On large deviations in testing
Ornstein-Uhlenbeck-type models
∗
Pavel V. Gapeev
†‡
Uwe K¨uchler
§
Statistical Inference for Stochastic Processes (2008) 11(2) (143–155)
We obtain exact large deviation rates for the log-likelihood ratio in testing models
with observed Ornstein-Uhlenbeck processes and get explicit rates of decrease for the
error probabilities of Neyman-Pearson, Bayes, and minimax tests. Moreover, we give
expressions for the rates of decrease for the error probabilities of Neyman-Pearson tests
in models with observed processes solving affine stochastic delay differential equations.
1 Introduction
Asymptotic properties of likelihood ratios play an important role in statistical testing problems.
Sometimes they can be studied by using large deviation results, for example, in the case of
binary statistical experiments. Chernoff [9] proved large deviation theorems for sums of i.i.d.
observations. Bahadur [1]-[3] studied asymptotic efficiency of tests and estimates for observed
sequences of random variables (see also Bahadur, Zabel and Gupta [4]). Birg´e [8] applied the
results of [9] to the investigation of the rate of decrease for error probabilities of Neyman-
Pearson tests. Generalizations of the large deviation results to the case of semimartingale
models and their applications are collected in the monograph [21]. Lin’kov [22] proved large
deviation theorems for extended random variables and applied them to the investigation of
general statistical experiments. Exact large deviation rates for the log-likelihood ratio in testing
models with fractional Brownian motion were derived in [23]. In the present paper we derive
an explicit form of large deviation theorems of Chernoff type for the log-likelihood ratio in
testing models with Ornstein-Uhlenbeck processes by applying the large deviation results from
the general continuous-time semimartingale framework of Lin’kov [21]. Note that the results
∗
This research was supported by Deutsche Forschungsgemeinschaft through the SFB 649 Economic Risk.
†
Weierstraß Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, D-10117 Berlin, Ger-
many, e-mail: gapeev@wias-berlin.de
‡
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Str. 65, 117997 Moscow, Russia
§
Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin, Germany,
e-mail: kuechler@mathematik.hu-berlin.de
Mathematics Subject Classification 2000. Primary 62F05, 60F10. Secondary 62C10, 62C20, 62M02, 62M07.
Key words and phrases: Likelihood ratio, Hellinger integral, Neyman-Pearson test, Bayes test, minimax
test, large deviation theorems, Girsanov formula for diffusion-type processes, Ornstein-Uhlenbeck-type process,
stochastic delay differential equation.
1
in [4] were obtained by using the large deviation techniques for sequences of random variables.
The problem of testing mean reversion for processes of Ornstein-Uhlenbeck type was earlier
studied by Szimayer and Maller [27]. Note that the Ornstein-Uhlenbeck processes play a key
role for modeling the behavior of interest rates in financial markets (see e.g. [29] or [5]).
In recent years, several statistical problems for models with time delay were studied. Dietz
[11] considered an Ornstein-Uhlenbeck-type model with exponential decreasing memory and
proved the local asymptotically mixed normality (in an extended sense) of the suitably nor-
malized model. Gushchin and K¨uchler [12] - [14] derived local asymptotic properties of the
likelihood process in (two-parameter) models connected with a special case of affine stochastic
delay differential equations. Putschke [25] continued this investigation for a multi-parametric
case of such affine delay equations. K¨uchler and Kutoyants [17] studied the asymptotic behavior
of the maximum likelihood and Bayesian estimators of delay in a simple Ornstein-Uhlenbeck-
type model. K¨uchler and Vasil’ev [18] investigated the almost sure consistency and asymptotic
normality of sequential estimators for multi-parametric affine delay equations. Gushchin and
K¨uchler [15] derived conditions under which a model with an affine stochastic delay differential
equation satisfies the local asymptotic normality property and where the maximum likelihood
and Bayesian estimators of a parameter are asymptotically normal and efficient. In this paper
we consider the problem of testing hypotheses and study the asymptotic behavior of the error
probabilities for Neyman-Pearson tests in Ornstein-Uhlenbeck-type models with delay. Asymp-
totic properties for tests of delay parameters in the cases of small noise and large sample size
were recently studied by Kutoyants [19] - [20].
The paper is organized as follows. In Section 2, we cite large deviation results for the log-
likelihood ratio process and their applications to the investigation of the rates of decrease for
error probabilities of Neyman-Pearson, Bayes, and minimax tests (cf. [21] - [23]). In Section 3,
by means of explicit expressions for Hellinger integrals, we obtain exact large deviation rates for
the log-likelihood ratio in a model of testing hypotheses about the parameter of an Ornstein-
Uhlenbeck process. We remark that there appears some kind of discontinuity in the solution
when the basic hypothesis is altered. The results are applied to the investigation of the rates
of decrease for error probabilities of the tests mentioned above. It seems possible to derive the
analogues of some of these results in the models with discretely observed data (see e.g. [4]). For
this, some essentially different techniques, which is applied by derivation of the large deviation
results for testing models with sequences of random variables, should be used. Then the initial
continuous-time assertions can be obtained as a limiting case of the corresponding results from
the discrete-time case by applying the invariance principle. In Section 4, we get the rates of
decrease for the error probabilities of Neyman-Pearson tests in models with processes that solve
affine stochastic delay differential equations and give two illustrating examples.
2 Large deviation theorems and their applications
Suppose that (Ω, F, P
0
, P
1
) is a binary statistical experiment and that X = (X
t
)
t≥0
is a real-
valued process. Let (F
t
)
t≥0
be the filtration generated by X , that is F
t
= σ(X
s
|0 ≤ s ≤ t),
t ≥ 0. Let H
0
and H
1
be two statistical hypotheses under which the distribution of the observed
process X is given by the different measures P
0
and P
1
, respectively. We will consider the
problem of testing the hypothesis H
0
against its alternative H
1
. In this section we cite some
2
results from Lin’kov [21] - [23] and use his notation.
2.1. Suppose that the measures P
0
and P
1
are locally equivalent on the filtration (F
t
)
t≥0
and introduce the log-likelihood ratio process Λ = (Λ
t
)
t≥0
defined as the logarithm of the
Radon-Nikodym derivative:
Λ
t
= log
d(P
1
|F
t
)
d(P
0
|F
t
)
(2.1)
for all t ≥ 0. Let the process H(ε) = (H
t
(ε))
t≥0
be the Hellinger integral of the order ε ∈
(−∞, ∞) of the restrictions P
1
|F
t
and P
0
|F
t
given by:
H
t
(ε) := H
t
(ε; P
1
, P
0
) = E
0
[exp(εΛ
t
)] (2.2)
for all t ≥ 0 (see e.g. [16; Chapter IV, Section 1]). Note that the relationship H
t
(ε; P
0
, P
1
) =
H
t
(1 − ε; P
1
, P
0
) holds for all ε ∈ (−∞, ∞) and t ≥ 0.
We will say that the Hellinger integral (2.2) satisfies the regularity condition if for some
function ψ
t
, t ≥ 0, such that ψ
t
→ ∞ as t → ∞, the (possibly infinite) limit
κ(ε) := lim
t→∞
ψ
−1
t
log H
t
(ε) (2.3)
exists for all ε ∈ (−∞, ∞). It is known (see e.g. [10; Chapter III, Section 3]) that the function
κ(ε) is a strictly convex and continuously differentiable function on (ε
−
, ε
+
) with
−∞ ≤ ε
−
:= inf{ε |κ(ε) < ∞} < ε
+
:= sup{ε |κ(ε) < ∞} ≤ ∞ (2.4)
where ε
−
≤ 0 and ε
+
≥ 1. If ε
−
< 0 or ε
+
> 1 then the derivatives κ
0
(0) and κ
0
(1) are
well-defined, respectively.
For every γ ∈ R let us define the function I(γ) as the Legendre-Fenchel transform of κ(ε)
by:
I(γ) := sup
ε∈(ε
−
,ε
+
)
(εγ − κ(ε)) (2.5)
(cf. e.g. [26]) with
κ
0
(ε
−
+) := lim
ε↓ε
−
κ
0
(ε) < κ
0
(ε
+
−) := lim
ε↑ε
+
κ
0
(ε) (2.6)
and define the values:
γ
0
:= κ
0
(0) if ε
−
< 0, γ
0
:= κ
0
(0+) = lim
ε↓ε
−
κ
0
(ε) if ε
−
= 0 (2.7)
γ
1
:= κ
0
(1) if ε
+
> 1, γ
1
:= κ
0
(1−) = lim
ε↑ε
+
κ
0
(ε) if ε
+
= 1 (2.8)
where by virtue of the convexity of κ(ε) on (ε
−
, ε
+
) we have γ
0
< γ
1
.
The following assertion is a large deviation theorem of Chernoff type for the log-likelihood
ratio process Λ = (Λ
t
)
t≥0
.
Proposition 2.1. Let the regularity condition (2.3) be satisfied. Then the following con-
clusions are valid:
(i) if γ
0
< κ
0
(ε
+
−) then for all γ ∈ (γ
0
, κ
0
(ε
+
−)) we have:
lim
t→∞
ψ
−1
t
log P
0
[ψ
−1
t
Λ
t
> γ] = lim
t→∞
ψ
−1
t
log P
0
[ψ
−1
t
Λ
t
≥ γ] = −I(γ); (2.9)
3
(ii) if ε
−
< 0 and κ
0
(ε
−
+) < κ
0
(0) then for all γ ∈ (κ
0
(ε
−
+), κ
0
(0)) we have:
lim
t→∞
ψ
−1
t
log P
0
[ψ
−1
t
Λ
t
< γ] = lim
t→∞
ψ
−1
t
log P
0
[ψ
−1
t
Λ
t
≤ γ] = −I(γ); (2.10)
(iii) if κ
0
(ε
−
+) < γ
1
then for all γ ∈ (κ
0
(ε
−
+), γ
1
) we have:
lim
t→∞
ψ
−1
t
log P
1
[ψ
−1
t
Λ
t
< γ] = lim
t→∞
ψ
−1
t
log P
1
[ψ
−1
t
Λ
t
≤ γ] = γ − I(γ); (2.11)
(iv) if ε
+
> 1 and κ
0
(1) < κ
0
(ε
+
−) then for all γ ∈ (κ
0
(1), κ
0
(ε
+
−)) we have:
lim
t→∞
ψ
−1
t
log P
1
[ψ
−1
t
Λ
t
> γ] = lim
t→∞
ψ
−1
t
log P
1
[ψ
−1
t
Λ
t
≥ γ] = γ − I(γ). (2.12)
This assertion is proved by using large deviation theorems for extended random variables
in [22].
2.2. The results cited above give an opportunity to investigate the rate of decrease of the
error probabilities for some statistical tests. In the rest of the section we refer some results about
the asymptotic behavior of the error probabilities for Neyman-Pearson, Bayes, and minimax
tests. The proofs of these results can be found in [22] (see also references in [23]).
Let α
t
, t ≥ 0, be an arbitrary function having values in (0, 1), and let δ
t
(α
t
) be a Neyman-
Pearson test of the level α
t
∈ (0, 1) for testing the hypotheses H
0
and H
1
under the observations
X
s
, 0 ≤ s ≤ t (see e.g. [21; Chapter II, Section 2.1]). The following assertion describes the rate
of decrease for the error probabilities of the first and second kind α
t
and β(α
t
), respectively,
for the test δ
t
(α
t
) under the regularity condition (2.3).
Proposition 2.2. Let (2.3) be satisfied with γ
0
< γ
1
. Then the following conclusions are
valid:
(i) for all a ∈ (I(γ
0
), I(γ
1
)) we have:
lim
t→∞
ψ
−1
t
log α
t
= −a if and only if lim
t→∞
ψ
−1
t
log β(α
t
) = −b(a) (2.13)
with
b(a) := a − γ(a) ∈ (I(γ
1
) − γ
1
, I(γ
0
) − γ
0
) (2.14)
and γ(a) is a unique solution of the equation I(γ) = a with respect to γ ∈ (γ
0
, γ
1
);
(ii) for all a ∈ [0, I(γ
0
)] we have:
lim
t→∞
ψ
−1
t
log α
t
= −a implies lim sup
t→∞
ψ
−1
t
log β(α
t
) ≤ γ
0
− I(γ
0
) (2.15)
and for all a ∈ [I(γ
1
), ∞] we have:
lim
t→∞
ψ
−1
t
log α
t
= −a implies lim inf
t→∞
ψ
−1
t
log β(α
t
) ≥ γ
1
− I(γ
1
); (2.16)
(iii) for all b ∈ [0, I(γ
1
) − γ
1
] we have:
lim
t→∞
ψ
−1
t
log β(α
t
) = −b implies lim sup
t→∞
ψ
−1
t
log α
t
≤ −I(γ
1
) (2.17)
4
and for all b ∈ [I(γ
0
) − γ
0
, ∞] we have:
lim
t→∞
ψ
−1
t
log β(α
t
) = −b implies lim inf
t→∞
ψ
−1
t
log α
t
≥ −I(γ
0
). (2.18)
These results under more restrictive conditions were proved in [21]. The only if part in
(2.13) for the sequence of observed i.i.d. random variables was proved by Birg´e [8].
Let δ
π
t
be a Bayes test for testing the hypotheses H
0
and H
1
based on the observations X
s
,
0 ≤ s ≤ t, where π and 1 − π , π ∈ [0, 1], are the a priori probabilities of the hypotheses H
0
and H
1
, respectively (see e.g. [21; Chapter II, Section 2.1]). The following assertion describes
the rate of decrease for the error probabilities of the first and second kind α
t
(δ
π
t
) and β(δ
π
t
),
and the risk e(δ
π
t
) for the test δ
π
t
under the regularity condition (2.3).
Proposition 2.3. Let (2.3) be satisfied with γ
0
< 0 < γ
1
. Then the following relationships
hold:
lim
t→∞
ψ
−1
t
log α(δ
π
t
) = lim
t→∞
ψ
−1
t
log β(δ
π
t
) = lim
t→∞
ψ
−1
t
log e(δ
π
t
) = −I(0). (2.19)
This assertion was proved by Chernoff [9] for the case of i.i.d. random variables. Under
some other conditions the last equality in (2.19) was proved by Vajda [28].
Let δ
∗
t
be a minimax test for testing the hypotheses H
0
and H
1
under the observations X
s
,
0 ≤ s ≤ t (see e.g. [7; Chapter III, Section 4]). The following assertion describes the rate of
decrease for the error probabilities of the first and second kind α
t
(δ
∗
t
) and β(δ
∗
t
), and the risk
e(δ
∗
t
) for the test δ
∗
t
under the regularity condition (2.3).
Proposition 2.4. Suppose that (2.3) is satisfied with γ
0
< 0 < γ
1
. Then we have:
lim
t→∞
ψ
−1
t
log α(δ
∗
t
) = lim
t→∞
ψ
−1
t
log β(δ
∗
t
) = lim
t→∞
ψ
−1
t
log e(δ
∗
t
) = −I(0). (2.20)
3 Ornstein-Uhlenbeck models
In this section we consider a model where the observation process X = (X
t
)
t≥0
satisfies the
following stochastic differential equation:
dX
t
= −θX
t
dt + dW
t
(X
0
= x) (3.1)
where W = (W
t
)
t≥0
is a standard Wiener process and θ ≥ 0, x ∈ R are some given constants.
We will study the problem of testing the simple hypothesis H
0
: θ = θ
0
against the simple
alternative H
1
: θ = θ
1
.
Here we specify the results of the previous section for Ornstein-Uhlenbeck processes in both
cases θ
1
> θ
0
= 0 and θ
1
> θ
0
> 0. It is remarkable that the first case cannot be obtained
from the second one by letting θ
0
↓ 0.
3.1. Since equation (3.1) has a pathwise unique continuous solution under both hypothe-
ses H
0
and H
1
, by means of the Girsanov formula for diffusion-type processes (see e.g. [24;
Chapter VII, Theorem 7.19]) we may conclude that the measures P
0
and P
1
are locally equiv-
alent on (F
t
)
t≥0
, and under the hypothesis H
0
the log-likelihood ratio process (2.1) admits the
representation:
Λ
t
= (θ
0
− θ
1
)
Z
t
0
X
s
dW
s
−
(θ
0
− θ
1
)
2
2
Z
t
0
X
2
s
ds. (3.2)
5