116 Handbook of Discrete-Valued Time Series
A realization of a bivariate Poisson series
Count
15
10
5
0
40200 60 80 100
Time
FIGURE 5.4
A realization of length 100 of a bivariate stationary time series, where each coordinate’s marginal distribution
is Poisson. Negative lag one autocorrelations exist in both coordinates; there is also negative cross-correlation
between the components.
its diagonal entries and −0.75 on off-diagonal entries. The renewal lifetimes for the two
components were both chosen as the three-point lifetime introduced below (5.13), with
= 0.1. The theoretical means are 6 for component one and 10 for component two. The
dotted lines demarcate sample means, which were 6.03 and 9.92 for the rst and second
components, respectively. This realization has marginal Poisson distributions and negative
correlation (at lag zero) between the two components. The negative correlation between
components is noticeable in Figure 5.4: large values in coordinate two usually accompany
small values of coordinate one, and vice-versa. Because of the lifetimes chosen, there is also
negative correlation at some lags in both component series.
5.4 Statistical Inference
Renewal count models are very parsimonious. For a univariate comparison, a Markov
chain on S states has S(S − 1) free parameters in its one-step-ahead transition matrix.
Renewal count models are described entirely through the parameters governing the life-
time L and M
t
—this does not change should the state space become innite. For example,
117 Renewal-Based Count Time Series
modeling a univariate Poisson series with long memory can be parameterized by a Pareto
lifetime L with parameter α and a Poisson {M
t
}, say with marginal mean λ. In this case,
only two parameters need to be estimated: α and λ.
Ideally, all parameters would be estimated via maximum likelihood. A true likelihood
approach would produce estimators that “feel the joint count distributional structure”
(rather than just say process moments). The issue increases in importance with small
counts. Unfortunately, likelihood methods have been very difcult to develop for count
models as they require calculation of the joint distributional structure. The difculties
encountered can be appreciated in Davis et al. (2003), where likelihood asymptotics are
pursued for the Poisson setting.
Kedem and Fokianos (2003) and Thavaneswaran and Ravishanker (2015; Chapter 7 in
this volume) have had success in estimating count time series parameters via a quasi-
likelihood (Godambe and Heyde, 1987) approach, which is a technique that we use here.
Suppose that Y
1
, ..., Y
n
is a sample from a univariate stationary renewal series {Y
t
}.Let
θ denote a vector containing all model parameters. One tractable estimation strategy
minimizes the sum of squared one-step-ahead prediction errors
n
(Y
t
− Y
ˆ
t
)
2
S(θ) =
v
t
t=1
in θ. Here, Y
ˆ
t
= P(Y
t
|1, Y
1
, ..., Y
t−1
) is the best (minimum mean squared error) linear
prediction of Y
t
from the process history Y
1
, ..., Y
t−1
and a constant (hence the one in the
set of predictands mentioned earlier) and v
t
= E[(Y
t
− Y
ˆ
t
)
2
] is its unconditional mean
squared error. In general, Y
ˆ
t
and v
t
will depend on θ. Since {Y
t
} is stationary, v
t
converges
monotonically downwards to a limit; often this convergence is geometric in t and there is
no asymptotic loss of precision in omitting v
t
altogether in the sum of squares. While one
might try to base inferences on the conditional expectation Y
ˆ
t
= E[Y
t
|Y
1
, ..., Y
t−1
],this
quantity also appears intractable in generality.
The stochastic structure of {Y
t
} in (5.18) is somewhat unwieldy. In general, it is not
Markov, conditional Markov, etc. One result that is useful in driving estimating equations
is (5.8) in the case where M
t
≡ M is xed. The scenario gets much more complicated when
M
t
is allowed to vary; however, the situation can be quantied (we will not do it here)
and the results allow us to evaluate E[Y
t
|Y
t−h
] for any h ≥ 1. This said, pilot computations
with Poisson cases indicate that linear predictions from all previous data predict Y
t
more
accurately than E[Y
t
|Y
t−1
], which is only based on the last observation Y
t−1
.
As the covariance structure of the process has been identied, best linear one-step-ahead
linear predictions are tractable. The one-step-ahead predictions have form
n−1
P(Y
t
|Y
1
, ..., Y
t−1
,1) = μ
Y
+ β
t
(Y
t
− μ
Y
),
t=1
where μ
Y
≡ E[Y
t
] and the coefcients β
1
, ..., β
t−1
are computed from the classical predic-
tion equations. Computation of one-step-ahead linear predictions and their mean squared
errors is a well-studied problem (see Chapter 5 of Brockwell and Davis, 1991), which can
be done rapidly and recursively in time t.
While no general results have yet to be proven, linear prediction inference meth-
ods for count series should yield consistent and asymptotically normal estimators of θ.
118 Handbook of Discrete-Valued Time Series
Cui and Lund (2010) derive such a theory, replete with an explicit asymptotic informa-
tion matrix, when {Y
t
} has binomial marginals. Here, M is constant and known and L is
quantied in terms of its hazard rates h
k
= P(L = k|L ≥ k), k = 1, 2, .... Specically, in
the case where h
k
is constant for k ≥ 2, Cui and Lund (2010) establish the joint asymptotic
normality of the two estimated hazard rates
h
ˆ
1
and
h
ˆ
2
that minimize the prediction sum of
squares; viz.,
h
ˆ
1
h
1
R
h
ˆ
2
∼ AN
2
h
2
,
n
.
The form of R is explicitly identied in terms of h
1
and h
2
. The notation here uses hats
to denote both estimators and one-step-ahead predictions (this should not cause confu-
sion). In cases where L has general hazard rates, these methods do not yield an explicit
information matrix. Even in simple cases, the computations are intense.
For the more general case, justifying asymptotic normality of the linear prediction
estimators may be feasible. If explicit forms for standard errors are not needed, numer-
ical standard error approximations could be obtained by inverting the Hessian matrix
associated with the sum of squares at its minimum.
5.5 Covariates and Periodicities
Covariate information often accompanies count data. Frequently, the goal is to explain the
counts in terms of the covariates. To modify the renewal paradigm for covariates, one can
allow M
t
in (5.10) to depend on the covariates. For example, consider the univariate Poisson
case and suppose that C
1,t
, ..., C
K,t
are K covariates at time t. To retain a Poisson marginal
distribution, M
t
is taken to be Poisson distributed; however, we now allow E[M
t
] to vary
with the time t covariates via
K
E[M
t
]=exp
β
0
+ β
C
,t
,
=1
where β
0
, ..., β
K
are regression coefcients. Of course, such a process has a time-varying
mean and is not technically stationary; however, such processes seem useful (Davis
et al. 2000). The exponential function is used to keep the Poisson mean positive. While the
resulting count series is no longer stationary, it is autocorrelated and can take on a wide
and exible range of covariance structures. It would be desirable to extend the Poisson
regression techniques of Davis et al. (2000) to the renewal setting.
Count series with periodicities could be devised in two ways. First, in the univariate
case, L could be allowed to depend on the season in which the last renewal occurred. This
is done in Fralix et al. (2012), where periodic Markov chains and renewal processes are
developed in generality. Second, one could allow M
t
to depend on time t in a periodic way.
Combinations of both approaches may prove useful.
119 Renewal-Based Count Time Series
5.6 Concluding Comments
Renewal superpositioning methods seem to generate more exible autocovariance
structures for count series than traditional ARMA-based approaches. They readily yield
stationary series with many marginal integer-valued distribution desired and their auto-
covariances can be positive and/or negative. Estimation can be conducted by minimizing
one-step-ahead prediction errors, which are easily calculated from process autocovari-
ances. Unfortunately, as with other count time series model classes, full likelihood
estimation approaches do not appear tractable at this time. Quasilikelihoods, composite
likelihood methods, etc. are currently being investigated.
Acknowledgments
Robert Lund’s research was partially supported by NSF Award DMS 1407480.
References
Al-Osh, M. and Alzaid, A.A. (1988). Integer-valued moving averages (INMA), Statistical Papers, 29,
281–300.
Blight, P.A. (1989). Time series formed from the superposition of discrete renewal processes, Journal
of Applied Probability, 26, 189–195.
Billingsley, P. (1995). Probability and Measure, 3rd edn., Springer-Verlag, New York.
Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd edn., Springer-Verlag,
New York.
Cui, Y. and Lund, R.B. (2009). A new look at time series of counts, Biometrika, 96, 781–792.
Cui, Y. and Lund, R.B. (2010). Inference for binomial AR(1) models, Statistics and Probability Letters,
80, 1985–1990.
Daley, D.J. and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes: Volume II: General
Theory and Structure, Springer, New York.
Davis, R.A., Dunsmuir, W.T.M., and Streett, S.B. (2003). Observation-driven models for Poisson
counts, Biometrika, 90, 777–790.
Davis, R.A., Dunsmuir, W.T.M., and Wang, Y. (2000). On autocorrelation in a Poisson regression
model, Biometrika, 87, 491–505.
Feller, W. (1968). An Introduction to Probability Theory and its Applications, 3rd edn., John Wiley & Sons,
New York.
Fralix, B., Livsey, J., and Lund, R.B. (2012). Renewal sequences with periodic dynamics, Probability in
the Engineering and Informational Sciences, 26, 1–15.
Godambe, V.P. and Heyde, C.C. (1987). Quasi-likelihood and optimal estimation, International
Statistical Review, 55, 231–244.
Jacobs, P.A. and Lewis, P.A.W. (1978a). Discrete time series generated by mixtures I: Correlational
and runs properties, Journal of the Royal Statistical Society, Series B, 40, 94–105.
Jacobs, P.A. and Lewis, P.A.W. (1978b). Discrete time series generated by mixtures II: Asymptotic
properties, Journal of the Royal Statistical Society, Series B, 40, 222–228.
120 Handbook of Discrete-Valued Time Series
Karlis, D. and Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models, Journal
of the Royal Statistical Society, Series D, 52, 381–393.
Kedem, B. and Fokianos, K. (2003). Regression theory for categorical time series, Statistical Science,
18, 357–376.
Kendall, D.G. (1959). Unitary dilations of Markov transition operators, and the corresponding
integral representations for transition-probability matrices. In: Probability and Statistics,
U. Grenander (ed.), Almqvist & Wiksell, Stockholm, Sweden, 138–161.
Lund, R.B., Holan, S. and Livsey, J. (2015). Long memory discrete-valued time series. In R. A.
Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook of Discrete-Valued Time Series,
pp. 447–458. Chapman & Hall, Boca Raton, FL.
Lund, R.B. and Tweedie, R. (1996). Geometric convergence rates for stochastically ordered Markov
chains, Mathematics of Operations Research, 21, 182–194.
McKenzie, E. (1985). Some simple models for discrete variate time series, Water Resources Bulletin, 21,
645–650.
McKenzie, E. (1986). Autoregressive-moving average processes with negative-binomial and geomet-
ric marginal distributions, Advances in Applied Probability, 18, 679–705.
McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts, Advances in
Applied Probability, 20, 822–835.
Ross, S.M. (1996). Stochastic Processes, 2nd edn., Wiley, New York.
Shin, K. and Pasupathy, R. (2010). An algorithm for fast generation of bivariate Poisson random
vectors, INFORMS Journal on Computing, 22, 81–92.
Smith, W.L. (1958). Renewal theory and its ramications, Journal of the Royal Statistical Society, Series
B, 20, 243–302.
Steutel, F.W. and Van Harn, K. (1979). Discrete analogues of self-decomposability and stability, Annals
of Probability, 7, 893–899.
Thavaneswaran, A. and Ravishanker, N. (2015). Estimating equation approaches for integer-valued
time series models. In R. A. Davis, S. H. Holan, R. Lund and N. Ravishanker, eds., Handbook of
Discrete-Valued Time Series, pp. 145–164. Chapman & Hall, Boca Raton, FL.
Yahav, I. and Shmueli, G. (2012). On generating multivariate Poisson data in management science
applications, Applied Stochastic Models in Business & Industry, 28, 91–102.
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