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Testing the Drift-Diffusion Model
\title{Testing the Drift-Diffusion Model}
\author{Drew Fudenberg \and Whitney Newey \and Philipp Strack \and Tomasz
Strzalecki}
\date{}
\maketitle
\section{Introduction}
The {\em drift diffusion model \ \/}(DDM) is a model of sequential sampling
with diffusion (Brownian) signals, where the decision maker accumulates
evidence until the process hits a stopping boundary, and then stops and
chooses the alternative that corresponds to that boundary. This model has
been widely used in psychology, neuroeconomics, and neuroscience to explain
the observed patterns of choice and response times in a range of binary
choice decision problems. One class of papers study \textquotedblleft
perception tasks\textquotedblright with an objectively correct answer (e.g.
\textquotedblleft are more of the dots on the screen moving left or moving
right?\textquotedblright; here the drift of the process is related to which
choice is objectively correct \cite{Ratcliff-review,ShadlenKiani13}. The
other class of papers study \textquotedblleft consumption
tasks\textquotedblright\ such as \textquotedblleft which of these snacks
would you rather eat?\textquotedblright; here the drift is related to the
relative appeal of the alternatives
\citep{FehrRangel,Roeetal,ClitheroRangel13,Krajbichetal10,Krajbichetal11,Krajbichetal12,Milosavljevicetal10,Krbahafe15,Reutskajaetal11}
.
The simplest version of the DDM\ assumes that the stopping boundaries are
constant over time \cite{Wald47,Stone60,Edwards65,Ratcliff78}. More recently
a number of papers use non-constant boundaries to better fit the data, and
in particular the observed correlation between response times and choice
accuracy, i.e., that correct responses are faster than error responses \cite
{Luce86,Milosavljevic10,Drugowitsch12,FSS2018}.
Constant stopping boundaries is the optimal solution for perception tasks
where the volatility of the signals and the flow cost of sampling are both
constant, and the prior belief is that the drift of the diffusion has only
two possible values, depending on which decision is correct. Even with
constant volatility and costs, non-constant boundaries are optimal for other
priors. \cite{FSS2018} characterize the optimal boundaries for the
consumption task: the decision maker is uncertain about the utility of each
choice, with independent normal priors on the value of each option. \cite
{Drugowitsch12} show how to computationally derive the optimal boundaries
for the perception task: the signal coherence varies from trial to trial, so
some decision problems are harder than others.
This paper provides a statistical test for DDM's with general boundaries. We
first prove a characterization theorem: we find a condition on choice
probabilities that is satisfied if and only if the choice probabilities are
generated by some DDM. Moreover, we show that the drift and the boundary are
uniquely identified. We then use our condition to nonparametrically estimate
the drift and the boundary and construct a test statistic based on finite
samples.
Recent related work on DDM includes \cite{Drugowitsch12} who conducted a
Bayesian estimation of a collapsing boundary model and \cite{FSS2018} who
conducted a maximum likelihood estimation. \cite
{hawkins2015revisiting} estimate collapsing boundaries in a parametric
class, allowing for a random nondecision time at the start. \cite
{Chiongetal18} estimate a version of DDM with constant boundaries but random
starting point of the signal accumulation process; \cite{Ratcliff02}
estimates a similar model where other parameters are made random. \cite
{Baldassi} partially characterize DDM with constant boundary.\footnote{
They ignore the issue of correlation between response times choices by
looking only at marginal distributions, which makes their conditions
necessary but not sufficient.}
Other work on DDM-like models includes the decision field theory of \cite
{BusemeyerTownsend,BT93,DFT} allows the signal process to be mean-reverting.
\cite{Alosetal18} and \cite{ES17} study models where response time is a
deterministic function of the utility difference. \cite
{HebertWoodford,Woodford14,CheMierendorff,Liangetal,LiangMu,Zhong19} study
dynamic costly optimal information acquisition.
\section{The Stochastic Choice Function}
Let $X$ be the universe of alternatives (actions) and $T=\mathbb{R}_{+}$ be
time. For every pair of objects $\{x, y\}$ the analyst observes pairwise
stochastic choices and decision times. In the limit as the sample size grows
large, the analyst will have access to the joint distribution over which
object is chosen and at which time a choice is made. We denote by $F^{xy}(t)$
the probability that the agent makes a choice by time $t$, and let $
p^{xy}(t) $ the probability that the agent picks $x$ conditional on stopping
at time $t $. Throughout, we restrict attention to cases where $F$ has full
support and no atoms at time $0$, so that $F(0)=0$, and we assume that $F$
is strictly increasing with $\lim_{t\rightarrow \infty} F(t)=1$. These
restrictions imply the agent never stops immediately, that there is a
positive probability of stopping in every time interval, and that the agent
always eventually stops. We call $(p^{xy},F^{xy})$, the {\em stochastic
choice function\/}.
An immediate restriction on the stochastic choice function is that the
choices of the agent are unaffected by which object we consider to be the
first and which object we consider to be the second. This is formally
equivalent to
\begin{equation*}
p^{xy}(t)\equiv 1-p^{yx}(t)\text{ for all }t\text{ and }F^{xy}\equiv F^{yx}
\text{ for all }x,y\in X.
\end{equation*}
Without loss of generality we only consider stochastic choice functions
which satisfy this restriction. We also assume that each option is chosen
with positive probability $0<p^{xy}(t)<1$ for all $t$.
Given $(p^{xy},F^{xy})$ we define the choice imbalance at each time $t$ to
be
\begin{equation*}
I^{xy}(t):=p^{xy}(t)\log \left( \frac{p^{xy}(t)}{1-p^{xy}(t)}\right)
+(1-p^{xy}(t))\log \left( \frac{1-p^{xy}(t)}{p^{xy}(t)}\right) \,.
\end{equation*}
This is the Kullback-Leibler divergence (or relative entropy) between the
Binomial distribution of the agent's time $t$ choice $P(t)=(p(t),1-p(t))$
and the permuted choice distribution $Q(t)=(1-p(t),p(t))$. As the
Kullback-Leibler divergence is a statistical measure of the similarity
between distributions $I(t)$ captures the imbalance of the agent's choice at
time $t$. Note that $I=0$ means that both choices are equally likely; $
I=\infty $ when $p$ equals $0$ or $1$, and that $I$ is symmetric about~0.5.
We define $\bar{I}^{xy}$ to be the average choice imbalance,
\begin{equation*}
\bar{I}^{xy}:=\int_{0}^{\infty }I^{xy}(t)\,dF^{xy}(t)\,,
\end{equation*}
and we define $\bar{T}^{xy}$ to be the average decision time,
\begin{equation*}
\bar{T}^{xy}:=\int_{0}^{\infty }t\,dF^{xy}(t)\,,
\end{equation*}
and define $\bar{p}^{xy}$ to be the average choice probability$,$
\begin{equation*}
\bar{p}^{xy}:=\int_{0}^{\infty }p^{xy}(t)\,dF^{xy}(t)\,,
\end{equation*}
and assume that all of these integrals exist. Finally, we relabel objects as
needed so that the first object is chosen weakly more often, i.e. $\bar{p}
^{xy}\geq 0.5$ for all $x,y$.
\section{DDM representation}
The drift diffusion model (DDM) is commonly used to explain the stochastic
choice data in neuroscience and psychology. The two main ingredients of a
DDM are the stimulus process $Z_{t}$ and a time-dependent stopping boundary $
b(t)$. In the DDM representation, the stimulus process $Z_{t}$ is a Brownian
motion with drift $\delta $ and volatility $\alpha $:
\begin{equation}
Z_{t}=\delta \,t+\alpha \,B_{t}, \label{eq:Z}
\end{equation}
where $B_{t}$ is a standard Brownian motion, so in particular $Z_{0}=0$.
Define the hitting time $\tau $
\begin{equation}
\tau =\inf \{t\geq 0:|Z_{t}|\geq b(t)\}, \label{eq:tau}
\end{equation}
i.e., the first time the absolute value of the process $Z_{t}$ hits the
boundary $b$. Let $F^{\ast }(t;\delta ,b,\alpha ):=\mathbb{P}\left[ \tau
\leq t\right]$ be the distribution of the stopping time $\tau $. Likewise,
let $p^{\ast }(t;\delta ,b,\alpha )$ be the conditional choice probability
induced by \eqref{eq:Z} and \eqref{eq:tau} and a decision rule that chooses $
x$ if $Z_{\tau }=b(\tau )$ and $y$ if $Z_{\tau }=-b(\tau )$.
Our goal in this paper is to determine which data is consistent with a DDM
representation, and when it is, when the representation is unique. When the
drift $\delta=0$, each alternative will be chosen half of the time
regardless of the shape of the boundary, so we will exclude this case going
forward.
The original formulation of the DDM was for \textquotedblleft perception
tasks\textquotedblright\ where the drift $\delta $ is either $+1$ or $-1$
depending on which decision is correct; more generally there can be a
distinct drift $\delta ^{xy}$ for each pair $x,y$. In consumption-choice
problems (otherwise known as value-based problems, see, e.g., \cite
{Milosavljevic10}) it is natural to assume that the net drift $\delta ^{xy}$
is the difference between two signals, an $x$-signal with drift $u(x)$ equal
to the utility of $x$ and a $y$-signal with drift $u(y)$ equal to the
utility of $y$, so that $\delta ^{xy}=u(x)-u(y).$ This imposes some
consistency conditions that we discuss below.
\begin{definition}[DDM Representation]
Stochastic choice data $(p^{xy},F^{xy})_{x,y\in X}$ has a DDM representation
if there exists a utility function $u:X\rightarrow \mathbb{R}$, a volatility
parameter $\alpha >0$ as well as a boundary $b:\mathbb{R}_{+}\rightarrow
\mathbb{R}_{+}$ such that for all $x,y\in X$ and $t\in \mathbb{R}$
\begin{align*}
p^{xy}(t)&=p^{\ast }\Big(t,u(x)-u(y),b,\alpha \Big) \\
\text{ and }F^{xy}(t)&=F^{\ast }\Big(t,u(x)-u(y),b,\alpha \Big)\,.
\end{align*}
\end{definition}
Note that this definition requires that the data from all of the menus $
\{x,y\}$ is generated with the {\em same\/} boundary function $b$. This
corresponds to cases where the agent treats each decision problem as a
random draw from a fixed environment.\footnote{
In an optimal stopping model, the shape of the boundary is determined by the
agent's prior over these draws.} We are interested in characterizing which
stochastic choice functions admits a DDM representation. The following
result follows immediately from rescaling $\delta $ and $b$.
\begin{lemma}
\label{lem:alpha} If a stochastic choice function exhibits a DDM
representation for some $\alpha $, then it also exhibits a DDM
representation for $\alpha =1$.
\end{lemma}
We will thus without loss of generality only consider the DDM model where we
normalized $\alpha =1$. We write $p^{\ast }(t,\delta ,b)$ and $F^{\ast
}(t,\delta ,b)$ as short-hands for $p^{\ast }(t,\delta ,b,1)$ and $F^{\ast
}(t,\delta ,b,1)$.
\section{Characterization}
\label{sec:theory}
Given a stochastic choice function $(p^{xy},F^{xy})$, define the {\em
revealed drift\/}
\begin{equation}
\widetilde{\delta }^{xy}:=\sqrt{\frac{\bar{I}^{xy}}{2\bar{T}^{xy}}}.
\label{eq:delta}
\end{equation}
When the revealed drift is is non zero, we define the {\em revealed boundary
\/} as
\begin{equation}
\widetilde{b}^{xy}(t):=\frac{\ln p^{xy}(t)-\ln (1-p^{xy}(t))}{2\widetilde{
\delta }^{xy}}. \label{eq:b}
\end{equation}
The revealed drift is high for a pair $x,y$ whenever the agent either makes
very imbalanced choices or decides quickly, and low for choices that are
slow and close to 50-50. Over time the boundary at time $t$ follows the
log-odds ratio of the agent's choice at time $t$ which is zero whenever the
agent's choice is balanced and and increases in the imbalance of the agent's
choice. The revealed boundary is smaller for pairs with a larger revealed
drift. In the knife-edge case when the revealed drift is 0 the revealed
boundary is not defined and our results do not apply.
Theorem \ref{thm:1} below says that if the true data generating process is a
DDM, then the revealed drift and boundary will exactly match the true
parameters. Moreover, Theorem \ref{thm:1} allows us to test whether the true
data generating process is indeed a DDM.
\subsection{Characterization for a fixed pair}
Our first result characterizes the DDM for a fixed pair $x, y\in X$.
\begin{theorem}
\label{thm:1} For a fixed pair $x,y$ with $\tilde{\delta}
^{xy}\neq 0$ the stochastic choice function $(p^{xy},F^{xy})$
admits a DDM representation if and only if for all $t\geq 0$
\begin{equation*}
F^{xy}(t)=F^{\ast }(t;\tilde{\delta}^{xy},\tilde{b}^{xy})
\end{equation*}
If such a representation exists it is unique (up to the choice of $\alpha $)
and given by $\tilde{\delta}^{xy},\tilde{b}^{xy}$.
\end{theorem}
Thus, the stochastic choice function $(p^{xy},F^{xy})$ is consistent with
DDM whenever the observed distribution of stopping times $F^{xy}$ equals to
the distribution of hitting times generated by the revealed drift $\tilde{
\delta}^{xy}$ and revealed boundary $\tilde{b}^{xy}$. Theorem \ref{thm:1}
shows that the revealed drift and boundary are the unique candidate for a
DDM representation. It thus allows us to identify the parameters of the DDM
model directly from choice data. This permits the model to be calibrated to
the data without computing the likelihood function, which requires
computationally costly Monte-Carlo simulations. More substantially, as
Theorem \ref{thm:1} connects the primitives of the model directly to data it
allows us to better understand their economic meaning. The drift in the DDM
model is a measure of how imbalanced and quick the agent's choices are and
the shape of the boundary follows the imbalance of the agent's choices over
time. We hope that this interpretation makes the empirical content of the
parameters of DDM model more transparent and the model thus more useful.
Note that this theorem shows that the distribution of stopping times
contains additional information that is not captured by the mean. For
example, choice data where $p^{xy}(t)$ and $\bar{T}^{xy}$ are any 2 given
constants is only consistent with one possible distribution of stopping
times $F^{xy}$\ \ However a test based only on the mean choice probability
and mean stopping time will accept any model that matches those two numbers,
and in particular regardless of $F^{xy}$ the data is consistent with a
constant stopping boundary. (See \cite{Baldassi}).
\subsection{Characterization for menus of pairs}
Our next result extends the characterization to all pairs $x, y\in X$.
\begin{theorem}
\label{thm:2} The stochastic choice function $(\{p^{xy}\},\{F^{xy}\})_{x,y
\in X})$ has a DDM representation iff
\begin{enumerate}
\item[(i)] $F^{xy}(t)=F^{*}(t; \tilde \delta^{xy}, \tilde b^{xy})$ for all $
t\geq0$,
\item[(ii)] $\widetilde{b}^{xy}(t)=\widetilde{b}^{xz}(t)$ for all $x,y,z\in X
$ and all $t\geq 0$.
\item[(iii)] $\widetilde{\delta }^{xy}+\widetilde{\delta }^{yz}=\widetilde{
\delta }^{xyz}$ for all $x,y,z\in X$,
\end{enumerate}
\end{theorem}
Thus, in addition to satisfying the condition from Theorem 1 pairwise, we
have two additional consistency conditions imposed across pairs. Condition
(ii) follows from our assumption that the agent uses the same stopping
boundary in every menu. Condition (iii) comes from the assumption that the
drift in a given menu depends on the difference of utilities, that is $
\delta ^{xy}=u(x)-u(y)$.\footnote{
The proof of the theorem follows from Theorem \ref{thm:1} and the Sincov
functional equation, see, e.g., \cite{Aczel66}.}
\section{An Econometric Test for a Fixed Pair of Alternatives}
\label{sec:metrics}
The idea for the test is based on Theorem \ref{thm:1}, which requires that
the observed distribution of stopping times matches the distribution induced
by the revealed boundary $\tilde{b}$ and drift $\tilde{d}$. We first
describe a nonparametric estimator of $\tilde{b}$ and $\tilde{\delta}$ based
on a finite data set. Next, we show how to test the distribution matching
condition. This test could be extended to multiple-alternatives settings
along the lines of Theorem \ref{thm:2}, but we do not do so here.
\subsection{Estimation of drift and boundary}
Suppose that we have a fixed pair $x,y\in X$. Define
\begin{equation*}
\gamma _{\tau }:=\left\{
\begin{array}{ll}
1, & \text{when choice $x$ is made,} \\
0, & \text{when choice $y$ is made.}
\end{array}
\right.
\end{equation*}
Each data point consists of the time $\tau _{i}$ at which the choice is made
and the choice $\gamma _{i}$ made at time $\tau _{i}.$
\begin{assumption}
The data $(\tau _{1},\gamma _{1}),\ldots ,(\tau _{n},\gamma _{n})$ are i.i.d.
\end{assumption}
The unknown features of the DDM model are the drift $\delta $ and the
boundary $b(t).$\textbf{\ }We use estimators based on equations
\eqref{eq:delta} and \eqref{eq:b} that identify the revealed drift and
boundary. Both of them depend on the choice probability, so we first give an
estimator of that. Here $p^{xy}\left( t\right) :=\Pr (\gamma _{i}=1|\tau
_{i}=t)$ is the probability of choice $x$ conditional on the choice being
made at $t$.
The nonparametric estimator we construct is a spline regression: that is, a
least squares regression of $\gamma _{i}$ on approximating functions of $
\tau _{i}.$ For simplicity, we use a linear probability estimator of $
p^{xy}(t)$.\footnote{
We reserve consideration of other estimators of the choice probability to
future work, including logit or probit with a series approximation inside
the logit or probit CDF.}
We first transform $\tau _{i}$ to the unit interval.\footnote{
In DDM models where $b(t)$ does not reach zero, there is no uniform bound on
realized decision times $\tau _{i}$. Because $\tau _{i}$ is the conditioning
variable (i.e. regressor) in the choice probability, it is important to
allow for an unbounded regressor.} For this purpose let $G(t)$ be a CDF of a
positive random variable with PDF that is positive on $(0,\infty ).$ Consider
\begin{equation*}
G_{i}=G\left( \tau _{i}\right) .
\end{equation*}
Because $G_{i}$ lies in the unit interval we can use standard series
estimation to estimate $p^{xy}(t)$. We consider regression spline estimation
of $p^{xy}(t)$. For this purpose let
\begin{equation*}
q^{K}\left( G\right) =\left( q_{1K}\left( G\right) ,\ldots ,q_{KK}\left(
G\right) \right) ^{\prime }
\end{equation*}
be a $B$-spline vector, say for evenly spaced knots on $(0,1)$. Let $\hat{
\beta}$ be OLS coefficients from regressing $\gamma _{i}$ on $
q_{i}^{K}=q^{K}\left( G_{i}\right) $. The choice probability estimator we
consider is
\begin{equation*}
\hat{p}\left( t\right) :=q^{K}\left( G(t)\right) ^{\prime }\hat{\beta},\text{
}\hat{\beta}:=\left[ \sum_{i=1}^{n}q_{i}^{K}q_{i}^{K}{}^{\prime }\right]
^{-1}\sum_{i=1}^{n}q_{i}^{K}\gamma _{i}.
\end{equation*}
We give conditions for this estimator to be consistent and have other
important large sample properties in Assumptions 2 and 3 to follow.
We can estimate the drift $\delta $ by plugging in $\hat{p}(t)$ for $
p^{xy}(t)$ in formula \eqref{eq:delta} and replacing expectations with
sample averages. Let
\begin{align*}
\hat{I}(t)& :=\hat{p}\left( t\right) \ln \left[ \frac{\hat{p}\left( t\right)
}{1-\hat{p}\left( t\right) }\right] +\left[ 1-\hat{p}\left( t\right) \right]
\ln \left[ \frac{1-\hat{p}\left( t\right) }{\hat{p}\left( t\right) }\right] ,
\\
\bar{I}& :=\frac{1}{n}\sum_{i=1}^{n}\hat{I}\left( \tau _{i}\right) ,\text{ }
\bar{\tau}:=\frac{1}{n}\sum_{i=1}^{n}\tau _{i}.
\end{align*}
The estimator of $\delta $ is then
\begin{equation*}
\hat{\delta}:=\sqrt{\frac{\bar{I}}{\bar{\tau}}}.
\end{equation*}
The estimator of the boundary $b\left( t\right) $ is obtained by plugging in
$\hat{\delta}$ and $\hat{p}(t)$ in the expression of equation \eqref{eq:b},
giving
\begin{equation*}
\hat{b}(t):=\frac{1}{\hat{\delta}}\ln \left[ \frac{\hat{p}\left( t\right) }{
1-\hat{p}\left( t\right) }\right] .
\end{equation*}
\subsection{Testing}
We now have to test whether the observed distribution of stopping times
matches the one induced by the revealed drift and boundary. We do this by
comparing sample moments of functions of the decision time with estimators
of the moments that predicted by the model. To describe such a test let $
m_{J}(\tau )=(m_{1J}(\tau ),...,m_{JJ}(\tau ))^{\prime }$ be a vector of
functions of $\tau $. Examples include indicator functions for intervals and
B-splines in $G(\tau ).$ The sample average vector will be $\bar{m}
=\sum_{i=1}^{n}m_{J}(\tau _{i})/n$.\footnote{
The Kolmogorov--Smirnoff test uses indicator functions but instead of the
the average of $m$ it takes the supremum. The Cramer--von Mises test takes
the sum of squares. We look at the average of $m$ because the target cdf we
are comparing with is not fixed, but involves estimates of the boundary and
drift, see \cite{Newey94}.} We use simulation to obtain model prediction. To
describe the simulated predictions, let $\{B_{t}^{1},...,B_{t}^{S}\}$ be $S$
independent copies of Browning motion and
\begin{equation*}
\hat{\tau}_{s}=\inf \{t\geq 0:\big\vert\hat{\delta}t+B_{t}^{s}\big\vert\geq
\hat{b}(t)\}.
\end{equation*}
A moment vector predicted by the model would be $\hat{m}_{S}=
\sum_{s=1}^{S}m_{J}(\hat{\tau}_{s})/S.$ A test of the model can be based on
comparing $\bar{m}$ and $\hat{m}.$ Let $\hat{V}$ be a consistent estimator
of the asymptotic variance of $\sqrt{n}(\bar{m}-\hat{m}_{S})$ when the model
is correctly specified, as we will describe below. A test statistic can be
formed as
\begin{equation*}
\hat{A}:=n(\bar{m}-\hat{m}_{S})^{\prime }\hat{V}^{-1}(\bar{m}-\hat{m}_{S}).
\end{equation*}
The model would be rejected if $\hat{A}$ exceeds the critical value of a $
\chi ^{2}(J)$ distribution. If $J$ is allowed to grow with $n$ and the $
m_{J}(\tau )$ is allowed to grow in dimension and richness as $n$ grows then
this approach will test all the restrictions implied by DDM as $n$ grows. In
Appendix A we describe the construction of $\hat{V}.$
In formulating conditions for the asymptotic distribution of this test we
will let $m_{jJ}(\tau ),$ $(j=1,...,J)$ be indicator functions for disjoint
intervals. Let $\tau _{jJ}=G^{-1}(j/(J+1)),$ $(j=0,...,J),$ $\tau
_{J+1,J}=\infty .$ Consider
\begin{equation*}
m_{jJ}(t)=\sqrt{J+1}\cdot \mathbbm{1}(\tau _{j,J}\leq t<\tau _{j+1,J}),\text{
}(j=1,...,J).
\end{equation*}
The test based on these functions will be based on comparing empirical
probabilities of intervals with those predicted by the model. The
normalization of multiplying by $\sqrt{J+1}$ is convenient in making the
second moment of these functions of the same magnitude for different values
of $J$. Note that we have left out the indicator for the interval $
(0,1/(J+1)).$ We have done this to account for the fact that the estimator
the drift parameter uses some information about $\tau _{i}$, so that we are
not able to test all of the implications of the DDM for the distribution of $
\tau _{i}.$ As usual we can only test overidentifying restrictions.
We derive results under the following conditions:
\begin{assumption}
\label{asst:2} The pdf of $G(\tau _{i})$ is bounded and bounded away from
zero.
\end{assumption}
This assumption is equivalent to the ratio of the pdf of $\tau _{i}$ to $
dG(t)/dt$ being bounded and bounded away from zero. It is straightforward to
weaken this condition to allow it to only hold on compact, connected
interval that is a subset of $(0,1),$ if we assume the $b(t)$ is constant on
known intervals near $0$ and where $\tau $ is large.
We also make a smoothness assumption on the boundary function.
\begin{assumption}
\label{asst:3} $b(G^{-1}(g))$ is bounded and $s\geq 1$ times differentiable
with bounded derivatives on $g\in \lbrack 0,1]$ and the $q_{kK}(G),$ $
k=1,...,K$ are b-splines of order $s-1.$
\end{assumption}
This condition requires that the derivatives of $b(t)$ go to zero in the
tails of the distribution of $\tau _{i}$ as fast as the pdf of $G(t)$ does.
We also require that the drift parameter be nonzero.
\begin{assumption}
\label{asst:4} $\delta \neq 0$.
\end{assumption}
We need to add other conditions about the smoothness of CDF of $\tau _{i}$
as a function of the drift $\delta $ and the boundary and about rates of
growth of $J$ and $K$. The involve much notation, so we state them in
Assumption \ref{asst:5} in Appendix \ref{app:smoothness}.
We can now state the following result on the limiting distribution of $\hat{A
}.$
\begin{theorem}
Suppose that Assumptions \ref{asst:2}, \ref{asst:3}, \ref{asst:4} and
Assumption \ref{asst:5} in Appendix \ref{app:smoothness} are satsified. Then
for the $1-\alpha $ quantile $c\left( \alpha ,J\right) $ of a chi-square
distribution with $J$ degrees of freedom
\begin{equation*}
\Pr \left( \hat{A}\geq c\left( \alpha ,J\right) \right) \longrightarrow
\alpha .
\end{equation*}
\end{theorem}