We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting
S
(
a
, 
b
) be the number of families observed in (
a
, 
b
), we study the expected sample variance and its asymptotics for
p
consecutive sequential samples
S
p
=
(
S
(
t
0
,
t
1
)
,
⋯
,
S
(
t
p
-
1
,
t
p
)
)
, for
0
=
t
0
<
t
1
<
⋯
<
t
p
. By conditioning on the sizes of the samples, we provide a connection between
S
p
and
p
sequential samples of sizes
n
1
,
n
2
,
⋯
,
n
p
, drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166–1194, 2023.
https://doi.org/10.3150/22-BEJ1494
). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of
t
1
,
t
2
,
⋯
,
t
p
, where the lengths of intervals are logarithmically equal, we revisit Fisher’s 1943 multi-sampling problem and give an...

Another view of sequential sampling in the birth process with immigration

10.1007/s00285-023-02041-0