingly, and specify these sets later. We set v f f(x) dx

L = {p : p = rV(f ;■), (r, f) 6GxF} Denote by Pn the empirical measure,

Consider the maximum likelihood estimator pn for unknown p 6 L,

It is clear, that pn — rri x v(fn] ■), where fn and fn are maximum likelihood estimators for r and f,

The estimator fn in general situation was suggested by Turnbull B.W. [Tur76], see also Finkelstein, D.M., Moore, D.F., Schoenfeld D.A. [FMD93].

6.1 The bracketing Hellinger e—entropy

Let (Y, B, ¡i) be a measurable space and Y\,...,Yn be i.i.d. random elements of Y with common distribution P 6 P and density f, dP dP

For nonnegative f, g let h(f, g) be the Hellinger distance,