# Estimation of the Directions for Unknown Parameters in Semiparametric Models

Jinyue Han1 Jun Wang2 Wei Gao1∗ Man-Lai Tang3
1\. School of Mathematics and Statistics, Northeat Normal University
2\. School of Mathematics and Statistics, Yunnan University
3\. Department of Mathematics, Brunel University

###### Abstract

Semiparametric models are useful in econometrics, social sciences and medicine
application. In this paper, a new estimator based on least square methods is
proposed to estimate the direction of unknown parameters in semi-parametric
models. The proposed estimator is consistent and has asymptotic distribution
under mild conditions without the knowledge of the form of link function.
Simulations show that the proposed estimator is significantly superior to
maximum score estimator given by Manski (1975) for binary response variables.
When the error term is long-tailed distributions or distribution with infinity
moments, the proposed estimator perform well. Its application is illustrated
with data of exporting participation of manufactures in Guangdong.

Key Words: Binary model, direction, least squares estimator, maximum score,
semi-parametric models, single index model.

##  1 Introduction

Considering the problem of estimating the regression model $E(Y\mid{\bf X})$,
where $Y$ denotes response variable, ${\bf X}$ is $p$-dimensional observable
covariates. Model $E(Y\mid{\bf X})$ has a significant applications in
economics, medicine and other fields, where the estimation of $E(Y\mid{\bf
X})$ is a key problem. While nonparametric methods are flexible, the price is
high: the estimation precision decreases rapidly as $p$ increasing and the
estimating results can be hard to interpret when dimension of covariate ${\bf
X}$ is greater. So to avoids the curse of dimensionality for nonparametric
model while still offering flexibility in the functional form of $E(Y\mid{\bf
X})$, a natural way is to assume that $E[Y\mid{\bf X}]$ is a semi-parametric
model. A popular semiparametric model is given by

$\displaystyle E[Y\mid{\bf X}]=E[Y\mid{\bf X}^{{}^{\prime}}{\bf\beta}]=g({\bf
X}^{{}^{\prime}}{\bf\beta}),$ (1)

where ${\bf\beta}\in R^{p}$ and $g:R\rightarrow R$ , an unknown link function.
When $g(\cdot)$ is known, the generalized moment estimation method could be
used to estimate unknown parameters ${\bf\beta}$. In this paper, we assume
that the link function $g(\cdot)$ is unknown and estimate the direction of
parameters ${\bf\beta}$.

A special case of models (1) is the binary response model, i.e.,

$\displaystyle Y=1\\{{\bf X}{{}^{\prime}}{\bf\beta}+\epsilon>0\\},$ (2)

where $\epsilon$ is unobservable random variable. The unknown parameters can
be estimated via maximum likelihood methods when the conditional distribution
of $\epsilon$ given ${\bf X}$ is known such as logistic Models or Probit
Models. When the conditional distribution of $\epsilon$ is unknow, Manski
(1975) proposed the maximum score estimator for binary response models with a
conditional median restriction, i.e., $\text{med}(\epsilon\mid{\bf X})=0$,
where $\text{med}(\epsilon\mid{\bf X})$ denotes the conditional median of
$\epsilon$ given ${\bf X}$. Base on results given by Manski (1975), Horowitz
(1992) proposed the smoothed maximum score estimator, proved it was
asymptotically normally distributed under certain assumptions and the
classical bootstrap was applied to make inference. Abrevaya and Huang (2005)
showed that the classical bootstrap was inconsistent for the maximum
estimator. Patra et al. (2018) proposed a model-based smooth bootstrap process
for making statistical inference on the maximum score estimator and proved its
consistency. Gao et al. (2022) proposed the two-stage maximum score estimator.

Another special form of model (1) is classical single index models

$\displaystyle Y=g({\bf X}^{{}^{\prime}}{\bf\beta})+\epsilon.$ (3)

There are mainly two kinds of techniques for estimation in single index
models. One is the M-estimation methods, which is based on kernel estimator
(Ichimura, 1993 ),regression splines (Park et al., 2020), local-linear
approximation (Zhou et al., 2019), penalized splines (Yu et al., 2002), and
smoothing splines (Kuchibhotla and Patra, 2020) to estimate $g(\cdot)$, and
minimize some appropriate criterion function such as quadratic loss (Yu and
Ruppert, 2002), quantile regression (Wu et al., 2010), estimating function
method (Cui et al., 2011), robust $L_{1}$ loss (Zou and Zhu, 2014), quasi-
likelihood (Wang and Guo, 2019), profiled likelihood (Patra et al., 2020) and
modal regression (Liu et al., 2013, Yang et al., 2020) to obtain ${\bf\beta}$.
The other kind is direct estimation methods such as maximum rank correlation
estimators (Han, 1987, Fan et al., 2020), average derivative estimators (
Chaudhuri et al., 1997, Hristache et al., 2001), dimension reduction
techniques (Li and Duan, 1989, Li and Racine, 2007 and Li, 2018), partial
least squares (Naik and Tsai, 2000, Zhou and He, 2008) and linearized maximum
rank correlation estimators (Shen et al., 2023). Recently Kuchibhotla et al.
(2021) introduced a convex and Lipschitz constrained least-square estimator
(CLSE) for both the parametric and the nonparametric components given
independent and identically distributed observations.

Model (1) contains the linear model where a least squares method is proposed
to estimate the direction of the parameter ${\bf\beta}$ in semiparametric
models for the response variable being ontinuous or discrete. Our proposed
method is computationally simple and theoretical reliable. Simulation results
shows that the proposed estimator is significantly superior to maximum score
estimation for response variables being discrete, and is comparable with
linearized maximum rank correlation(LMRC) and the maximum likelihood
estimation methods for Probit Models. When the distribution of error term is
long-tailed (i.e., Student t) and distributions with no existing moments
(i.e., Cauchy), the proposed estimator and LMRC estimator perform better than
standard estimator. When the dimension of covariates is relatively high, the
proposed method is still feasible. The proposed estimation is superior to the
linearized maximum rank correlation estimation with nonlinear models.

This paper is organized as follows: estimators for the direction of the
parameter ${\bf\beta}$ are considered in Section 2 with their theoretical
properties presented. A simulation study is conducted, and the results will be
reported in Section 3. In Section 4, we apply our methodology to a real data
set that studies the influence of series exporting determined variables on the
export-market participation of specialized and transport facility manufactures
in the province of Guangdong, China in 2006. Conclusions and discussions will
be discussed in Section 5. Proofs of lemma and theorems will be presented in
the Appendix

##  2 The Proposed estimator

Consider the samples $(Y_{i},{\bf X}_{i})(i=1,\cdots,n)$ are observed from the
following models

$\displaystyle E[Y_{i}\mid{\bf X}_{i}]=E[Y_{i}\mid{\bf
X}_{i}^{T}{\bf\beta}]=g({\bf X}_{i}^{T}{\bf\beta}),$ (4)

where $g(\cdot)$ is an unknown and monotonically increasing function and
${\bf\beta}$ is a parameter in $R^{p}$..

In this paper, we mainly consider the estimation of the direction of
parameters ${\bf\beta}\ ({\bf\beta}\neq 0)$, in case of ${\bf\beta}=0$ which
is simple. Because models (4) includes the linear model, so the estimator for
the direction of ${\bf\beta}$ in model (4) is obtained via the least squares
method

$D(\hat{{\bf\beta}})=\frac{\hat{\Sigma}_{X}^{-1}\times\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-{\bar{\bf
X}})}{\left|\left|\hat{\Sigma}_{X}^{-1}\times\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-{\bar{\bf X}})\right|\right|}$ (5)

where

${\hat{\Sigma}}_{X}=\frac{1}{n}\sum\limits_{i=1}^{n}({{\bf X}}_{i}-\bar{{\bf
X}})({{\bf X}}_{i}-\bar{{\bf X}})^{{}^{\prime}},$

and $\bar{{\bf X}}$ is sample mean value of ${\bf X}$. The least squares
method can be used to estimate the direction of unknown parameters
${\bf\beta}$ in semi-parametric model for the response variable being
continuous or discrete.

Under some regular conditions, the estimator of direction of ${\bf\beta}$ is
consistent. The proof is given in the Appendix.

Theorem 1.  When ${\bf X}$ is distributed by the elliptical distributions with
mean $\bf\mu$ and covariance $Var({\bf X})=\Sigma$( positive definite) for
Models (4) and $E(g^{2}({{\bf X}}^{{}^{\prime}}{\bf\beta}))<+\infty$, then

$D(\hat{{\bf\beta}})\xrightarrow{P}\frac{{\bf\beta}}{\|{\bf\beta}\|}.$

When the mean of ${\bf X}$ is $E{\bf X}=\bf 0$, the direction of ${\bf\beta}$
can be given by

$D^{*}(\hat{{\bf\beta}})=\frac{\hat{\Sigma}_{X}^{-1}\times\sum\limits_{i=1}^{n}Y_{i}{{\bf
X}}_{i}}{\left|\left|\hat{\Sigma}_{X}^{-1}\times\sum\limits_{i=1}^{n}Y_{i}{{\bf
X}}_{i}\right|\right|},$ (6)

and the following theorem will give the asymptotic distribution of
$D^{*}(\hat{\bf\beta})$.

Theorem 2.  When ${\bf X}$ is distributed by the elliptical distributions with
mean $\bf 0$ and covariance $Var({\bf X})=\Sigma$( positive definite) for
Models (4), $E(g^{2}({{\bf X}}^{{}^{\prime}}{\bf\beta}))<+\infty$ and
$E(X_{j}^{4})<+\infty$ for $j=1,\cdots,p$, then

$\sqrt{n}\left(D^{*}(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)\xrightarrow{d}\frac{1}{\|{\bf\beta}\|}\Bigg{\\{}\lambda^{-1}\Sigma^{-1}{\bf
U}-\Sigma^{-1}{\bf
V}-\frac{\left[\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
U}-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
V}\right]}{\|{\bf\beta}\|^{2}}{\bf\beta}\Bigg{\\}}$

where

$\lambda=E(Y{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf X})/\|{\bf\beta}\|^{2}$

and

$\begin{pmatrix}\bf U\\\ \bf V\end{pmatrix}\sim N\left(\begin{pmatrix}\bf 0\\\
\bf 0\\\ \end{pmatrix},\Omega\right),\;\;\;\Omega=Var\begin{pmatrix}Y{\bf
X}\\\ {\bf X}{\bf X}^{{}^{\prime}}{\bf\beta}\end{pmatrix}.$

Remark. $D^{*}(\hat{{\bf\beta}})-{\bf\beta}$ is close to 0 with the increase
of samples, therefor, $D^{*}(\hat{{\bf\beta}})$ is in the tangent plane of
${\bf\beta}$. Therefor,
$\frac{1}{\|{\bf\beta}\|}\Bigg{\\{}\lambda^{-1}\Sigma^{-1}{\bf
U}-\Sigma^{-1}{\bf
V}-\frac{\left[\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
U}-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
V}\right]}{\|{\bf\beta}\|^{2}}{\bf\beta}\Bigg{\\}}$ is a degenerate normal
distribution in $R^{p-1}$.

Similar to Theorem 2, $D(\hat{{\bf\beta}})$ has the following results.

Corollary 1.  When ${\bf X}$ is distributed by the elliptical distributions
with mean $\bf\mu$ and covariance $Var({\bf X})=\Sigma$( positive definite)
for Models (4), $E(g^{2}({{\bf X}}^{{}^{\prime}}{\bf\beta}))<+\infty$ and
$E(X_{j}^{4})<+\infty$ for $j=1,\cdots,p$, then

$\displaystyle\sqrt{n}\left(D(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$
$\displaystyle\xrightarrow{d}$
$\displaystyle\frac{1}{\|{\bf\beta}\|}\Bigg{\\{}\lambda^{-1}\Sigma^{-1}{\bf
U}-\Sigma^{-1}{\bf V}-\lambda^{-1}\gamma\Sigma^{-1}{\bf Z}$
$\displaystyle-\frac{\left[\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
U}-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{{\bf
V}-\lambda^{-1}\gamma{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
Z}}\right]}{\|{\bf\beta}\|^{2}}{\bf\beta}\Bigg{\\}}$

where

$\lambda=E(Y{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
X})/\|{\bf\beta}\|^{2},\;\gamma=EY$

and

$\begin{pmatrix}\bf Z\\\ \bf U\\\ \bf V\end{pmatrix}\sim
N\left(\begin{pmatrix}\bf 0\\\ \bf 0\\\ \bf 0\\\
\end{pmatrix},\Xi\right),\;\;\;\Xi=Var\begin{pmatrix}{\bf X}\\\ Y{\bf X}\\\
{\bf X}{\bf X}^{{}^{\prime}}{\bf\beta}\end{pmatrix}.$

We consider the following Wald statistic for testing the direction for
$H_{0}:\beta=\beta_{0}$ based on Corollary 1 or Theorem 2,

$W=\sqrt{n}\left(D(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)^{{}^{\prime}}\Sigma_{{\bf\beta}}^{+}\sqrt{n}\left(D(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right),$

where $\Sigma_{{\bf\beta}}$ is the covariance of
$\sqrt{n}\left(D(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$,
and $A^{+}$ denotes the Moore-Penrose inverse of matrix $A$. Since
$\|{\bf\beta}\|$ is involved in covariance matrix $\Sigma_{{\bf\beta}}$, so we
impose constraints $\|{\bf\beta}\|=1$ for estimating the covariance matrix
$\Sigma_{{\bf\beta}}$. Under the $H_{0}$ and $\|{\bf\beta}_{0}\|=1$, the Wald
statistic is

$W^{*}=n\left(D(\hat{{\bf\beta}})-{\bf\beta}_{0}\right)^{{}^{\prime}}\Sigma_{D(\hat{{\bf\beta}})}^{+}\left(D(\hat{{\bf\beta}})-{\bf\beta}_{0}\right)\sim\chi^{2}(m),$

where the degree of freedom $m$ is the rank of matrix
$\Sigma_{D(\hat{{\bf\beta}})}$ which is a estimator of $\Sigma_{{\bf\beta}}$,
and $m=p-1$. Therefore, the acceptance region under the confidence level
$1-\alpha$ is

$p\Big{(}W^{*}\leq\chi^{2}_{1-\alpha}(m)\Big{)}=1-\alpha.$

##  3 Simulation studies

In this section, we conduct several simulation studies to evaluate the finite
sample performance of the proposed parameter direction estimator
$D(\hat{{\bf\beta}})$ in section 2.

In the first simulation study, we assume that the response variable $Y$ is
discrete, and the dimension of $X$ is $p=3$, i.e., $X\sim
N(0,\Sigma=(\sigma_{ij}))$, $\sigma_{ij}=\rho^{|i-j|}$, $i,j=1,2,3$, and
${\bf\beta}=({\bf\beta}_{1},{\bf\beta}_{2},{\bf\beta}_{3})^{{}^{\prime}}=(s_{1},s_{2},s_{3})^{{}^{\prime}}/\|\bf
s\|$, $s_{i}\sim U(-1,1)$, $i=1,2,3$. We mainly consider that the following
data generation scenarios in models (4).

  * •

Case I. $Y=1\\{{\bf X}^{{}^{\prime}}{\bf\beta}+\epsilon>0\\}$, where
$\epsilon\sim N(0,1)$.

  * •

Case II. $Y=1\\{{\bf X}^{{}^{\prime}}{\bf\beta}+\epsilon>0\\}$, where
$\epsilon\sim t(1)$.

  * •

Case III. $Y=1\\{{\bf X}^{{}^{\prime}}{\bf\beta}+\epsilon>0\\}$, where
$\epsilon\sim 0.4*N(-3,1)+0.6*N(2,2)$.

When the distributions of error term $\epsilon$ is unknown with a conditional
median restriction, Manski (1975) proposed the maximum score (MS) estimator
for the parameter ${\bf\beta}$. Shen et al. (2023) proposed the linearized
maximum rank correlation (LMRC) estimator. Here, we compared the proposed
estimator with MS estimators, LMRC estimator and the standard method (probit
regression) estimator. To measure the distance between the real direction
${\bf\beta}$ and the estimated direction $\hat{{\bf\beta}}$ for parameters
$\beta$, we use the cosine value of the angle $\theta$ between the two
directions (Bandiera et al., 2007, Rong et al., 2021), namely

$\cos(\theta)=\frac{{\bf\beta}*D(\hat{{\bf\beta}})}{\|{\bf\beta}\|*\|D(\hat{{\bf\beta}})\|}.$

We obtain that ${\bf\cos}$ value and standard error (SE) of estimators of
${\bf\beta}$ for different distributions of $\epsilon$ with sample sizes equal
to 100, 300 and 500 for Case I-III. The results based on 100 repetitions and
different $\rho$ are reported in Table (1).

Table 1: The SE and cos value of proposed estimators with ${\bf\beta}$ with sample size n=100, 300, 500 based on 100 repetitions for Case I-III.  | n | $\rho$ | New | MS | LMRC |  | Standard
---|---|---|---|---|---|---|---
|  |  | cos | SE | cos | SE | cos | SE | cos | SE
| 100 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9612
---
0.9705
0.9805
0.9713
0.9733
| 0.0372
---
0.0307
0.0173
0.0339
0.0276
| 0.7326
---
0.7545
0.7879
0.7546
0.7402
| 0.3018
---
0.2930
0.2580
0.2747
0.3136
| 0.9602
---
0.9700
0.9807
0.9711
0.9730
| 0.0382
---
0.0314
0.0175
0.0342
0.0276
| 0.9629
---
0.9720
0.9810
0.9713
0.9751
| 0.0369
---
0.0292
0.0186
0.0341
0.0255
Case I | 300 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9877
---
0.9918
0.9922
0.9929
0.9883
| 0.0148
---
0.0087
0.0069
0.0073
0.0141
| 0.7819
---
0.7799
0.7966
0.7821
0.7528
| 0.2633
---
0.2593
0.2457
0.2680
0.2918
| 0.9877
---
0.9918
0.9922
0.9929
0.9883
| 0.0149
---
0.0086
0.0070
0.0073
0.0142
| 0.9886
---
0.9922
0.9922
0.9928
0.9880
| 0.0127
---
0.0084
0.0071
0.0074
0.0148
| 500 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9932
---
0.9963
0.9958
0.9957
0.9930
| 0.0073
---
0.0035
0.0036
0.0036
0.0082
| 0.7902
---
0.8358
0.8464
0.8848
0.8229
| 0.3238
---
0.2370
0.2280
0.2192
0.2367
| 0.9931
---
0.9962
0.9958
0.9957
0.9930
| 0.0073
---
0.0035
0.0036
0.0036
0.0082
| 0.9932
---
0.9961
0.9960
0.9958
0.9932
| 0.0072
---
0.0036
0.0035
0.0038
0.0077
| 100 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9142
---
0.9449
0.9478
0.9420
0.9186
| 0.1054
---
0.0780
0.0638
0.0708
0.0843
| 0.6699
---
0.7181
0.7513
0.7228
0.6363
| 0.3730
---
0.3208
0.2626
0.2859
0.3763
| 0.9144
---
0.9449
0.9472
0.9427
0.9194
| 0.1048
---
0.0773
0.0648
0.0699
0.0832
| 0.9133
---
0.9449
0.9475
0.9419
0.9183
| 0.1059
---
0.0785
0.0632
0.0702
0.0857
Case II | 300 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9753
---
0.9832
0.9559
0.9813
0.9708
| 0.0319
---
0.0172
0.0467
0.0228
0.0353
| 0.7289
---
0.7902
0.7826
0.7919
0.7514
| 0.3403
---
0.2512
0.2481
0.2321
0.2748
| 0.9752
---
0.9831
0.9558
0.9814
0.9711
| 0.0323
---
0.0174
0.0468
0.0228
0.0351
| 0.9749
---
0.9829
0.9559
0.9811
0.9709
| 0.0328
---
0.0178
0.0459
0.0229
0.0353
| 500 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9862
---
0.9909
0.9899
0.9902
0.9851
| 0.0172
---
0.0093
0.0086
0.0119
0.0181
| 0.7942
---
0.8342
0.8421
0.8315
0.7999
| 0.2779
---
0.2266
0.2118
0.2081
0.2247
| 0.9861
---
0.9909
0.9899
0.9903
0.9851
| 0.0174
---
0.0093
0.0086
0.0119
0.0181
| 0.9860
---
0.9907
0.9895
0.9900
0.9846
| 0.0169
---
0.0094
0.0091
0.0117
0.0178
| 100 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9454
---
0.9664
0.9704
0.9712
0.9518
| 0.0600
---
0.0365
0.0342
0.0268
0.0508
| 0.6863
---
0.7421
0.7461
0.7338
0.6830
| 0.3439
---
0.2865
0.2860
0.2931
0.3096
| 0.9441
---
0.9657
0.9708
0.9705
0.9511
| 0.0649
---
0.0396
0.0343
0.0276
0.0501
| 0.9463
---
0.9672
0.9713
0.9717
0.9533
| 0.0630
---
0.0363
0.0349
0.0280
0.0493
Case III | 300 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9824
---
0.9875
0.9886
0.9886
0.9825
| 0.0185
---
0.0129
0.0136
0.0119
0.0206
| 0.7791
---
0.8214
0.8241
0.8193
0.7812
| 0.2321
---
0.2012
0.2285
0.2012
0.2299
| 0.9823
---
0.9875
0.9886
0.9887
0.9826
| 0.0186
---
0.0129
0.0136
0.0118
0.0206
| 0.9829
---
0.9879
0.9886
0.9887
0.9826
| 0.0185
---
0.0126
0.0142
0.0123
0.0213
| 500 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9908
---
0.9942
0.9941
0.9940
0.9906
| 0.0098
---
0.0051
0.0065
0.0053
0.0130
| 0.8249
---
0.8329
0.8101
0.7729
0.7424
| 0.2252
---
0.2004
0.2230
0.2847
0.3193
| 0.9908
---
0.9942
0.9941
0.9940
0.9906
| 0.0098
---
0.0051
0.0065
0.0053
0.0131
| 0.9910
---
0.9944
0.9942
0.9942
0.9907
| 0.0092
---
0.0047
0.0062
0.0051
0.0123

A main drawback of the maximum score estimator is its computational
difficulty, because the objective function of optimization is non-convex and
non-smooth, which makes it a difficult task to find the global optimal
solution. In addition, the calculation difficulty is more serious with the
dimension of $X$ being larger (Khan et al., 2021). To consider the influence
of the increase in the dimension of covariates on the estimation results, we
consider the following data generation scenarios, i.e., $X\sim
N(0,\Sigma=(\sigma_{ij}))$, $\sigma_{ij}=\rho^{|i-j|}$, $i,j=1,\cdots,p$.
$Y=1\\{X^{T}{\bf\beta}+\epsilon>0\\}$, where $p=10,15$, $\epsilon\sim t(1)$
and
${\bf\beta}=({\bf\beta}_{1},\cdots,{\bf\beta}_{p})^{{}^{\prime}}=(s_{1},\cdots,s_{p})^{{}^{\prime}}/\|\bf
s\|$, $s_{i}\sim U(-1,1)$, $i=1,\cdots,p$. Compared the proposed method with
the LMRC method and standard method with the dimension of $X$ is relatively
large. Simulation results are listed in Table (2) with n=500 and 100
repetitions for two different $p$ values. Results show that our proposed
method is still feasible when the dimension of X is relatively high, e.g.,
$p=10,15$. The proposed estimator is comparable with LMRC estimator, and
performs better than standard estimator.

Table 2: The SE and cos value of proposed estimators with ${\bf\beta}$ with sample size n=500 based on 100 repetitions for different dimensions of $X$ .  Dimension | $\rho$ | New | LMRC | Standard
---|---|---|---|---
|  | cos | SE | cos | SE | cos | SE
10 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.9265
---
0.9490
0.9588
0.9506
0.9150
| 0.0397
---
0.0275
0.0195
0.0284
0.0504
| 0.9263
---
0.9489
0.9589
0.9506
0.9147
| 0.0397
---
0.0277
0.0194
0.0283
0.0507
| 0.9259
---
0.9483
0.9581
0.9500
0.9141
| 0.0395
---
0.0280
0.0200
0.0295
0.0514
15 |  | -0.6
---
-0.3
0
0.3
0.6
| 0.8950
---
0.9283
0.9351
0.9264
0.8883
| 0.0516
---
0.0327
0.0215
0.0332
0.0438
| 0.8951
---
0.9283
0.9350
0.9263
0.8882
| 0.0518
---
0.0326
0.0216
0.0332
0.0439
| 0.8942
---
0.9273
0.9348
0.9255
0.8858
| 0.0525
---
0.0330
0.0218
0.0334
0.0455

Finally, we assume that the response variable $Y$ is continuous and $X\sim
N(0,\Sigma=(\sigma_{ij}))$, $\sigma_{ij}=\rho^{|i-j|}$, $i,j=1,2,3$,
${\bf\beta}=({\bf\beta}_{1},{\bf\beta}_{2},{\bf\beta}_{3})^{{}^{\prime}}=(s_{1},s_{2},s_{3})^{{}^{\prime}}/\|\bf
s\|$, $s_{i}\sim U(-1,1)$, $i=1,2,3$, $\epsilon\sim N(0,1)$. and following
data generation scenarios are considered.

  * •

Case 1. $Y={\bf X}^{{}^{\prime}}{\bf\beta}+\epsilon$.

  * •

Case 2. $Y=\Phi({\bf X}^{{}^{\prime}}{\bf\beta})+\epsilon$.

  * •

Case 3. $Y=\log(1+\exp({\bf X}^{{}^{\prime}}{\bf\beta}))+\epsilon$.

  * •

Case 4. $Y=\frac{1}{1+\exp(-{\bf X}^{{}^{\prime}}{\bf\beta})}+\epsilon$.

We compare the proposed estimator with LMRC estimator. The ${\bf\cos}$ value
and SE of proposed estimators with ${\bf\beta}$ for different values of $\rho$
with different distributions of $\epsilon$ and simple size n=100, 300, 500
based on 100 repetitions are reported in Table (3). From Table (3), we observe
that as the sample size increases, all SE for ${\bf\cos}$ decrease and the
values of ${\bf cos}$ is closer to $1$. Although the proposed estimator is
slightly inferior to LMRC estimator when the model is lineal model, the
proposed estimator performs better than LMRC estimator when the model is
nonlinear.

Table 3: The SE and cos value of proposed estimators with ${\bf\beta}$ with sample size n=100, 300, 500 based on 100 repetitions for Case 1-4.  n | $\rho$ |  | Case 1 | Case 2 | Case 3 | Case 4
---|---|---|---|---|---|---
|  |  | cos | SE | cos | SE | cos | SE | cos | SE
| -0.3 |  | New
---
LMRC
| 0.9893
---
0.9880
| 0.0109
---
0.0117
| 0.8640
---
0.8621
| 0.1500
---
0.1537
| 0.9558
---
0.9513
| 0.0492
---
0.0507
| 0.7680
---
0.7633
| 0.2516
---
0.2666
100 | 0 |  | New
---
LMRC
| 0.9902
---
0.9896
| 0.0110
---
0.0108
| 0.8841
---
0.8813
| 0.1186
---
0.1298
| 0.9590
---
0.9565
| 0.0465
---
0.0467
| 0.7844
---
0.7858
| 0.2522
---
0.2562
| 0.3 |  | New
---
LMRC
| 0.9899
---
0.9882
| 0.0097
---
0.0126
| 0.8855
---
0.8790
| 0.1264
---
0.1479
| 0.9607
---
0.9570
| 0.0432
---
0.0494
| 0.7913
---
0.7834
| 0.2719
---
0.2865
| -0.3 |  | New
---
LMRC
| 0.9960
---
0.9953
| 0.0046
---
0.0052
| 0.9487
---
0.9450
| 0.0677
---
0.0670
| 0.9828
---
0.9808
| 0.0216
---
0.0242
| 0.9046
---
0.8939
| 0.1164
---
0.1311
300 | 0 |  | New
---
LMRC
| 0.9967
---
0.9964
| 0.0031
---
0.0034
| 0.9595
---
0.9586
| 0.0375
---
0.0412
| 0.9867
---
0.9860
| 0.0127
---
0.0133
| 0.9252
---
0.9245
| 0.0704
---
0.0777
| 0.3 |  | New
---
LMRC
| 0.9965
---
0.9960
| 0.0038
---
0.0044
| 0.9561
---
0.9545
| 0.0491
---
0.0509
| 0.9855
---
0.9838
| 0.0169
---
0.0187
| 0.9198
---
0.9152
| 0.0875
---
0.0964
| -0.3 |  | New
---
LMRC
| 0.9979
---
0.9978
| 0.0020
---
0.0023
| 0.9739
---
0.9733
| 0.0261
---
0.0260
| 0.9912
---
0.9907
| 0.0085
---
0.0091
| 0.9512
---
0.9500
| 0.0514
---
0.0487
500 | 0 |  | New
---
LMRC
| 0.9981
---
0.9980
| 0.0017
---
0.0020
| 0.9775
---
0.9763
| 0.0198
---
0.0209
| 0.9923
---
0.9917
| 0.0070
---
0.0076
| 0.9589
---
0.9559
| 0.0357
---
0.0384
| 0.3 |  | New
---
LMRC
| 0.9979
---
0.9977
| 0.0019
---
0.0021
| 0.9726
---
0.9718
| 0.0284
---
0.0280
| 0.9914
---
0.9905
| 0.0088
---
0.0092
| 0.9488
---
0.9468
| 0.0553
---
0.0532

##  4 Real data

In this section, we apply our proposed methodology to a real dataset that
studies the influence of series exporting determined variables on the export-
market participation of specialized and transport facility manufactures in the
province of Guangdong, China in 2006 (Baltagi et al., 2022). The data is
available on the National Bureau of Statistics of China (NBS).

In the subsequent analyses, the variables we mainly consider include expd-ford
(=1 if the company is an exporter; 0 otherwise), lemp (log firm sizes), lprod
(log output per worker), lcapint (capital divided by total sales), intastr
(intangible assets over total assets), cmp  (log sales over operating
profits), ${\bf cmp^{2}}$ (Square of cmp), ltastx (Fixed export costs), and
sez (=1 if the firm is located in the Special Economic Zone; =0 otherwise).
Finally, we obtained that a total of 1614 companies with total annual sales of
at least 5 mn. RMB (about 700,000 US dollars) in 2006. As mentioned in Khan et
al. (2021), the maximum score method was extremely difficult to calculate for
the dimension of the covariate being large. There are a total of 8 covariates
in this analysis, therefore the maximum score estimator is ignored here.

To illustrate our methodologies, we let the response variable (i.e., Y) be
expd-ford and $X$ includes the remaining variables. The results estimated by
the proposed method, LMRC method and standard method are presented in Table
(4).

Table 4: The influence of series exporting determined variables on the export-market participation of specialized and transport facility manufactures based on various methods. Parameter | Proposed method | Probit | LMRC
---|---|---|---
| lemp
---
lprod
lcapint
intastr
Cmp
$\mbox{Cmp}^{2}$
ltastx
sez
| 0.7621(0.0911)
---
0.0236(0.0586)
0.1463(0.0592)
-0.0844(0.0456)
-0.1228(0.2357)
0.0846(0.2393)
-0.1766(0.0552)
0.5802(0.1081)
| 0.8785(0.1210)
---
0.0231(0.0677)
0.1796(0.0703)
-0.0960(0.0524)
-0.2154(0.3172)
0.1478(0.3494)
-0.1981(0.0641)
0.2806(0.0623)
| 0.8934(0.0885)
---
0.0187(0.0668)
0.1957(0.0671)
-0.1032(0.0514)
-0.2823(0.2461)
0.1878(0.2540)
-0.1925(0.0601)
0.0228(0.0364)

From Table (4), we can explain some economic hypotheses. Obviously, a simple
test $T=\hat{{\bf\beta}}_{i}/\hat{\sigma}_{i},i,\cdots,8$ suggests that lemp,
lcapint, ltastx, sez have a significant impact on response variable at a
significance level of 0.05 for the proposed method. Large size of firms where
fixed costs are more important (through a higher capital intensity) tend to be
more likely to export. Higher the Fixed export costs is, lower the
profitability of exporting will be, which explain the decrease influence of
export. Compared with the LMRC, the proposed method finds that sez is a new
influencing factors on response variables. Since the distance between firms
and Special Economic Zone should also affect the exporting, the binary
variable do have a significant influence, hence our proposed method is more
reasonable.

##  5 Conclusion and discussion

In this paper, we consider that the estimation of unknown parameter direction
in semi-parametric models for response variables, which can be continuous or
discrete. The least square method are proposed to estimate the direction of
unknown parameters in semi-parametric models. The proposed estimator is
computationally simple and has a closed-form expression. It is proved that the
proposed estimator is consistent and asymptotically normal. The proposed
estimation is significantly superior to the maximum score estimation for
binary response variables and comparable with the linearized maximum rank
correlation(LMRC) and the maximum likelihood estimation for Probit models.
When the distribution of error term is long-tailed (i.e., Student t) and
distributions with infinity moments (i.e., Cauchy), the proposed estimator
perform well. The proposed estimation is superior to the linearized maximum
rank correlation estimation for continuous response variable with nonlinear
models. Furthermore if one is interested in the estimation of the link
function $g(\cdot)$, it can be directly estimated by non-parametric methods
with $E[Y\mid X^{T}\hat{{\bf\beta}}]$.

In this paper, we mainly consider that the link function $g(\cdot)$ is
monotonically increasing. The estimator of unknown parameter direction can be
obtained by a similar method when the link function is monotonically
decreasing, the direction estimation of unknown parameters is opposite to the
true direction of parameters. In practice, the observable covariates are
always high-dimensional for various reasons, we extend the proposed method to
handle parameter direction estimation for model (4) with high-dimensional
covariates. We can estimate the direction of parameters ${\bf\beta}$ via
minimize the sum of loss function and penalty function. This research problem
will be considered in the future.

## Appendix

The following similar result is also given by Brillinger(1983), Li and
Duan(1989) and Li(2018,Theorem 8.3) and here another way is given.

Lemma A. When ${\bf X}$ is distributed by the elliptical distributions with
mean $\bf 0$ and covariance $\text{Var}({\bf X})=\Sigma$( positive definite)
for Models (4), then

$E(Y{\bf X})=\lambda\Sigma{\bf\beta},$

where $\lambda>0$.

Proof:

$E(Y{\bf X})=E[g({\bf X}^{{}^{\prime}}{\bf\beta}){\bf X}]=E[(g({\bf
X}^{{}^{\prime}}{\bf\beta})-g(0)){\bf X}].$

(I) in case ${\bf\beta}=0$, it is obviously proved. (II) in case
${\bf\beta}\neq 0$ and $\Sigma=\sigma^{2}I$. Let $A$ be the orthogonal matrix
with the first row ${\bf\beta}^{{}^{\prime}}/\|{\bf\beta}\|$ and

${\bf W}=\left(\begin{array}[]{cc}W_{1}\\\ W_{2}\\\ \vdots\\\ W_{p}\\\
\end{array}\right)=A{\bf X}.$

Then from definition of ${\bf W}$, one has
$W_{1}={\bf\beta}^{{}^{\prime}}/\|{\bf\beta}\|{\bf X}$ and

$E(Y{\bf X})=E[(g({\bf X}^{{}^{\prime}}{\bf\beta})-g(0)){\bf
X}]=A^{{}^{\prime}}E[(g({\bf X}^{{}^{\prime}}{\bf\beta})-g(0))A{\bf
X}]=A^{{}^{\prime}}E[(g(W_{1}\|{\bf\beta}\|)-g(0)){\bf W}].$

Since ${\bf X}$ is distributed by the elliptical distributions with mean $\bf
0$ and variance $\text{Var}({\bf X})=\sigma^{2}I$, ${\bf W}=A{\bf X}$ is
distributed by the elliptical distributions with mean $\bf 0$ and variance
$\sigma^{2}AA^{{}^{\prime}}=\sigma^{2}I$ and then

$E(W_{i}\mid W_{1})=0,i=2,\cdots,p,$

by Theorem 6 given by Frahm (2004). So

$E[(g(\|{\bf\beta}\|W_{1})-g(0))W_{i}]=0,i=2,\cdots,p,$
$E[(g(\|{\bf\beta}\|W_{1})-g(0))W_{1}]>0$

by $g(\cdot)$ is strictly increasing and

$\displaystyle E(Y{\bf X})=$ $\displaystyle
A^{{}^{\prime}}E[(g(\|{\bf\beta}\|W_{1})-g(0))W]$ $\displaystyle=$
$\displaystyle
A^{{}^{\prime}}\left(\begin{array}[]{cc}E[(g(\|{\bf\beta}\|W_{1})-g(0))W_{1}]\\\
0\\\ \vdots\\\ 0\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{{\bf\beta}}{\|{\bf\beta}\|}\times
E[(g(\|{\bf\beta}\|W_{1})-g(0))W_{1}]$ $\displaystyle=$
$\displaystyle\lambda\sigma^{2}I{\bf\beta}$ $\displaystyle=$
$\displaystyle\lambda\Sigma{\bf\beta},$

where

$\lambda=\frac{E[(g(\|{\bf\beta}\|W_{1})-g(0))W_{1}]}{\sigma^{2}\|{\bf\beta}\|}>0.$

(III) in general ${\bf\beta}\neq 0$ and $\text{Var}(X)=\Sigma$. Let ${\bf
X}^{*}=\Sigma^{-1/2}{\bf X}$ and ${\bf\beta}^{*}=\Sigma^{1/2}{\bf\beta}$. Then
${\bf X}^{*}$ is distributed by the elliptical distributions with mean $\bf 0$
and variance $I$. By Models (4), one has

$E[Y\mid{\bf X}]=E[Y\mid{\bf X}^{*}]=g({{\bf
X}^{*}}^{{}^{\prime}}{\bf\beta}^{*}).$

and so

$E(Y{\bf X})=\Sigma^{1/2}E(Y{\bf X}^{*})=\Sigma^{1/2}\times\lambda
I{\bf\beta}^{*}=\lambda\Sigma{\bf\beta}$

by the case (II) and $\lambda>0$.

Proof of Theorem 1: By the Law of Large Number, we have

$\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}({{\bf X}}_{i}-\bar{{\bf
X}})=\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-{\bf\mu})+({\bf\mu}-\bar{{\bf X}})\bar{Y}\xrightarrow{p}E(Y{{\bf
X}})=\lambda\Sigma{\bf\beta}.$

According to the Lemma A and

$\hat{\Sigma}\xrightarrow{p}\Sigma.$

Hence, we can obtain that

${\hat{\Sigma}}_{X}^{-1}\times\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-\bar{{\bf X}})\xrightarrow{\ p\
}\Sigma^{-1}\times\lambda\Sigma{\bf\beta}=\lambda{\bf\beta},\lambda>0.$

that is

$D(\hat{{\bf\beta}})=\frac{{\hat{\Sigma}}_{X}^{-1}\times\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-\bar{{\bf
X}})}{\|{\hat{\Sigma}}_{X}^{-1}\times\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}({{\bf
X}}_{i}-\bar{{\bf X}})\|}\xrightarrow{\ p\
}\frac{\lambda{\bf\beta}}{\|\lambda{\bf\beta}\|}=\frac{{\bf\beta}}{\|{\bf\beta}\|}.$

Therefor, the proposed estimator $D(\hat{{\bf\beta}})$ is consistent.

Proof of Theorem 2: Let

${\bf U}_{n}=\frac{1}{n}\sum\limits_{i=1}^{n}Y{{\bf X}}_{i},\;\;{\bf
V}_{n}={\hat{\Sigma}}_{X}{\bf\beta}=\frac{1}{n}\sum\limits_{i=1}^{n}({\bf
X}_{i}-\bar{{\bf X}})({\bf X}_{i}-\bar{{\bf X}})^{{}^{\prime}}{\bf\beta}.$

By Lemma A and the Central Limit Theorem, one know $E{\bf
U}_{n}=\lambda\Sigma{\bf\beta}$ and

$\displaystyle\sqrt{n}\begin{pmatrix}{\bf U}_{n}-E[U_{n}]\\\ {\bf
V}_{n}-\Sigma{\bf\beta}\end{pmatrix}=\begin{pmatrix}{\bf
U}_{n}-\lambda\Sigma{\bf\beta}\\\ {\bf V}_{n}-\Sigma{\bf\beta}\\\
\end{pmatrix}\xrightarrow{d}\begin{pmatrix}\bf U\\\ \bf V\end{pmatrix}\sim
N\left(\begin{pmatrix}\bf 0\\\ \bf 0\\\ \end{pmatrix},\Omega\right),$ (7)

where

$\Omega=Var\begin{pmatrix}Y{\bf X}\\\ {\bf X}{\bf
X}^{{}^{\prime}}{\bf\beta}\end{pmatrix}.$

According to the form of the proposed estimator, we have

$\displaystyle D^{*}(\hat{{\bf\beta}})=\frac{{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf U}_{n}\right\|}$ $\displaystyle=$
$\displaystyle\frac{{\hat{\Sigma}}_{X}^{-1}\left({\bf U}_{n}-E{\bf
U_{n}}\right)+\left({\hat{\Sigma}}_{X}^{-1}E{\bf
U}_{n}-\lambda{\bf\beta}\right)+\lambda{\bf\beta}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}$ $\displaystyle=$
$\displaystyle\frac{{\hat{\Sigma}}_{X}^{-1}\left({\bf U}_{n}-E{\bf
U}_{n}\right)+\left({\hat{\Sigma}}_{X}^{-1}E{\bf
U}_{n}-\lambda{\bf\beta}\right)}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}+\frac{\lambda{\bf\beta}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}.$

In order to obtain the asymptotic distribution of
$D^{*}(\hat{{\bf\beta}})-{\bf\beta}$, we first require to prove the asymptotic
distribution of $\sqrt{n}({\bf U}_{n}-E{\bf U}_{n})$,
$\sqrt{n}\left({\hat{\Sigma}}_{X}^{-1}E{\bf U}_{n}-\lambda{\bf\beta}\right)$
and $\sqrt{n}\left(\frac{\lambda{\bf\beta}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$. From (7), we have
$\sqrt{n}({\bf U}_{n}-E{\bf U}_{n})\xrightarrow{d}{\bf U}\sim N(0,Var(Y{\bf
X})$. Next, we will prove the asymptotic properties of
$\sqrt{n}\left({\hat{\Sigma}}_{X}^{-1}E{\bf U}_{n}-\lambda{\bf\beta}\right)$
and $\sqrt{n}\left(\frac{\lambda{\bf\beta}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$ respectively.

$\displaystyle\sqrt{n}\left({\hat{\Sigma}}_{X}^{-1}E[{\bf
U}_{n}]-\lambda{\bf\beta}\right)$ $\displaystyle=$
$\displaystyle\sqrt{n}\left({\hat{\Sigma}}_{X}^{-1}\lambda\Sigma{\bf\beta}-\lambda{\bf\beta}\right)$
$\displaystyle=$
$\displaystyle\lambda{\hat{\Sigma}}_{X}^{-1}\sqrt{n}\left(\Sigma{\bf\beta}-{\hat{\Sigma}}_{X}{\bf\beta}\right)$
$\displaystyle=$ $\displaystyle-\lambda{\hat{\Sigma}}_{X}^{-1}\sqrt{n}({\bf
V}_{n}-\Sigma{\bf\beta})$ $\displaystyle\xrightarrow{d}$
$\displaystyle-\lambda\Sigma^{-1}{\bf V}.$

Let

${\bf S}_{n}({\bf
U}_{n},\Sigma_{X}^{-1})=\sqrt{n}\left(\frac{\lambda{\bf\beta}}{\left\|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right\|}-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)=\sqrt{n}\left(\frac{\lambda}{\sqrt{{\bf
U}_{n}^{{}^{\prime}}{\hat{\Sigma}}_{X}^{-2}{\bf
U}_{n}}}-\frac{1}{\|{\bf\beta}\|}\right){\bf\beta}.$

and expand ${\bf S}_{n}({\bf U}_{n},\Sigma_{X}^{-1})$ at
$(\lambda\Sigma{\bf\beta},\Sigma^{-1})$, we can obtain that

$\displaystyle{\bf S}_{n}({\bf U}_{n},\Sigma_{X}^{-1})$ $\displaystyle=$
$\displaystyle\sqrt{n}\left[-\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}({\bf
U}_{n}-E{\bf
U}_{n})-{\bf\beta}^{{}^{\prime}}\Sigma\Sigma^{-1}(\hat{\Sigma}_{X}^{-1}-\Sigma^{-1})\Sigma{\bf\beta}+o_{p}(n^{-1/2})\right]\frac{{\bf\beta}}{\|{\bf\beta}\|^{3}}$
$\displaystyle=$
$\displaystyle\sqrt{n}\left[-\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}({\bf
U}_{n}-E{\bf
U}_{n})-{\bf\beta}^{{}^{\prime}}(\hat{\Sigma}_{X}^{-1}-\Sigma^{-1})\Sigma{\bf\beta}+o_{p}(n^{-1/2})\right]\frac{{\bf\beta}}{\|{\bf\beta}\|^{3}}$
$\displaystyle=$
$\displaystyle\sqrt{n}\left[-\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}({\bf
U}_{n}-E{\bf
U}_{n})-{\bf\beta}^{{}^{\prime}}\hat{\Sigma}^{-1}_{X}(\Sigma-\hat{\Sigma}_{X}){\bf\beta}+o_{p}(n^{-1/2})\right]\frac{{\bf\beta}}{\|{\bf\beta}\|^{3}}$
$\displaystyle=$
$\displaystyle\sqrt{n}\left[-\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}({\bf
U}_{n}-E{\bf
U}_{n})-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}_{n}(\Sigma{\bf\beta}-{\bf
V}_{n})+o_{p}(n^{-1/2})\right]\frac{{\bf\beta}}{\|{\bf\beta}\|^{3}}$
$\displaystyle\xrightarrow{d}$
$\displaystyle-\left[\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
U}-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
V}\right]\frac{{\bf\beta}}{\|{\bf\beta}\|^{3}}.$

Therefore,

$\displaystyle\sqrt{n}\left(D^{*}(\hat{{\bf\beta}})-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$
$\displaystyle=$ $\displaystyle\frac{\hat{\Sigma}_{X}}{\|\hat{\Sigma}_{X}{\bf
U}_{n}\|}\sqrt{n}({\bf U}_{n}-E[U_{n}])+\frac{1}{\|\hat{\Sigma}_{X}{\bf
U}_{n}\|}\sqrt{n}(\left({\hat{\Sigma}}_{X}^{-1}E[{\bf
U}_{n}]-\lambda{\bf\beta}\right))$
$\displaystyle+\sqrt{n}\left(\frac{\lambda{\bf\beta}}{\left|{\hat{\Sigma}}_{X}^{-1}{\bf
U}_{n}\right|}-\frac{{\bf\beta}}{\|{\bf\beta}\|}\right)$
$\displaystyle\xrightarrow{d}\frac{1}{\|{\bf\beta}\|}\Bigg{\\{}\lambda^{-1}\Sigma^{-1}{\bf
U}-\Sigma^{-1}{\bf
V}-\frac{\left[\lambda^{-1}{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
U}-{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
V}\right]}{\|{\bf\beta}\|^{2}}{\bf\beta}\Bigg{\\}},$

where

$\lambda=E(Y{\bf\beta}^{{}^{\prime}}\Sigma^{-1}{\bf
X})/\|{\bf\beta}\|^{2},\;\;\begin{pmatrix}\bf U\\\ \bf V\end{pmatrix}\sim
N\left(\begin{pmatrix}\bf 0\\\ \bf 0\\\
\end{pmatrix},\Omega\right),\;\;\;\Omega=Var\begin{pmatrix}Y{\bf X}\\\ {\bf
X}{\bf X}^{{}^{\prime}}{\bf\beta}\end{pmatrix}.$

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