# Asymptotic Behaviour of the Modified Likelihood Root

Yanbo Tang  Nancy Reid  Department of Statistical Sciences, University of
Toronto, Toronto, Canada  Vector Institute, Toronto, Canada

###### Abstract

We examine the normal approximation of the modified likelihood root, an
inferential tool from higher-order asymptotic theory, for the linear
exponential and location-scale family. We show that the $r^{\star}$ statistic
can be thought of as a location and scale adjustment to the likelihood root
$r$ up to $O_{p}(n^{-3/2})$, and more generally $r^{\star}$ can be expressed
as a polynomial in $r$. We also show the linearity of the modified likelihood
root in the likelihood root for these two families.

, and

##  1  Introduction

The use of $p$-values, although sometimes controversial, has become a key part
of modern statistical science, they are the building block of various multiple
testing correction procedures used in statistical genetics, where millions of
hypotheses are simultaneously considered. In most circumstances $p$-values are
not exact but are calculated from the limiting distribution of a test
statistic. The usual test statistics provided in statistical software, such as
the likelihood ratio test, Wald test and score test, all have a common known
limiting distribution and are accurate to the first order, meaning that the
approximation error behaves like $O(n^{-1/2})$. However, in the small sample
setting or when the number of nuisance parameters is high relative to the
number of observations, this trio of tests may perform poorly. An improved
test statistic, $r^{\star}$, a modified version of the likelihood root, can be
used for likelihood-based inference for scalar parameters of interest. It
produces more accurate $p$-values compared to the first order approximations
of the test statistics. The accuracy of the $p$-values generated by
$r^{\star}$ can be quite remarkable; in some models the approximation is
accurate with very few observations (Brazzale et al., 2007, §3.2).

Given the importance that $p$-values play in statistical inference, the exact
mechanism through which $r^{\star}$ generates more accurate $p$-values
warrants a careful examination. We provide insight into the behaviour of
$r^{\star}$ by expressing it as a formal asymptotic expansion, showing that it
is asymptotically linear the likelihood root, which we introduce below.

We assume the data $y=(y_{1},\cdots,y_{n})^{\top}$ are generated independently
from a model parametrized by $\theta=(\psi,\lambda)$ where $\psi$ is a scalar
parameter of interest, $\lambda$ is the nuisance parameter and
$\hat{\lambda}_{\psi}$ denotes the constrained maximum likelihood estimate;
i.e. the value of $\lambda$ that maximises the log-likelihood function for a
fixed $\psi$. We denote the log-likelihood function by $l(\psi,\lambda;y)$,
and the the data generating parameter by $\theta_{0}=(\psi_{0},\lambda_{0})$.
The profile log-likelihood function,

$l_{\text{p}}(\psi;y):=\sup_{\lambda}l(\psi,\lambda;y)=l(\psi,\hat{\lambda}_{\psi};y),$

accounts for the presence of nuisance parameters through constrained
maximization. Under suitable regularity conditions (Barndorff-Nielsen and Cox,
1994, §3.4),

$w(\psi_{0};y):=2\\{l_{\text{p}}(\hat{\psi})-l_{\text{p}}(\psi_{0})\\}\xrightarrow{d}\chi^{2}_{1},$

where $\chi^{2}_{1}$ is a random variable distributed as chi-squared with one
degree of freedom. Equivalently, the log-likelihood root

$\displaystyle r(\psi_{0};y)$
$\displaystyle:=\text{sign}(\hat{\psi}-\psi_{0})[2\\{l_{\text{p}}(\hat{\psi};y)-l_{\text{p}}(\psi_{0};y)\\}]^{\frac{1}{2}}\xrightarrow{d}Z,$
(1)

where the random variable $Z$ has the standard normal distribution (Barndorff-
Nielsen and Cox, 1994, §2.3).

By adding a correction term to $r$, we obtain the modified likelihood root

$\displaystyle
r^{\star}(\psi_{0};y)=r(\psi_{0};y)+\frac{1}{r(\psi_{0};y)}\log\left\\{\frac{q(\psi_{0};y)}{r(\psi_{0};y)}\right\\};$
(2)

the form of $q$ depends on the model. It has been shown under regularity
conditions that the normal approximation to the distribution of $r^{\star}$ is
accurate to $O(n^{-3/2})$ (Barndorff-Nielsen and Cox, 1994), whereas the
normal approximation to $r$ is only accurate to $O(n^{-1/2})$.

The mechanism through which $r^{\star}$ achieves this accuracy is not entirely
transparent. Cakmak et al. (1998) show that in models with no nuisance
parameters, the leading term in the adjustment factor $r^{\star}$ is a
location and scale correction to $r$. We generalize this result to models with
nuisance parameters. We then use a formal expansion to discuss the asymptotic
behaviour of $r^{\star}$ when the number of parameters increases with the
number of observations. We show that the adjustment factor $(1/r)\log(q/r)$ is
potentially of the same asymptotic order as $r$ in the high-dimensional
regime, agreeing with results in Tang and Reid (2020).

Though similar techniques as in the proof of the main theorems. We also show
that $r=q+q^{2}A/n^{1/2}+q^{3}B/n+O_{p}(n^{-3/2})$, in the linear exponential
and location scale families, where $A$ and $B$ are $O_{p}(1)$, which may
depend on $y$ and $\theta_{0}$. This result underlies the proof that the
normal approximation to the distribution of $r^{\star}$ has relative error
$O(n^{-3/2})$ (Brazzale et al., 2007, §8.5), but to our knowledge this has not
been established in the vector parameter setting.

We focus our analysis on the location-scale and linear exponential family, as
the expression for $q$ in (2) are explicitly available. We begin with
background details on $r^{\star}$, the linear exponential family and the
location-scale family in §2. We present out main theorems in §3, briefly
discuss the impact of the results if $p$ scales with $n$ in §4 and show that
$r$ can be expressed as a third-order polynomial in $q$ in §5. We present some
simulations in §6 which illustrate the accuracy of the approximations to
$r^{\star}$ and conclude with some additional proof details in §7.

###  1.1 Notation

Derivatives of the log-likelihood function are denoted by subscripts placed on
“$l$”, for example $l_{\psi\lambda\lambda}(\theta)$ represents the matrix with
components are
$[l_{\psi\lambda\lambda}(\theta)]_{rs}=\partial^{3}l(\theta)/\partial\psi\partial\lambda_{r}\partial\lambda_{s}$.
We use $j$ to denote the observed information matrix,
$j(\theta)=-l_{\theta\theta}(\theta)$; subscripts placed on $j$ denote sub-
matrices of $j$ and we let $j_{p}(\psi)=-d^{2}l_{\text{p}}(\psi)/d\psi^{2}$. A
tilde placed on any quantity denotes that it is evaluated at the constrained
maximum likelihood estimate, $(\psi,\hat{\lambda}_{\psi})$, for example
$\tilde{\jmath}_{\lambda\lambda}=j_{\lambda\lambda}(\psi,\hat{\lambda}_{\psi})$,
and a hat denotes that it is evaluated at the global maximum likelihood
estimate, $\hat{\theta}$, thus
$\hat{\jmath}=j(\hat{\psi},\hat{\lambda})=j(\hat{\theta})$.

We use $d/d\psi$ to denote the total derivative with respect to $\psi$ and
$\partial/\partial\psi$ to denote the partial derivative with respect to
$\psi$. The $k$-th derivative of the profile log-likelihood is
$\zeta_{k}(\psi)=d^{k}l_{p}(\psi)/d\psi^{k}$ and $k$-th total derivative of
the log determinant of the information matrix is

$\gamma_{k}(\psi)=d^{k}\log\\{|j_{\lambda\lambda}(\psi,\hat{\lambda}_{\psi})|\\}/d\psi^{k}.$

We define the $k$-th quasi-cumulant of the profile log-likelihood function as:

$\displaystyle\kappa_{k}(\psi)=\frac{\zeta_{k}(\psi)}{\\{-\zeta_{2}(\psi)\\}^{k/2}}.$
(3)

In the sequel we suppress the dependence of functions on the data $y$ in the
notation, for example $r(\psi_{0};y)$ will simply be written as $r(\psi_{0})$.
Finally $\sigma_{\max}(A)$ is the maximum singular value of a matrix $A$. The
following inequalities will prove useful, for two square matrices $A,B$ and
$B$ positive definite:

$\displaystyle|\operatorname{\text{tr}}(AB)|\leq\left\lVert
A\right\rVert_{F}\left\lVert
B\right\rVert_{F}\quad\text{and,}\quad|\operatorname{\text{tr}}(AB)|\leq\sigma_{\max}(A)\operatorname{\text{tr}}(B),$

where $\left\lVert A\right\rVert_{F}=(\sum_{i,j}A_{ij}^{2})^{1/2}$ is the
Frobenius norm and the latter inequality is a consequence of the von Neumann
trace inequality (Mirsky, 1975). For a vector $v$, we let $\left\lVert
v\right\rVert_{p}$ denote its $L^{p}$ norm.

##  2 Background and Assumptions

We assume the following conditions on the model:

######  Assumption 1.

The $k$-th order partial derivatives of the log-likelihood function with
respect to the elements of $\theta$ are $O_{p}(n)$ for all integers $k>1$ when
evaluated at $\hat{\theta}$.

######  Assumption 2.

$\hat{\theta}-\theta_{0}=O_{p}(n^{-1/2})$

######  Assumption 3.

The eigenvalues of $\hat{\jmath}/n$, $j(\hat{\theta}_{\psi_{0}})/n$,
$n\\{\hat{\jmath}\\}^{-1}$ and $nj^{-1}(\hat{\theta}_{\psi_{0}})$ are positive
and $O_{p}(1)$.

Assumption 1 ensures that the likelihood derivatives grow at the usual rate
when evaluated at the maximum likelihood estimate. Assumption 2 states that
the maximum likelihood estimate is $n^{1/2}$ consistent for the true parameter
value: this rate of consistency is typically achieved for most well behaved
parametric models (van der Vaart, 1998, §5). Finally Assumption 3 ensures that
the asymptotic covariance matrix for the maximum likelihood estimate is well
defined. For the regression problems that we consider, Assumption 3 is
equivalent to a restriction on the eigenvalues on the Gramian matrix,
$X^{\top}X$, as well as a lower bound on the variance of the fitted values.

###  2.1 Linear Exponential Family

Let $X$ be a $n\times p$ matrix of covariates, $x_{i,j}$ denote the
$(i,j)^{th}$ entry in $X$, and $x^{\top}_{i}$ denote the $i$-th row of $X$. We
assume the density of $y_{i}$ is a full exponential family model with log-
likelihood function

$\displaystyle\log\\{f(y_{i};\psi,\lambda,x_{i})\\}=\psi
u(x_{i,p},y_{i})+\lambda^{\top}v(x_{i},y_{i})-c_{i}(\psi,\lambda)+h(x_{i},y_{i}),$
(4)

where $u(x_{i,p},y_{i})$ is a scalar sufficient statistic associated with
$\psi$ and

$v(x_{i},y_{i})=\\{v_{1}(x_{i,1},y_{i}),\cdots,v_{p-1}(x_{i,p-1},y_{i})\\}^{\top},$

is the vector of sufficient statistics associated with the nuisance
parameters.

In this model, $q$ in (2) takes the form

$\displaystyle q(\psi_{0})=t(\psi_{0})\rho(\psi_{0}),$

where

$\displaystyle t$
$\displaystyle=(\hat{\psi}-\psi_{0})j_{\text{p}}^{1/2}(\hat{\psi}),$ (5)

where $t$ is the Wald statistic for testing $\psi=\psi_{0}$,

$\displaystyle\rho$
$\displaystyle=\\{|j_{\lambda\lambda}(\hat{\theta})|/|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|\\}^{1/2},$

where $j_{\lambda\lambda}(\theta)$ is the $(p-1)\times(p-1)$ sub-matrix of
$j(\theta)$ associated with the nuisance parameters (Brazzale et al., 2007,
§8.6.1). We follow Pierce and Peters (1992) and write

$\displaystyle r^{\star}=r+r_{np}+r_{inf},$

where $r_{np}$ is a nuisance parameter adjustment and $r_{inf}$ is an
information adjustment. This partitioning of the adjustments will be helpful
to the analysis of the asymptotic behaviour of $r^{\star}$. For this model,

$\displaystyle r_{np}=\frac{1}{r}\log(\rho)\text{,}\quad r_{inf}$
$\displaystyle=\frac{1}{r}\log(\frac{t}{r}).$ (6)

###  2.2 Location-Scale Family

For a linear regression model based on the location scale-family, the model is

$\displaystyle y_{i}=x^{\top}_{i}\beta+\sigma\epsilon_{i},$ (7)

where the errors $\epsilon_{i}$ are assumed independent and identically
distributed from a known distribution with continuous density $f(\epsilon)$.
The model is parametrized by $\theta=(\beta,\sigma)$, we assume that the
parameter of interest is a component of $\beta$. For this model

$\displaystyle q(\psi_{0})$ $\displaystyle=s(\psi_{0})/\rho(\psi_{0})$
$\displaystyle s$
$\displaystyle=l_{\text{p}}^{\prime}(\psi_{0})/j_{\text{p}}^{1/2}(\hat{\psi}),$
(8)

$s$ is the standard score test statistic and $\rho$ is defined above. For this
model

$\quad r_{np}=-\frac{1}{r}\log(\rho),\quad
r_{inf}=\frac{1}{r}\log(\frac{s}{r}).$

###  2.3 General Models

For general models $r_{np}$ and $r_{inf}$ are more difficult to work with, the
additional difficulty lies in the necessity of conditioning on an ancillary
statistic. It may be possible to use certain variants of $r^{\star}$, where it
is written in a similar form to a Wald or a Score statistic (Reid, 2003, 3.4)
and proceed in a similar fashion as in proof of Theorem 3.2 and 3.3, we do not
pursue this however.

##  3 Formal expansions of $r_{inf}$ and $r_{np}$

In this section we obtain formal asymptotic expansions for $r_{inf}$ and
$r_{np}$, which detail the relationship between $r$ and $r^{\star}$ in the
linear exponential and location scale families, respectively. Analogous
results to Theorem 3.2 and 3.3 were obtained by Cakmak et al. (1998) in the
case of no nuisance parameters. The expansions for $r_{inf}$ and $r_{np}$ show
that $r^{\star}$ is asymptotically equivalent to a location and scale
adjustment to the likelihood root $r$.

We begin by showing the relationships between $r$, $s$ and $t$.

######  Lemma 3.1.

Under Assumptions 1–3, for $r$, $t$ and $s$ defined in (1), (5) and (8):

$\displaystyle
t=r\left\\{1+\frac{A_{1}}{n^{1/2}}r+\frac{B_{1}}{n}r^{2}+O_{p}(n^{-3/2})\right\\},$
$\displaystyle
s=t\left\\{1+\frac{A_{2}}{n^{1/2}}t+\frac{B_{2}}{n}t^{2}+O_{p}(n^{-3/2})\right\\},$

where,

$\displaystyle A_{1}$
$\displaystyle=-\frac{n^{1/2}}{6}\kappa_{3}(\hat{\psi}),\quad
B_{1}=\frac{n}{24}\kappa_{4}(\hat{\psi})+\frac{5n}{72}\kappa_{3}^{2}(\hat{\psi}),$
$\displaystyle A_{2}$
$\displaystyle=\frac{n^{1/2}\kappa_{3}(\hat{\psi})}{2},\quad\quad
B_{2}=-\frac{n\kappa_{4}(\hat{\psi})}{6}.$

###### Proof.

We begin by deriving the relationship between $r$ and $t$, defined in (1) and
(5) :

$\displaystyle r^{2}$
$\displaystyle=2\Big{\\{}l_{\text{p}}(\hat{\psi})-l_{\text{p}}(\psi_{0})\Big{\\}},$
$\displaystyle=2\Big{\\{}l_{\text{p}}(\hat{\psi})-l_{\text{p}}(\hat{\psi})+(\hat{\psi}-\psi_{0})\zeta_{1}(\hat{\psi})-\frac{(\hat{\psi}-\psi_{0})^{2}}{2}\zeta_{2}(\hat{\psi})$
$\displaystyle\quad+\frac{(\hat{\psi}-\psi_{0})^{3}}{6}\zeta_{3}(\hat{\psi})+\frac{(\hat{\psi}-\psi_{0})^{4}}{24}\zeta_{4}(\hat{\psi})++O_{p}(n^{-3/2})\Big{\\}}$
$\displaystyle=t^{2}\Big{\\{}1+\frac{\kappa_{3}(\hat{\psi})}{3}t-\frac{\kappa_{4}(\hat{\psi})}{12}t^{2}+O_{p}(n^{-3/2})\Big{\\}}.$

The Taylor-series expansion for $(1+x)^{1/2}$ gives

$\displaystyle r$
$\displaystyle=t\Big{\\{}1+\frac{\kappa_{3}(\hat{\psi})}{6}t-\frac{\kappa_{4}(\hat{\psi})}{24}t^{2}-\frac{\kappa^{2}_{3}(\hat{\psi})}{72}t^{2}+O_{p}(n^{-3/2})\Big{\\}},$

which implies

$\displaystyle t$
$\displaystyle=r\Big{\\{}1+\frac{\kappa_{3}(\hat{\psi})}{6}t-\frac{\kappa_{4}(\hat{\psi})}{24}t^{2}-\frac{\kappa^{2}_{3}(\hat{\psi})}{72}t^{2}+O_{p}(n^{-3/2})\Big{\\}}^{-1}$
$\displaystyle=r\Big{\\{}1-\frac{\kappa_{3}(\hat{\psi})}{6}t+\frac{\kappa_{4}(\hat{\psi})}{24}t^{2}+\frac{\kappa^{2}_{3}(\hat{\psi})}{72}t^{2}+\frac{\kappa^{2}_{3}(\hat{\psi})}{36}t^{2}+O_{p}(n^{-3/2})\Big{\\}}$
$\displaystyle=r\Big{\\{}1-\frac{\kappa_{3}(\hat{\psi})}{6}t+\frac{\kappa_{4}(\hat{\psi})}{24}t^{2}+\frac{\kappa^{2}_{3}(\hat{\psi})}{24}t^{2}+O_{p}(n^{-3/2})\Big{\\}}.$
(9)

As $t$ appears on both sides of the equation, we iteratively solve the
equation by substitution.

$\displaystyle t$
$\displaystyle=r\Big{[}1-\frac{\kappa_{3}(\hat{\psi})}{6}\Big{\\{}\Big{(}1-\frac{\kappa_{3}(\hat{\psi})}{6}t\Big{)}r\Big{\\}}+\frac{1}{24}\Big{\\{}\kappa_{3}^{2}(\hat{\psi})+\kappa_{4}(\hat{\psi})\Big{\\}}r^{2}+O_{p}(n^{-3/2})\Big{]}$
(10)
$\displaystyle=r\Big{[}1-\frac{\kappa_{3}(\hat{\psi})}{6}r+\Big{\\{}\frac{\kappa^{2}_{3}(\hat{\psi})}{36}r\Big{\\}}t+\frac{1}{24}\Big{\\{}\kappa_{3}^{2}(\hat{\psi})+\kappa_{4}(\hat{\psi})\Big{\\}}r^{2}+O_{p}(n^{-3/2})\Big{]}$
$\displaystyle=r\Big{\\{}1-\frac{\kappa_{3}(\hat{\psi})}{6}r+\frac{5}{72}\kappa^{2}_{3}(\hat{\psi})r^{2}+\frac{1}{24}\kappa_{4}(\hat{\psi})r^{2}+O_{p}(n^{-3/2})\Big{\\}}$
$\displaystyle=r\Big{\\{}1+\frac{A_{1}}{n^{1/2}}r+\frac{B_{1}}{n}r^{2}+O_{p}(n^{-3/2})\Big{\\}},$
(11)

where

$\displaystyle A_{1}=-\frac{n^{1/2}}{6}\kappa_{3}(\hat{\psi}),\quad
B_{1}=\frac{n}{24}\kappa_{4}(\hat{\psi})+\frac{5n}{72}\kappa_{3}^{2}(\hat{\psi}).$

While for the expansion of $s$, write

$\displaystyle s$
$\displaystyle=\frac{\zeta_{1}(\psi_{0})}{j_{p}^{1/2}(\hat{\psi})}$
$\displaystyle=\frac{1}{j_{p}^{1/2}(\hat{\psi})}\Big{\\{}\zeta_{1}(\hat{\psi})-\zeta_{2}(\hat{\psi})(\hat{\psi}-\psi_{0})+\frac{\zeta_{3}(\hat{\psi})}{2}(\hat{\psi}-\psi_{0})^{2}-\frac{\zeta_{4}(\hat{\psi})}{6}(\hat{\psi}-\psi_{0})^{3}+O_{p}(n^{-3/2})\Big{\\}}$
$\displaystyle=t+\frac{\kappa_{3}(\hat{\psi})}{2}t^{2}-\frac{\kappa_{4}(\hat{\psi})}{6}t^{3}+O_{p}(n^{-3/2}),$
$\displaystyle=t\left\\{1+\frac{A_{2}}{n^{1/2}}t+\frac{B_{2}}{n}t^{2}+O_{p}(n^{-3/2})\right\\},$

where

$\displaystyle A_{2}=\frac{n^{1/2}}{2}\kappa_{3}(\hat{\psi}),\quad
B_{2}=-\frac{n}{6}\kappa_{4}(\hat{\psi}).$

∎

######  Theorem 3.2.

Under Assumptions 1–3, for the linear exponential family

$\displaystyle
r_{np}=\frac{1}{2}\frac{\gamma_{1}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})^{1/2}}-\Big{\\{}\frac{1}{12}\frac{\kappa_{3}(\hat{\psi})\gamma_{1}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})^{1/2}}-\frac{1}{4}\frac{\gamma_{2}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})}\Big{\\}}r+O_{p}(n^{-3/2}),$
(12)

and for the location-scale family

$\displaystyle
r_{np}=-\frac{1}{2}\frac{\gamma_{1}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})^{1/2}}+\Big{\\{}\frac{1}{12}\frac{\kappa_{3}(\hat{\psi})\gamma_{1}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})^{1/2}}-\frac{1}{4}\frac{\gamma_{2}(\hat{\psi})}{j_{\text{p}}(\hat{\psi})}\Big{\\}}r+O_{p}(n^{-3/2}).$
(13)

######  Theorem 3.3.

Under Assumptions 1–3, for the linear exponential family,

$\displaystyle
r_{inf}=-\frac{1}{6}\kappa_{3}(\hat{\psi})+\Big{\\{}\frac{1}{24}\kappa_{4}(\hat{\psi})+\frac{4}{72}\kappa_{3}^{2}(\hat{\psi})\Big{\\}}r+O_{p}(n^{-3/2}),$
(14)

and for the location-scale family

$\displaystyle
r_{inf}=\frac{1}{3}\kappa_{3}(\hat{\psi})-\Big{\\{}\frac{3}{24}\kappa_{4}(\hat{\psi})+\frac{11}{72}\kappa_{3}^{2}(\hat{\psi})\Big{\\}}r+O_{p}(n^{-3/2}).$
(15)

###### Proof.

Linear Exponential Family: Using (6) and Lemma 3.1 for a linear exponential
family we have

$\displaystyle r_{inf}$
$\displaystyle=\frac{1}{r}\log\Big{(}\frac{t}{r}\Big{)}$
$\displaystyle=\frac{A_{1}}{n^{1/2}}+\left(\frac{B_{1}}{n}-\frac{A_{1}^{2}}{2n}\right)r+O_{p}(n^{-3/2})$
$\displaystyle=-\frac{1}{6}\kappa_{3}(\hat{\psi})+\Big{\\{}\frac{1}{24}\kappa_{4}(\hat{\psi})+\frac{4}{72}\kappa_{3}^{2}(\hat{\psi})\Big{\\}}r+O_{p}(n^{-3/2}).$
(16)

A similar expansion can be developed for $r_{np}$:

$\displaystyle r_{np}$
$\displaystyle=\frac{1}{2r}\log\Big{\\{}\frac{|j_{\lambda\lambda}(\hat{\psi},\hat{\lambda})|}{|j_{\lambda\lambda}(\psi_{0},\hat{\lambda}_{\psi_{0}})|}\Big{\\}}$
$\displaystyle=\frac{1}{2r}\left[\frac{\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}t+\frac{\gamma_{2}(\hat{\psi})}{2\zeta_{2}(\hat{\psi})}t^{2}+O_{p}\Big{(}n^{-3/2}\Big{)}\right]$
(17)
$\displaystyle=\frac{1}{2}\left[\Big{(}1+\frac{A_{1}}{n^{1/2}}r\Big{)}\frac{\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}+\frac{\gamma_{2}(\hat{\psi})}{2\zeta_{2}(\hat{\psi})}r+O_{p}\Big{(}n^{-3/2}\Big{)}\right]$
$\displaystyle=\frac{1}{2}\frac{\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}+\left[\frac{1}{2}\frac{A_{1}\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})n\\}^{1/2}}+\frac{\gamma_{2}(\hat{\psi})}{4\zeta_{2}(\hat{\psi})}\right]r+O_{p}\Big{(}n^{-3/2}\Big{)},$
(18)

where the third equality uses Lemma 3.1.

Location-Scale Family: Using Lemma 3.1, we obtain that for the location-scale
family:

$\displaystyle r_{inf}$
$\displaystyle=\frac{1}{r}\log\Big{[}\frac{1}{r}\Big{\\{}t+\frac{\kappa_{3}(\hat{\psi})}{2}t^{2}-\frac{\kappa_{4}(\hat{\psi})}{6}t^{3}+O_{p}(n^{-3/2})\Big{\\}}\Big{]}$
$\displaystyle=\frac{1}{r}\log\Big{[}1+\frac{A_{2}}{n^{1/2}}r+\frac{B_{2}}{n}r^{2}+\frac{\kappa_{3}(\hat{\psi})}{2}\Big{\\{}1+\frac{A_{2}}{n^{1/2}}r\Big{\\}}^{2}r-\frac{\kappa_{4}(\hat{\psi})}{6}r^{2}+O_{p}(n^{-3/2})\Big{]}$
$\displaystyle=\frac{1}{3}\kappa_{3}(\hat{\psi})-\Big{\\{}\frac{3}{24}\kappa_{4}(\hat{\psi})+\frac{11}{72}\kappa_{3}^{2}(\hat{\psi})\Big{\\}}r+O_{p}(n^{-3/2}).$

The expansion for the nuisance parameter adjustment $r_{np}$ is the same as in
the exponential family, except for a change in sign.

∎

######  Remark 1.

From Theorems 3.2 and 3.3, for the linear exponential family,

$\displaystyle r^{\star}$
$\displaystyle=-\frac{1}{6}\kappa_{3}(\hat{\psi})+\frac{1}{2}\frac{\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$
$\displaystyle+\left[1+\frac{1}{24}\kappa_{4}(\hat{\psi})+\frac{4}{72}\kappa_{3}^{2}(\hat{\psi})-\frac{1}{12}\frac{\kappa_{3}(\hat{\psi})\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}-\frac{1}{4}\frac{\gamma_{2}(\hat{\psi})}{\zeta_{2}(\hat{\psi})}\right]r+O_{p}(n^{-3/2}),$

and for the location-scale family,

$\displaystyle r^{\star}$
$\displaystyle=\frac{1}{3}\kappa_{3}(\hat{\psi})-\frac{1}{2}\frac{\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$
$\displaystyle+\left[1-\frac{3}{24}\kappa_{4}(\hat{\psi})-\frac{11}{72}\kappa_{3}^{2}(\hat{\psi})+\frac{1}{12}\frac{\kappa_{3}(\hat{\psi})\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}+\frac{1}{4}\frac{\gamma_{2}(\hat{\psi})}{\zeta_{2}(\hat{\psi})}\right]r+O_{p}(n^{-3/2}).$

After some algebraic manipulations using the above equations, we obtain for
both families:

$\displaystyle
r^{\star}=\frac{r-\tilde{A}/n^{1/2}}{\\{1+\tilde{B}/n\\}}+O_{p}(n^{-3/2}),$

which shows that $r^{\star}$ is a shift and rescaling of $r$, where
$\tilde{A}$ and $\tilde{B}$ are $O_{p}(1)$ terms.

######  Corollary 1.

Under Assumptions 1–3, the expansions of $r_{inf}$ and $r_{np}$ can be
extended to:

$\displaystyle
r_{inf}=\sum_{k=1}^{m}\frac{A_{k}}{n^{k/2}}r^{k-1}+O_{p}(n^{-(m+1)/2}),\quad
r_{np}=\sum_{k=1}^{m}\frac{B_{k}}{n^{k/2}}r^{k-1}+O_{p}(n^{-(m+1)/2}),$ (19)

for arbitrary $m\in\mathbb{N}$ where the coefficients $A_{k}=O_{p}(1)$ and
$B_{k}=O_{p}(1)$.

From the substitution argument employed in (10) to (11), we can deduce that
the coefficient of $r^{k-1}$ are of order $O_{p}(n^{-k/2})$ and

$A_{k}=n^{k/2}\sum_{j=1}^{k}\sum_{\\{i_{1},\cdots,i_{j}\\}\in
S_{j}}Z_{j}(\\{i_{1},\cdots,i_{j}\\})\prod_{l=1}^{j}\hat{\kappa}_{i_{l}},$

where indices $\\{i_{1},\cdots,i_{j}\\}$ take values in $\\{3,\cdots,k+2\\}$,
$S_{j}$ is the set of all indices such that $\sum_{l=1}^{j}(i_{l}-2)=k$, and
$Z_{j}(\cdot)$ is a function of a set of indices that returns a numerical
constant.

From equation (18), when grouping terms in powers of $r$, we obtain the
following expressions for the coefficients in the expansion of $r_{np}$:

$B_{k}=\sum_{j=0}^{k-1}n^{(k-j)/2}C_{j}\frac{\gamma_{k-j}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{(k-j)/2}}\quad\text{
and }\quad C_{k}=\sum_{m=1}^{k}\sum_{\\{i_{1},\cdots,i_{m}\\}\in
D_{m,k}}\prod_{l=1}^{m}A_{i_{l}},$

where $C_{0}=1$, the indices $\\{i_{1},\cdots,i_{m}\\}$ range from $1$ to $j$.
The set $D_{m,k}$ is the set of all indices $\\{i_{1},\cdots,i_{m}\\}$ such
that $\sum_{l=1}^{m}i_{l}=k$. Some examples of terms which appear in $A_{k}$,
$B_{k}$ and $C_{k}$ are given in Figure 1.

$k$ |  $A_{k}$ |  $B_{k}$ |  $C_{k}$
---|---|---|---
1 |  $n^{1/2}\kappa_{3}(\hat{\psi})$ |  $\frac{n^{1/2}\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$ |  $A_{1}$
2 |  $n\kappa_{3}^{2}(\hat{\psi})$, $n\kappa_{4}(\hat{\psi})$ |  $\frac{n^{1/2}C_{1}\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$, $\frac{n\gamma_{2}(\hat{\psi})}{-\zeta_{2}(\hat{\psi})\ }$ |  $A_{2}$, $A_{1}^{2}$
3 |  $n^{3/2}\kappa_{3}^{3}(\hat{\psi})$, $n^{3/2}\kappa_{3}(\hat{\psi})\kappa_{4}(\hat{\psi})$ , $n^{3/2}\kappa_{5}(\hat{\psi})$ |  $\frac{C_{2}n^{1/2}\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$, $\frac{C_{1}n\gamma_{2}(\hat{\psi})}{-\zeta_{2}(\hat{\psi})}$, $\frac{n^{3/2}\gamma_{3}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{3/2}}$ |  $A_{3}$,$A_{1}A_{2}$, $A_{1}^{3}$
4 |  $n^{2}\kappa_{3}^{4}(\hat{\psi})$, $n^{2}\kappa^{2}_{3}(\hat{\psi})\kappa_{4}(\hat{\psi})$ , $n^{2}\kappa^{2}_{4}(\hat{\psi})$, $n^{2}\kappa_{5}(\hat{\psi})\kappa_{3}(\hat{\psi})$, $n^{2}\kappa_{6}(\hat{\psi})$ |  $\frac{C_{3}n^{1/2}\gamma_{1}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{1/2}}$, $\frac{C_{2}n\gamma_{2}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}}$, $\frac{C_{1}n^{3/2}\gamma_{3}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{3/2}}$, $\frac{n^{2}\gamma_{4}(\hat{\psi})}{\\{-\zeta_{2}(\hat{\psi})\\}^{2}}$ |  $A_{4}$, $A_{3}A_{1}$, $A_{2}^{2}$, $A_{1}^{2}$, $A_{2}$, $A_{1}^{4}$
Figure 1: The first four terms of $A_{k}$, $B_{k}$ and $C_{k}$ for reference.
The order of the $\kappa_{j}$ terms are given in Lemma 2 in §7.1.

##  4 $r^{\star}$ in High Dimensions

We discuss the behaviour of $r^{\star}$ in the high dimensional setting, when
$p$ increases with $n$. Recently this asymptotic paradigm has become
increasingly popular as the datasets observed in practice are increasing in
size not only in terms of number of samples but also in number of covariates.
It is known that some traditional statistics, such as the likelihood ratio
test, does not perform well in this setting (Sur and Candès, 2019). Thus, it
is reasonable to question how $r^{\star}$ performs in this setting. We
quantify the behaviour of the order of the adjustment factors $r_{inf}$ and
$r_{np}$ within this asymptotic regime, and show that the order is potentially
much larger than in the $p$-fixed asymptotic regime. A similar analysis is
performed in Tang and Reid (2020), studying the behaviour of $r_{inf}$ and
$r_{np}$, although the approach differs from the one taken here as it relies
on a direct expansion of the adjustment factor instead of the iterative
substitution argument employed. The results obtained from the two approaches
coincide in the two families considered.

In addition to the assumptions made in §3, we assume that
$j_{\psi\lambda}(\theta)=O_{p}(n^{1/2})$ for all
$\theta\in\\{\theta:\left\lVert\theta-\theta_{0}\right\rVert_{2}<\delta\\}$
for some $\delta>0$. This will be possible if the parametrization
$(\psi,\lambda)$ is orthogonal in the Cox and Reid (1987) sense. We also
assume that the $k$-th order derivatives of the constrained maximum likelihood
estimate are $O_{p}(1)$ for $k\geq 2$. We also place certain restrictions on
the sizes of the eigenvalues of the third likelihood derivative matrices,
these restrictions vary by the family we consider.

###  4.1 Linear Exponential Family

In the linear exponential family, there exists an orthogonal parametrization
$\hat{\lambda}_{\psi}=\hat{\lambda}$, so that under this parametrization
$d\hat{\lambda}_{\psi}^{k}/d\psi^{k}=0$ for for all integer-valued $k$ as the
observed and expected information coincide. For this sub-section assume that
the maximum singular value of $j_{\psi\lambda\lambda}(\hat{\theta})$ is
$O_{p}(n)$. First consider the leading terms which appear in $r_{np}$ and
$r_{inf}$ in (12) and (14). We make the strong assumption that the higher-
order terms are smaller. We find that, $\kappa_{3}(\hat{\psi})$ and
$\kappa_{4}(\hat{\psi})$, as defined in (3), satisfy

$\kappa_{3}(\hat{\psi})=O_{p}(n^{-1/2}),\quad\kappa_{4}(\hat{\psi})=O_{p}(n^{-1}),$

which implies that

$r_{inf}=O_{p}(n^{-1/2}).$

For $r_{np}$, we have

$\displaystyle\left|\gamma_{1}(\hat{\psi})\right|=\left|\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\psi})j_{\psi\lambda\lambda}(\hat{\psi})\right]\right|\leq\sigma_{\max}\\{j_{\psi\lambda\lambda}(\hat{\theta})\\}\operatorname{\text{tr}}[j_{\lambda\lambda}^{-1}(\hat{\theta})]=O_{p}(p),$

using von Neumann’s inequality and

$\displaystyle\left|\gamma_{2}(\hat{\psi})\right|$
$\displaystyle=\left|\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\psi})j_{\psi\lambda\lambda}(\hat{\psi})j_{\lambda\lambda}^{-1}(\hat{\psi})j_{\psi\lambda\lambda}(\hat{\psi})\right]+\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\psi})j_{\psi\psi\lambda\lambda}(\hat{\theta})\right]\right|$
$\displaystyle\leq\sigma^{2}_{\max}\\{j_{\psi\lambda\lambda}(\hat{\theta})\\}\operatorname{\text{tr}}[\\{j_{\lambda\lambda}^{-1}(\hat{\theta})\\}^{2}]+\left\lVert
j^{-1}_{\lambda\lambda}(\hat{\theta})\right\rVert_{F}\left\lVert
j_{\psi\psi\lambda\lambda}(\hat{\theta})\right\rVert_{F}$
$\displaystyle\leq\sigma^{2}_{\max}\\{j_{\psi\lambda\lambda}(\hat{\theta})\\}p\sigma_{\max}[\\{j_{\lambda\lambda}^{-1}(\hat{\theta})\\}^{2}]+p^{1/2}\sigma_{\max}\\{j_{\lambda\lambda}^{-1}(\hat{\theta})\\}\left\lVert
j_{\psi\psi\lambda\lambda}(\hat{\theta})\right\rVert_{F}$
$\displaystyle=O_{p}(p)+O_{p}(p^{3/2}).$

In the above we have used von Neumann’s trace inequality, and the fact that
the largest singular value is bounded by the Frobenius norm. This implies that

$r_{np}=O_{p}\left(pn^{-1/2}\right),$

from Theorem 3.2.

If $p=o(n^{1/2})$, these results are consistent with Tang and Reid (2020). The
above implies that $r$ coincides with $r^{\star}$ asymptotically in
distribution if $p=o(n^{1/2})$.

###  4.2 Location-Scale Family

We only consider the leading terms which appear in $r_{inf}$ and $r_{np}$ in
(15) and (13), making the strong assumption that the other terms are of
smaller order, as was done for the exponential family. Under the orthogonal
parametrization (Tang and Reid, 2020, Lemma 1):

$\left\lVert\frac{d\hat{\lambda}_{\psi}}{d\psi}|_{\psi=\hat{\psi}}\right\rVert_{2}=O_{p}(p^{1/2}n^{-1/2}),$

showing that $\kappa_{3}(\hat{\psi})=O_{p}(p/n^{1/2})$, and
$\kappa_{4}(\hat{\psi})=O_{p}(p^{2}/n)$, which further implies that

$r_{inf}=O_{p}\\{(\max(p/n^{1/2},p^{2}/n)\\},$

showing a dependence in $p$ not present in the linear exponential family. Next
we examine the size of $r_{np}$. As the derivatives of the constrained maximum
likelihood estimate are not 0, we make the assumptions that
$\max_{i=1,\cdots,p}\sigma_{\max}\\{j_{\theta_{i}\lambda\lambda}(\hat{\theta})\\}=O_{p}(n)$.
Then

$\displaystyle|\gamma_{1}(\hat{\psi})|$
$\displaystyle=\left|\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\theta})\frac{d}{d\psi}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right]\right|\leq\sigma_{\max}\left\\{\frac{d}{d\psi}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right\\}\operatorname{\text{tr}}[j_{\lambda\lambda}^{-1}(\hat{\theta})]$
$\displaystyle=\sigma_{\max}\left\\{j_{\psi\lambda\lambda}(\hat{\theta})+\sum_{i=1}^{p-1}\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\
j_{\lambda_{i}\lambda\lambda}(\hat{\theta})\right\\}\operatorname{\text{tr}}[j_{\lambda\lambda}^{-1}(\hat{\theta})]$
$\displaystyle\leq\left[\sigma_{\max}\left\\{j_{\psi\lambda\lambda}(\hat{\theta})\right\\}+\left\lVert\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right\rVert_{1}\
\sigma_{\max}\left\\{j_{\lambda_{i}\lambda\lambda}(\hat{\theta})\right\\}\right]\operatorname{\text{tr}}[j_{\lambda\lambda}^{-1}(\hat{\theta})]$
$\displaystyle\leq\left[\sigma_{\max}\left\\{j_{\psi\lambda\lambda}(\hat{\theta})\right\\}+p^{1/2}\left\lVert\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right\rVert_{2}\sigma_{\max}\left\\{j_{\lambda_{i}\lambda\lambda}(\hat{\theta})\right\\}\right]\operatorname{\text{tr}}[j_{\lambda\lambda}^{-1}(\hat{\theta})]$
$\displaystyle=\\{O_{p}(n)+O_{p}(pn^{1/2})\\}O_{p}(p/n)=O_{p}\left\\{\max(p,p^{2}/n^{1/2})\right\\}.$

Also,

$\displaystyle\left|\gamma_{2}(\hat{\psi})\right|$
$\displaystyle=\left|\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\theta})\frac{d}{d\psi}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\
j_{\lambda\lambda}^{-1}(\hat{\theta})\frac{d}{d\psi}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right]+\operatorname{\text{tr}}\left[j_{\lambda\lambda}^{-1}(\hat{\theta})\frac{d^{2}}{d\psi^{2}}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right]\right|$
$\displaystyle\leq\sigma^{2}_{\max}\left\\{\frac{d}{d\psi}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right\\}\operatorname{\text{tr}}[\\{j_{\lambda\lambda}^{-1}(\hat{\theta})\\}^{2}]+\operatorname{\text{tr}}[j^{-1}_{\lambda\lambda}(\hat{\theta})]\sigma_{\max}\left\\{\frac{d^{2}}{d\psi^{2}}j_{\lambda\lambda}(\hat{\theta}_{\psi})|_{\psi=\hat{\psi}}\right\\}$
$\displaystyle=O_{p}\\{\max(p^{2},p^{4}/n)\\}+O_{p}\\{\max(pn,p^{2}n^{1/2},p^{3})\\}.$

Detailed calculation of the maximum singular value of
$d^{2}j_{\lambda\lambda}(\hat{\theta}_{\psi})/d\psi^{2}|_{\psi=\hat{\psi}}$ is
deferred to §7.2 If $p=o(n^{1/2})$, then

$\displaystyle r_{np}=O_{p}(p/n^{1/2}),$

which agrees with the result in Tang and Reid (2020, §5.2).

##  5 $r$ as a series in $q$

In this section, we obtain an expansion of $r$ as a polynomial in $q$, which
is used to justify the normal approximation to the distribution of
$r^{\star}$, see Brazzale et al. (2007, §8). We do not give the exact forms of
the coefficients which appear in the polynomial expansion, as they are not as
simple as those in §3.

######  Theorem 5.1.

Under Assumptions 1–3, for the linear exponential and location-scale families,

$r=q+\frac{A}{n^{1/2}}q^{2}+\frac{B}{n}q^{3}+O_{p}\left(n^{-3/2}\right),$

where $A$ and $B$ are $O_{p}(1)$.

###### Proof.

Note that $\rho^{-1}=O_{p}(1)$ by Assumption 3. We prove the result in the
case for the linear exponential family; the proof for the location-scale
family is similar. We use capital letters to denote terms of order $O_{p}(1)$.
From (9),

$\displaystyle t$
$\displaystyle=r\Big{\\{}1-\frac{\kappa_{3}(\hat{\psi})}{6}t+\frac{\kappa_{4}(\hat{\psi})}{24}t^{2}+\frac{\kappa^{2}_{3}(\hat{\psi})}{24}t^{2}+O_{p}(n^{-3/2})\Big{\\}}$
$\displaystyle=r\left[1-\frac{\kappa_{3}(\hat{\psi})}{6\rho}q+\left\\{\frac{1}{24\rho^{2}}\kappa_{4}(\hat{\psi})+\frac{1}{24\rho^{2}}\kappa^{2}_{3}(\hat{\psi})\right\\}q^{2}+O_{p}(n^{-3/2})\right]$
$\displaystyle=r\left\\{1+\frac{C}{n^{1/2}}q+\frac{D}{n}q^{2}+O_{p}(n^{-3/2})\right\\}.$
(20)

We expand $|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|$,

$\displaystyle|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|$
$\displaystyle=|j_{\lambda\lambda}(\hat{\theta})|+(\psi_{0}-\hat{\psi})\frac{d|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|}{d\psi}|_{\psi=\hat{\psi}}+\frac{1}{2}(\psi_{0}-\hat{\psi})^{2}\frac{d|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|}{d\psi^{2}}|_{\psi=\hat{\psi}}+\cdots$
$\displaystyle=|j_{\lambda\lambda}(\hat{\theta})|\left\\{1+(\hat{\psi}-\psi_{0})\gamma_{1}(\hat{\psi})+(\hat{\psi}-\psi_{0})^{2}\gamma_{2}(\hat{\psi})+O_{p}(n^{-3/2})\right\\}$
$\displaystyle=|j_{\lambda\lambda}(\hat{\theta})|\left[1+\frac{\gamma_{1}(\hat{\psi})}{\\{\rho
j_{\text{p}}^{1/2}(\hat{\psi})\\}}q+\frac{\gamma_{2}(\hat{\psi})}{\rho^{2}j_{\text{p}}(\hat{\psi})}q^{2}+O_{p}(n^{-3/2})\right]$
$\displaystyle=|j_{\lambda\lambda}(\hat{\theta})|\left\\{1+\frac{E}{n^{1/2}}q+\frac{F}{n}q^{2}+O_{p}(n^{-3/2})\right\\}.$

Therefore,

$\displaystyle\rho$
$\displaystyle=\left\\{\frac{|j_{\lambda\lambda}(\hat{\theta})|}{|j_{\lambda\lambda}(\hat{\theta}_{\psi_{0}})|}\right\\}^{1/2}=\left[\frac{1}{1+Cq/n^{1/2}+Dq^{2}/n+O_{p}(n^{-3/2})}\right]^{1/2}$
$\displaystyle=1+\frac{G}{n^{1/2}}q+\frac{H}{n}q^{2}+O_{p}(n^{-3/2}).$ (21)

Note that $\gamma_{1}(\psi)=O_{p}(1)$ and $\gamma_{2}(\psi)=O_{p}(1)$ by Lemma
7.1 in §7.2. Combining (20) and (21),

$\displaystyle q$
$\displaystyle=t\rho=r\left\\{1+\frac{C}{n^{1/2}}q+\frac{D}{n}q^{2}+O_{p}(n^{-3/2})\right\\}\left\\{1+\frac{G}{n^{1/2}}q+\frac{H}{n}q^{2}+O_{p}(n^{-3/2})\right\\}.$
$\displaystyle r$
$\displaystyle=q\left\\{1+\frac{qC}{n^{1/2}}+\frac{q^{2}D}{n}+O_{p}(n^{-3/2})\right\\}^{-1}\left\\{1+\frac{G}{n^{1/2}}q+\frac{H}{n}q^{2}+O_{p}(n^{-3/2})\right\\}^{-1}$
$\displaystyle=q\left\\{1+\frac{A}{n^{1/2}}q+\frac{B}{n}q^{2}+O_{p}(n^{-3/2})\right\\}=q+\frac{A}{n^{1/2}}q^{2}+\frac{B}{n}q^{3}+O_{p}(n^{-3/2}),$

which shows the desired result. For the location scale family, we use the same
arguments and apply them to $s$ instead of $t$. ∎

##  6 Simulations

We perform numerical simulations to illustrate the results in §3; for the
linear exponential and location scale family

$\displaystyle r^{\star}=A^{\star}/n^{1/2}+(1+B^{\star}/n)r+O_{p}(n^{-3/2}),$
(22)

by Remark 1. To provide numerical evidence that (22) holds, note that,

$\displaystyle r^{\star}-A^{\star}/n^{1/2}-(1+B^{\star}/n)r=O_{p}(n^{-3/2}),$
(23)

and a sufficient condition for a random variable to be $O_{p}(n^{-3/2})$ is
for both its mean and standard deviation to be $O(n^{-3/2})$. We illustrate
the relationship in (23) graphically by plotting the value of the mean and
standard deviation of $r^{\star}-A^{\star}-(1+B^{\star})r$ as a function of
$n$; we expect the mean and standard deviation to roughly follow a linear
trend when plotted against $\log(n)$, with slope $-3/2$ or smaller.

###  6.1 Logistic Regression

We first consider an example based on logistic regression in which there are
$5$ covariates associated with each $y_{i}$, taken to be independent and
identically distributed standard normals. The true regression coefficients are
$\beta=(0,1,1,1,1)$, and the intercept is $\beta_{0}=1$. We are interested in
testing for $H_{0}:\beta_{1}=0$. The number of samples is
$n=150,300,600,1200,2400$; we obtain estimates of $A^{\star}$ and $B^{\star}$
in (23) by numerical differentiation of the profile log-likelihood and the
log-determinant of the information matrix. For each $n$, we simulate 2000
values of $r^{\star}-A^{\star}-(1+B^{\star})r$ and plot the 95% bootstrap
confidence intervals of the empirical mean and standard deviation from 1000
bootstrap simulations. In Figure, 2, the mean and standard deviation of
$r^{\star}-A^{\star}-(1+B^{\star})r$ are linear with slope $-3/2$, although
the fit for the standard deviation is a bit off.

(a) Empirical mean of $r^{\star}-A^{\star}-(1+B^{\star})r$.

(b) Empirical standard deviation of $r^{\star}-A^{\star}-(1+B^{\star})r$

Figure 2: Plots for logistic regression illustrating order of
$r^{\star}-A^{\star}-(1+B^{\star})r$. We see that both the mean and standard
deviation roughly follows a slop of $-3/2$ as a function of $n$ on the log
scale, suggesting that the difference is of order $O_{p}(n^{-3/2})$. The solid
line has slope $-3/2$ and the intercept is fitted.

###  6.2 Linear Regression with $t$ Errors

We consider an example based on a location-scale regression model where the
error follows a $t_{5}$-distribution. There are $5$ covariates $x_{i}$, which
are taken to be independent and identically distributed standard normal. We
are interested in testing for $H_{0}:\beta_{1}=0$ The true regression
coefficients are $\beta=(0,1,1,1,1)$, and the intercept is $\beta_{0}=1$. The
number of samples is $n=150,300,600,1200,2400$. For each $n$, we simulate 2000
values of $r^{\star}-A^{\star}-(1+B^{\star})r$ and plot the 95% bootstrap
confidence intervals of the empirical mean and standard deviation from 1000
bootstrap simulations. The results of the simulations are given in Figure 3.

(a) Empirical mean of $r^{\star}-A^{\star}-(1+B^{\star})r$ .

(b) Empirical standard deviation of $r^{\star}-A^{\star}-(1+B^{\star})r$

Figure 3: Plots for $t_{5}$ based regression illustrating the order of
$r^{\star}-A^{\star}-(1+B^{\star})r$. We see that both the mean and standard
deviation roughly follows a slope of $-3/2$ as a function of $n$ on the log
scale, suggesting that the difference is of order $O_{p}(n^{-3/2})$. The solid
line has slope $-3/2$ and the intercept is fitted.

##  7 Additional Proof Details

###  7.1 Order of $\gamma_{k}$ and $\kappa_{k}$

We first establish the order of $\gamma_{k}(\hat{\psi})$ and
$\zeta_{k}(\hat{\psi})$ for arbitrary integer $k$. All results are proved for
$\psi$ and $\lambda$ scalar parameters; generalization to vector $\lambda$ is
straightforward but notationally tedious.

######  Lemma 7.1.

For all integers $k$,

$\displaystyle\frac{\partial^{k}\hat{\lambda}_{\psi}}{\partial\psi^{k}}|_{\psi=\hat{\psi}}=O_{p}(1),\quad\frac{d^{k}}{d\psi^{k}}\tilde{\jmath}_{\lambda\psi}|_{\psi=\hat{\psi}}=O_{p}(n),\quad\frac{d^{k}}{d\psi^{k}}\tilde{\jmath}_{\lambda\lambda}|_{\psi=\hat{\psi}}=O_{p}(n),$
$\displaystyle\kappa_{k}(\hat{\psi})=O_{p}\\{n^{-(k-2)/2}\\},\quad\gamma_{k}(\hat{\psi})=O_{p}(1).$

###### Proof.

Differentiating the expression

$\displaystyle 0$ $\displaystyle=\tilde{l}_{\lambda},$

we obtain,

$\displaystyle\tilde{\jmath}_{\lambda\lambda}\frac{\partial\hat{\lambda}_{\psi}}{\partial\psi}$
$\displaystyle=-\tilde{\jmath}_{\psi\lambda},$ (24)

thus,

$\displaystyle\frac{\partial\hat{\lambda}_{\psi}}{\partial\psi}|_{\psi=\hat{\psi}}$
$\displaystyle=-\hat{\jmath}^{-1}_{\lambda\lambda}\hat{\jmath}_{\psi\lambda}=O_{p}(1),$
(25)

by Assumptions 1 and 3, we use the above to show

$\frac{d}{d\psi}\tilde{\jmath}_{\lambda\lambda}|_{\psi=\hat{\psi}}=\hat{\jmath}_{\psi\psi}+\frac{\partial\hat{\lambda}}{\partial\psi}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\psi\lambda}=O_{p}(n),$

and by the same argument show that
$d\tilde{\jmath}_{\lambda\psi}/d\psi|_{\psi=\hat{\psi}}=O_{p}(n)$. This proves
the first three claims hold for $k=1$.

Induction Step: We assume that for all $k_{1}\leq k$ the result holds, we need
to show that for $k+1$

$\frac{\partial^{k+1}\hat{\lambda}_{\psi}}{\partial\psi^{k+1}}|_{\psi=\hat{\psi}}=O_{p}(1),\quad\frac{d^{k+1}}{d\psi^{k+1}}\tilde{\jmath}_{\lambda\lambda}|_{\psi=\hat{\psi}}=O_{p}(n),\quad\frac{d^{k+1}}{d\psi^{k+1}}\tilde{\jmath}_{\lambda\psi}|_{\psi=\hat{\psi}}=O_{p}(n).$

First, we differentiate (25) $k$ times to obtain

$\frac{\partial^{k+1}\hat{\lambda}_{\psi}}{\partial\psi^{k+1}}=-\sum_{i=1}^{k}{k\choose
i}\frac{d^{i}(\tilde{\jmath}_{\lambda\lambda})^{-1}}{d\psi^{i}}\frac{d^{k-i}\tilde{\jmath}_{\psi\lambda}}{d\psi^{k-i}}.$

Using Faa di Bruno’s formula for the differentiation of a composition of
functions we obtain

$\displaystyle\frac{d^{i}}{d\psi^{i}}(\tilde{\jmath}_{\lambda\lambda})^{-1}=\sum_{k=1}^{i}(-1)^{k}k!\\{\tilde{\jmath}_{\lambda\lambda}\\}^{(-k-1)}B_{i,k}\left(\frac{d}{d\psi}\tilde{\jmath}_{\lambda\lambda},\cdots,\frac{d}{d\psi^{i-k+1}}\tilde{\jmath}_{\lambda\lambda}\right),$
(26)

where

$B_{i,k}(x_{1},\cdots,x_{i-k+1})=\sum\frac{i!}{j_{1}!j_{2}!\cdots
j_{i-k+1}!}\left(\frac{x_{1}}{1!}\right)^{j_{1}}\left(\frac{x_{2}}{2!}\right)^{j_{2}}\cdots\left(\frac{x_{i-k+1}}{(i-k+1)!}\right)^{j_{i-k+1}},$

and the summation in the above expression is taken over all sets of
${j_{1},\dots,j_{i-k+1}}$ such that,

$\displaystyle j_{1}+j_{2}+\cdot+j_{i-k+1}=k,\quad
j_{1}+2j_{2}+\cdots+(i-k+1)j_{i-k+1}=i.$

The polynomials $B_{i,k}$ are the partial Bell polynomials. From (26) we
deduce that
$d(\tilde{\jmath}_{\lambda\lambda})^{-1}/d\psi|_{\psi=\hat{\psi}}=O_{p}(n^{-1})$,
since the constraint $j_{1}+j_{2}+\cdot+j_{n-l+1}=k$ implies that

$B_{i,k}(d\tilde{\jmath}_{\lambda\lambda}/d\psi|_{\psi=\hat{\psi}},\cdots,d^{i-k+1}\tilde{\jmath}_{\lambda\lambda}/d\psi^{i-k+1}|_{\psi=\hat{\psi}})=O_{p}(n^{k}),$

and $(\hat{\jmath}_{\lambda\lambda})^{-k-1}=O_{p}(n^{-k-1})$, which implies
that every term in the summation is $O_{p}(n^{-1})$.

Thus, by the induction assumption, we have
$d^{k-i}\tilde{\jmath}_{\psi\lambda}/d\psi^{k-i}|_{\psi=\hat{\psi}}=O_{p}(n)$
for $i=1,\cdots,k-1$. Therefore we have the desired result for the constrained
derivative of the maximum likelihood estimate. Next we show that

$\frac{d^{k+1}}{d\psi^{k+1}}\tilde{\jmath}_{\lambda\lambda}|_{\psi=\hat{\psi}}=O_{p}(n),\quad\frac{d^{k+1}}{d\psi^{k+1}}\tilde{\jmath}_{\lambda\psi}|_{\psi=\hat{\psi}}=O_{p}(n).$

For this,

$\displaystyle\frac{d^{k+1}}{d\psi^{k+1}}\tilde{\jmath}_{\lambda\psi}=\sum_{i,j,l,m=1}^{k+1}a_{i,j,l,m}\frac{\partial^{i+j}\tilde{\jmath}_{\psi\lambda}(\psi,\hat{\lambda}_{\psi})}{\partial\psi^{i}\partial\lambda^{j}}\left(\frac{\partial^{l}\hat{\lambda}_{\psi}}{\partial\psi^{l}}\right)^{m}=O_{p}(n),$
(27)

which can be obtained through successive applications of the chain rule, some
of the coefficients $a_{i,j,l,m}$ may be $0$. The result follows from the fact
that all derivatives of the constrained maximum likelihood estimate are
$O_{p}(1)$ up to the $(k+1)$ order when evaluated at $\hat{\psi}$ and log-
likelihood derivatives are assumed to be $O_{p}(n)$ when evaluated at
$\hat{\theta}$. A similar argument can be made for the derivatives of
$\tilde{\jmath}_{\lambda\lambda}$.

Order of $\kappa_{k}(\psi)$

The total derivative of the profile log-likelihood function is a summation of
partial derivatives multiplied by the derivative of the constrained maximum
likelihood estimate, so the result is obtained from arguments used in (27).

Order of $\gamma_{k}(\psi)$

We have

$\displaystyle\gamma_{k}(\psi)=\sum_{i+j=k,\ j\geq
1}\operatorname{\text{tr}}\left[\frac{d^{i}}{d\psi^{i}}(\tilde{\jmath}_{\lambda\lambda})^{-1}\frac{d^{j}}{d\psi^{j}}\tilde{\jmath}_{\lambda\lambda}\right].$
(28)

Using
$d^{i}(\tilde{\jmath}_{\lambda\lambda})^{-1}/d\psi^{i}|_{\psi=\hat{\psi}}=O_{p}(n^{-i})$
from (26) and
$d^{j}\tilde{\jmath}_{\lambda\lambda}/d\psi^{j}|_{\psi=\hat{\psi}}=O_{p}(n)$
from (27) we conclude $\gamma_{k}(\hat{\psi})=O_{p}(1)$. ∎

###  7.2 Order of Maximum Singular Value in §4

We obtain the order of the maximum singular value of the second derivative of
the information matrix for the location-scale model in the high-dimensional
setting. We have

$\displaystyle\frac{d^{2}}{d\psi^{2}}\tilde{\jmath}_{\lambda\lambda}|_{\psi=\hat{\psi}}$
$\displaystyle=\hat{\jmath}_{\psi\psi\lambda\lambda}+2\sum_{i=1}^{p-1}\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\psi\lambda_{i}\lambda\lambda}+\sum_{i=1}^{p-1}\frac{\partial^{2}\hat{\lambda}_{\psi,i}}{\partial\psi^{2}}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\lambda_{i}\lambda\lambda}$
$\displaystyle+\sum_{i=1}^{p-1}\sum_{j=1}^{p-1}\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\frac{\partial\hat{\lambda}_{\psi,j}}{\partial\psi}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\lambda_{i}\lambda_{j}\lambda\lambda}.$

Now the maximal singular values of the matrices of interest are:

$\displaystyle\sigma_{\max}\\{\hat{\jmath}_{\psi\psi\lambda\lambda}\\}\leq\left\lVert\hat{\jmath}_{\psi\psi\lambda\lambda}\right\rVert_{F}=O_{p}(pn).$
$\displaystyle\sigma_{\max}\left\\{\sum_{i=1}^{p-1}\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\psi\lambda_{i}\lambda\lambda}\right\\}$
$\displaystyle\leq\sum_{i=1}^{p-1}\left|\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\sigma_{\max}\left\\{\hat{\jmath}_{\psi\lambda_{i}\lambda\lambda}\right\\}$
$\displaystyle\leq\sum_{i=1}^{p-1}\left|\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\left\lVert\hat{\jmath}_{\psi\lambda_{i}\lambda\lambda}\right\rVert_{F}$
$\displaystyle\leq
p^{1/2}\left\lVert\frac{\partial\hat{\lambda}_{\psi}}{\partial\psi}|_{\psi=\hat{\psi}}\right\rVert_{2}\max_{i=1,\dots,p}\left\lVert\hat{\jmath}_{\psi\lambda_{i}\lambda\lambda}\right\rVert_{F}=O_{p}(p^{2}n^{1/2}).$
$\displaystyle\sigma_{\max}\left\\{\sum_{i=1}^{p-1}\frac{\partial^{2}\hat{\lambda}_{\psi,i}}{\partial\psi^{2}}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\lambda_{1}\lambda\lambda}\right\\}$
$\displaystyle\leq\sum_{i=1}^{p-1}\left|\frac{\partial^{2}\hat{\lambda}_{\psi,i}}{\partial\psi^{2}}|_{\psi=\hat{\psi}}\right|\
\sigma_{\max}\left\\{\hat{\jmath}_{\lambda_{i}\lambda\lambda}\right\\}=O_{p}(pn)$
$\displaystyle\sigma_{\max}\left\\{\sum_{i=1}^{p-1}\sum_{j=1}^{p-1}\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\frac{\partial\hat{\lambda}_{\psi,j}}{\partial\psi}|_{\psi=\hat{\psi}}\
\hat{\jmath}_{\lambda_{i}\lambda_{j}\lambda\lambda}\right\\}$
$\displaystyle\leq\sum_{i=1}^{p-1}\sum_{j=1}^{p-1}\left|\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\left|\frac{\partial\hat{\lambda}_{\psi,j}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\
\sigma_{\max}\left\\{\hat{\jmath}_{\lambda_{i}\lambda_{j}\lambda\lambda}\right\\}$
$\displaystyle\leq\sum_{i=1}^{p-1}\sum_{j=1}^{p-1}\left|\frac{\partial\hat{\lambda}_{\psi,i}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\left|\frac{\partial\hat{\lambda}_{\psi,j}}{\partial\psi}|_{\psi=\hat{\psi}}\right|\left\lVert\hat{\jmath}_{\lambda_{i}\lambda_{j}\lambda\lambda}\right\rVert_{F}$
$\displaystyle=p\left\lVert\frac{\partial\hat{\lambda}_{\psi}}{\partial\psi}|_{\psi=\hat{\psi}}\right\rVert_{2}^{2}\max_{i,j=1,\dots,p}\left\lVert\hat{\jmath}_{\lambda_{i}\lambda_{j}\lambda\lambda}\right\rVert_{F}=O_{p}(p^{3}).$

Using the triangle inequality results in the rates obtained for
$\gamma_{1}(\hat{\psi})$ and $\gamma_{2}(\hat{\psi})$ given in §5.2.

## Acknowledgements

We thank Nicola Sartori, Michele Lambardi di San Miniato, Ioannis Kosmidis,
Heather Battey and Michaël Lalancette for helpful discussions. This research
was partially supported by the Natural Sciences and Engineering Research
Council of Canada and the Vector Institute.

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Player-Game Fixed Effects | $Yes$ | $Yes$ | $Yes$ | $Yes$
Move Observations | $106391$ | $106391$ | $106391$ | $106391$
Player-Game Observations | $3963$ | $3963$ | $3963$ | $3963$

_Note:_ OLS estimates. Evaluations of performance are based on the Stockfish 8
chess engine. The variable White player is a dummy variable indicating the
player that plays with white pieces. The variable Favorite is a dummy variable
indicating the player with the higher ELO number prior to the game. The
variable Num. previous moves is calculated as the number of previous moves per
player. Standard errors are clustered on the game level. ∗: $p<0.1$, ∗∗:
$p<0.05$, ∗∗∗: $p<0.01$.

Table A17: Behavioral Heterogeneity – Opponent’s Remaining Time and Time Spent

| Dependent Variable:
---|---
|  $\Delta^{E}$ (binary) |  $\Delta^{P}$ (binary) |  $\Delta^{N}$ (binary) |  $\Delta^{C}$ (categ.)
| (1) | (2) | (3) | (4)
_Current Position_ |  |  |  |
Better position (>0.5 pawnunits) | $0.0792^{***}$ | $0.0005$ | $0.0787^{***}$ | $-0.0782^{***}$
| $(0.0060)$ | $(0.0043)$ | $(0.0048)$ | $(0.0069)$
Worse position (<-0.5 pawnunits) | $0.0015$ | $0.0304^{***}$ | $-0.0289^{***}$ | $0.0594^{***}$
| $(0.0061)$ | $(0.0047)$ | $(0.0051)$ | $(0.0076)$
_Time Pressure_ |  |  |  |
Remaining time (hours) | $-0.0002$ | $0.0262^{***}$ | $-0.0263^{***}$ | $0.0525^{***}$
| $(0.0102)$ | $(0.0074)$ | $(0.0086)$ | $(0.0124)$
_Fatigue_ |  |  |  |
Num. previous moves | $-0.0017^{***}$ | $-0.0016^{***}$ | $-0.0001$ | $-0.0015^{***}$
| $(0.0002)$ | $(0.0002)$ | $(0.0002)$ | $(0.0002)$
_Complexity_ |  |  |  |
Seconds to reach fixed depth | $0.0028^{***}$ | $0.0012^{***}$ | $0.0015^{***}$ | $-0.0003^{**}$
| $(0.0001)$ | $(0.0001)$ | $(0.0001)$ | $(0.0001)$
_Time Opponent_ |  |  |  |
Remaining time (opp.) | $0.0193^{*}$ | $0.0164^{**}$ | $0.0029$ | $0.0135$
| $(0.0103)$ | $(0.0076)$ | $(0.0086)$ | $(0.0125)$
Time spent on move (opp.) | $-0.0003$ | $0.0002$ | $-0.0005^{*}$ | $0.0008$
| $(0.0004)$ | $(0.0003)$ | $(0.0003)$ | $(0.0005)$
Player-Game Fixed Effects | $Yes$ | $Yes$ | $Yes$ | $Yes$
Move Observations | $104409$ | $104409$ | $104409$ | $104409$
Player-Game Observations | $3954$ | $3954$ | $3954$ | $3954$

_Note:_ OLS estimates. Evaluations of performance are based on the Stockfish 8
chess engine. The variable Num. previous moves is calculated as the number of
previous moves per player. Standard errors are clustered on the game level. ∗:
$p<0.1$, ∗∗: $p<0.05$, ∗∗∗: $p<0.01$.

Table A18: Time Spent Per Move – Interactions

| Dependent Variable:
---|---
|  $\Delta^{E}$ (binary) |  $\Delta^{P}$ (binary) |  $\Delta^{N}$ (binary) |  $\Delta^{C}$ (categ.)
| (1) | (2) | (3) | (4)
Time spent on move (min.) | $0.0390^{***}$ | $0.0149^{***}$ | $0.0241^{***}$ | $-0.0092^{***}$
| $(0.0022)$ | $(0.0018)$ | $(0.0020)$ | $(0.0030)$
_Current Position_ |  |  |  |
Better position (>0.5 pawnunits) | $0.0780^{***}$ | $-0.0009$ | $0.0789^{***}$ | $-0.0798^{***}$
| $(0.0066)$ | $(0.0048)$ | $(0.0052)$ | $(0.0075)$
Worse position (<-0.5 pawnunits) | $0.0213^{***}$ | $0.0334^{***}$ | $-0.0122^{**}$ | $0.0456^{***}$
| $(0.0066)$ | $(0.0050)$ | $(0.0053)$ | $(0.0079)$
Better position $\times$ Time spent on move | $-0.0042^{***}$ | $-0.0009$ | $-0.0033^{***}$ | $0.0024$
| $(0.0011)$ | $(0.0009)$ | $(0.0009)$ | $(0.0014)$
Worse position $\times$ Time spent on move | $-0.0044^{***}$ | $-0.0004$ | $-0.0040^{***}$ | $0.0037^{**}$
| $(0.0013)$ | $(0.0010)$ | $(0.0011)$ | $(0.0017)$
_Time Pressure_ |  |  |  |
Remaining time (hours) | $-0.0410^{***}$ | $0.0207^{***}$ | $-0.0616^{***}$ | $0.0823^{***}$
| $(0.0082)$ | $(0.0056)$ | $(0.0064)$ | $(0.0088)$
Remaining time $\times$ Time spent on move | $-0.0179^{***}$ | $-0.0058^{***}$ | $-0.0121^{***}$ | $0.0063^{***}$
| $(0.0011)$ | $(0.0009)$ | $(0.0010)$ | $(0.0015)$
_Fatigue_ |  |  |  |
Num. previous moves | $-0.0014^{***}$ | $-0.0015^{***}$ | $0.0001$ | $-0.0016^{***}$
| $(0.0003)$ | $(0.0002)$ | $(0.0002)$ | $(0.0002)$
Num. previous moves $\times$ Time spent on move | $0.0001$ | $-0.0000$ | $0.0001^{**}$ | $-0.0001^{**}$
| $(0.0001)$ | $(0.0000)$ | $(0.0000)$ | $(0.0001)$
_Complexity_ |  |  |  |
Seconds to reach fixed depth | $0.0026^{***}$ | $0.0012^{***}$ | $0.0014^{***}$ | $-0.0001$
| $(0.0001)$ | $(0.0001)$ | $(0.0001)$ | $(0.0001)$
Seconds to reach fixed depth $\times$ Time spent on move | $-0.0000$ | $-0.0000^{**}$ | $0.0000$ | $-0.0000$
| $(0.0000)$ | $(0.0000)$ | $(0.0000)$ | $(0.0000)$
Player-Game Fixed Effects | $Yes$ | $Yes$ | $Yes$ | $Yes$
Move Observations | $106391$ | $106391$ | $106391$ | $106391$
Player-Game Observations | $3963$ | $3963$ | $3963$ | $3963$

_Note:_ OLS estimates. Evaluations of performance are based on the Stockfish 8
chess engine. The variable Num. previous moves is calculated as the number of
previous moves per player. Standard errors are clustered on the game level. ∗:
$p<0.1$, ∗∗: $p<0.05$, ∗∗∗: $p<0.01$.