A Test for Multivariate Location Parameter in Elliptical Model Based on Forward Search Method

Abstract

In this article, we develop a test for multivariate location parameter in elliptical model based on the forward search estimator for a specified scatter matrix. Here, we study the asymptotic power of the test under contiguous alternatives based on the asymptotic distribution of the test statistics under such alternatives. Moreover, the performances of the test have been carried out for different simulated data and real data, and compared the performances with more classical ones.

Appendices

Appendix A: Proofs

In this section, we present the proofs of the theorems and some related lemmas.

Lemma 1.

Asn →∞underH₀, $\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})$convergesweakly to ad-dimensional normal distribution with the location parameter = 0and the scatter parameter Σ₁, where${\Sigma }_{1} =\left [\frac {1}{d\gamma } \frac {\pi ^{\frac {d}{2}}}{{\Gamma }{(\frac {d}{2})}}\int \limits _{0}^{\infty } x^{\frac {d}{2}} g(x)dx\right ]{\Sigma }$.

Proof of Lemma 1.

A straightforward application of polar transformation for an elliptical distribution and the construction of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ along with the central limit theorem and Slutsky’s theorem (see, e.g., Billingsley (1999)), it follows that $\sqrt {n}(\dot {{\boldsymbol {\mu }}}_{\gamma ,n}-{\boldsymbol {\mu }}_{0})$ converges weakly to d-dimensional Gaussian distribution with mean = 0 and variance-covariance matrix $= \left [\frac {1}{d\gamma } \frac {\pi ^{\frac {d}{2}}}{{\Gamma }{(\frac {d}{2})}}\int \limits _{0}^{\infty } x^{\frac {d}{2}} g(x)dx\right ]{\Sigma }$ under H₀. This implies ${\dot {\boldsymbol {\mu }}}_{\gamma ,n}$ is a consistent estimator of μ, which follows from Prohorov’s theorem (see, e.g., Van Der Vaart (1998)). □

Proof of Theorem 2.1.

To test H₀ : μ = μ₀ against H₁ : μ≠ μ₀, the power of the test based on ${T_{n}^{1}}$ is given by $P_{H_{1}}[{T_{n}^{1}} > c_{\alpha }]$, where c_α is the (1 − α)-th (0 < α < 1) quantile of the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}{Z_{i}^{2}}$. Here, λ_is are the eigen values of ${\Sigma }_{1} = \left [\frac {1}{d\gamma } \frac {\pi ^{\frac {d}{2}}}{\Gamma }{(\frac {d}{2})}\int \limits _{0}^{\infty } x^{\frac {d}{2}} g(x)dx\right ]{\Sigma }$, and Z_is are the i.i.d. N(0,1) random variables. In view of the orthogonal decomposition of multivariate normal distribution, ${T_{n}^{1}}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}{Z_{i}^{2}}$, and hence, the asymptotic size of the test based on ${T_{n}^{1}}$ is α. Let us now denote μ = μ₁(≠ μ₀) under H₁, and we now consider

$$\begin{array}{@{}rcl@{}} &&\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[T_{n}^{1} > c_{\alpha}\right]\\ \!&=&\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{0})\right|\right|^2 > c_{\alpha}\right]\\ \!&=&\! \lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1}+{\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^2 >c_{\alpha}\right]\\ \!&=&\! \lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} + \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^{2} + 2 \left<\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1}), \sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right> \!>\! c_{\alpha}\right]\\ \!&=&\!\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1})\right|\right|^2 > c_{\alpha} - \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^2 - 2n \left<(\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1}), ({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0}) \right>\right] \\ \!&=&\! \displaystyle\lim\limits_{n\rightarrow\infty}P_{H_{1}} \left[\left|\left|\sqrt{n} (\dot{{\boldsymbol{\mu}}}_{\gamma,n} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} > c_{\alpha} - n\left|\left|({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^{2} \right]~\text{since under}~H_{1}, \dot{{\boldsymbol{\mu}}}_{\gamma,n} \xrightarrow{a.s.} {\boldsymbol{\mu}}_{1} \\ \!&\rightarrow&\! 1~\text{as}~n\rightarrow\infty. \end{array} $$

The last implication follows from the fact that $\left |\left |\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{1})\right |\right |^2$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}{Z_{i}^{2}}$ under H₁, and c_α − n ||(μ₁ −μ₀)||² converges to −∞ as n →∞. This fact leads to the result. □

Proof of Proposition 2.1.

For testing H₀ : μ = μ₀ against H₁ : μ ≠ μ₀, the power of the test based on ${T_{n}^{2}}$ will be $P_{H_{1}}[{T_{n}^{2}} > c^{*}_{\alpha }]$, where $c^{*}_{\alpha }$ is the (1 − α)-th (0 < α < 1) quantile of the distribution of $\sum \limits _{i = 1}^{d}\lambda ^{*}_{i}Z_{i}^{*2}$. Here, $\lambda ^{*}_{i}$s are the eigen values of ${\sigma _{2}^{2}}{\Sigma }$, and $Z^{*}_{i}$’s are i.i.d. N(0,1) random variables. In view of the orthogonal decomposition of multivariate normal distribution, ${T_{n}^{2}}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}^{*}Z_{i}^{*2}$, and hence, the asymptotic size of the test based on ${T_{n}^{2}}$ is α. As earlier, let us again denote μ = μ₁(≠ μ₀) under H₁, and we now have

$$\begin{array}{@{}rcl@{}} &&\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[T_{n}^{2} > c^{*}_{\alpha}\right] \\ &=& \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{0})\right|\right|^{2} > c^{*}_{\alpha}\right]\\ &=& \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1}+{\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^{2} >c^{*}_{\alpha}\right]\\ & =& \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1})\right|\right|^2 + \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^2 + 2 \left<\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1}), \sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right> \!>\! c^{*}_{\alpha}\right]\\ &=&\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} > c^{*}_{\alpha} - \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^{2} - 2n \left<(\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1}), ({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0}) \right>\right]\\ &=& \displaystyle\lim\limits_{n\rightarrow\infty}P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{SM} - {\boldsymbol{\mu}}_{1})\right|\right|^2 > c^{*}_{\alpha} - n\left|\left|({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^2 \right]~(\text{since under}~H_{1}, \hat{{\boldsymbol{\mu}}}_{SM}\xrightarrow{a.s.}{\boldsymbol{\mu}}_{1})\\ &\rightarrow& 1~\text{as} n\rightarrow\infty. \end{array} $$

The last implication follows from the fact that $\left |\left |\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{SM} - {\boldsymbol {\mu }}_{1})\right |\right |^{2}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda ^{*}_{i}Z_{i}^{*2}$ under H₁, and $c^{*}_{\alpha } - n\left |\left |({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{0})\right |\right |^{2}$ converges to −∞ as n →∞. This completes the proof. □

Proof of Proposition 2.2.

In order to test H₀ : μ = μ₀ against H₁ : μ≠ μ₀, the form of the power function of the test based on ${T_{n}^{3}}$ is $P_{H_{1}}[{T_{n}^{3}} > c^{**}_{\alpha }]$, where $c^{**}_{\alpha }$ is same as described in the statement of the proposition. In view of the orthogonal decomposition of multivariate normal distribution, ${T_{n}^{3}}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}^{**}Z_{i}^{**2}$, and hence, the asymptotic size of the test based on ${T_{n}^{3}}$ is α Further, we denote that μ = μ₁(≠ μ₀) under H₁, and we then have

$$\begin{array}{@{}rcl@{}} &&\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[T_{n}^{3} > c^{**}_{\alpha}\right] \\ \!&=&\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{0})\right|\right|^2 > c^{**}_{\alpha}\right]\\ \!&=&\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1}+{\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^2 >c^{**}_{\alpha}\right]\\ \!&=&\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} + \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^{2} + 2 \left<\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1}), \sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right> \!>\! c^{**}_{\alpha}\right] \\ \!&=&\!\!\!\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1})\right|\right|^2 \!>\! c^{**}_{\alpha} - \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^2 - 2n \left<(\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1}), ({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right>\right] \\ \!&=&\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty}P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{CM} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} \!>\! c^{**}_{\alpha} - n\left|\left|({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^{2} \right]~(\text{since under}~H_{1}, \hat{{\boldsymbol{\mu}}}_{CM} \xrightarrow{a.s.} {\boldsymbol{\mu}}_{1}) \\ &\rightarrow&\!\! 1~\text{as}~n\rightarrow\infty. \end{array} $$

The last implication follows from the fact that $\left |\left |\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{CM} - {\boldsymbol {\mu }}_{1})\right |\right |^2$ converges weakly to the same distribution as described in the statement of the proposition, and $c^{**}_{\alpha } - n\left |\left |({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{0})\right |\right |^2$ converges to −∞ as n →∞. This completes the proof. □

Proof of Proposition 2.3.

In order to test H₀ : μ = μ₀ against H₁ : μ≠ μ₀, the form of the power function of the test based on ${T_{n}^{4}}$ is $P_{H_{1}}[{T_{n}^{4}} > c^{***}_{\alpha }]$, where $c^{***}_{\alpha }$ is same as described in the statement of the proposition. In view of the orthogonal decomposition of multivariate normal distribution, ${T_{n}^{4}}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}^{***}Z_{i}^{***2}$, and hence, the asymptotic size of the test based on ${T_{n}^{4}}$ is α Further, we denote that μ = μ₁(≠ μ₀) under H₁, and we then have

$$\begin{array}{@{}rcl@{}} &&\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[T_{n}^{4} > c^{***}_{\alpha}\right] \\ \!&=&\!\!\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{0})\right|\right|^{2} > c^{***}_{\alpha}\right]\\ \!&=&\!\!\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1}+{\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^{2} >c^{***}_{\alpha}\right]\\ \!& =&\!\!\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty} P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1})\right|\right|^2 + \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0})\right|\right|^2 + 2 \left<\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1}), \sqrt{n}({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right> \!>\! c^{***}_{\alpha}\right] \\ \!&=&\!\!\!\!\displaystyle\lim\limits_{n\rightarrow\infty} P_{H_{1}} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1})\right|\right|^{2} > c^{***}_{\alpha} - \left|\left|\sqrt{n}({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^{2} - 2n \left<(\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1}), ({\boldsymbol{\mu}}_{1} - {\boldsymbol{\mu}}_{0}) \right>\right] \\ \!&=&\!\!\!\!\! \displaystyle\lim\limits_{n\rightarrow\infty}P_{H_1} \left[\left|\left|\sqrt{n} (\hat{{\boldsymbol{\mu}}}_{HL} - {\boldsymbol{\mu}}_{1})\right|\right|^2 > c^{***}_{\alpha} - n\left|\left|({\boldsymbol{\mu}}_{1}-{\boldsymbol{\mu}}_{0})\right|\right|^2 \right]~(\text{since under}~H_{1}, \hat{{\boldsymbol{\mu}}}_{HL} \xrightarrow{a.s.} {\boldsymbol{\mu}}_{1})\\ &\rightarrow 1&~\text{as}~n\rightarrow\infty. \end{array} $$

The last implication follows from the fact that $\left |\left |\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{HL} - {\boldsymbol {\mu }}_{1})\right |\right |^{2}$ converges weakly to the same distribution as described in the statement of the proposition, and $c^{***}_{\alpha } - n\left |\left |({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{0})\right |\right |^{2}$ converges to −∞ as n →∞. This completes the proof. □

Proof of Theorem 2.2.

At first, one needs to know the form of g(x) in the expressions of e₁(d) and e₂(d) that provided in the first paragraph in Section 2.2. Note that for a d-dimensional Gaussian distribution, $g(x) = e^{-\frac {x}{2}}$, in the case of a d-dimensional Cauchy distribution, $g(x) = \frac {1}{(1 + x)^{\frac {d + 1}{2}}}$, and for the given d-dimensional Spherical distribution, $g(x)= e^{-x^{100}}$. Now, using the form of g(x), we have $e_{1}(d) = \frac {\gamma }{{2\pi }^{\frac {d}{2}}}$, $e_{2}(d) = \frac {\gamma {\pi }^{1-\frac {d}{2}}}{{2}^(1+\frac {d}{2})}$ and $e_{3}(d) = \frac {\gamma \pi ^{1-\frac {d}{2}}}{3 \times {(2)}^{\frac {d}{2}}}$ for a d-dimensional Gaussian distribution. Since π > 1, one can conclude that $\displaystyle \lim _{d\rightarrow \infty } e_{1}(d) = 0$. Next, note that $e_{2}(d) = \frac {\gamma {\pi }^{1-\frac {d}{2}}}{{2}^(1+\frac {d}{2})} = \frac {\gamma \pi }{2\times (2\pi )^{\frac {d}{2}}}\rightarrow 0$ and $e_{3}(d) = \frac {\gamma {\pi }^{1-\frac {d}{2}}}{3 \times {(2)}^{\frac {d}{2}}} = \frac {\gamma \pi }{3\times (2\pi )^{\frac {d}{2}}}\rightarrow 0$ as d →∞ since 2π > 1. Further, note that since Var(Y₁) = ∞ for a Cauchy distribution, we have e₁(d) = ∞ for all d, and hence, $\displaystyle \lim _{d\rightarrow \infty } e_{1}(d) = \infty$ for a d-dimensional Cauchy distribution. Similarly, using the form of g(x) associated with Cauchy distribution, we have $e_{2}(d) = \frac {\gamma d {\pi }^{\frac {3-d}{2}} {\Gamma }({\frac {d + 1}{2}})}{4}$ and $e_{3}(d) = \frac {\gamma d {\pi }^{\frac {3-d}{2}} {\Gamma }({\frac {d + 1}{2}})}{12}$. In order to investigate the limiting properties of e₂(d), we consider $\frac {e_{2}(d + 1)}{e_{2}(d)} = \frac {d + 1}{d}\times \frac {1}{\sqrt {\pi }}\times \frac {{\Gamma }(d/2 + 1)}{{\Gamma }(d/2 + 1/2)}$. Next, using Stirling’s approximation formula: ${\Gamma }(n + 1) = \sqrt {2\pi }e^{-n} n^{n + 1/2}$ when n is an integer and n →∞, we have

$$\begin{array}{@{}rcl@{}} \lim\limits_{d\rightarrow\infty}\frac{e_{2}(d + 1)}{e_{2}(d)} & =& \lim\limits_{d\rightarrow\infty}\frac{d + 1}{d}\times\frac{1}{\sqrt{\pi}}\times\frac{{\Gamma}(\frac{d}{2} + 1)}{{\Gamma}(\frac{d}{2} + \frac{1}{2})}\\ & = & 1\times\frac{1}{\sqrt{\pi}} \times\lim\limits_{d\rightarrow\infty}\frac{{\Gamma}(\frac{d}{2} + 1)}{{\Gamma}(\frac{d}{2} + \frac{1}{2})}\\ & = & 1\times\frac{1}{\sqrt{\pi}} \times\lim\limits_{d\rightarrow\infty}\frac{\sqrt{2\pi} e^{-\frac{d}{2}} (\frac{d}{2})^{\frac{d}{2} + 1}}{\sqrt{2\pi}e^{-(\frac{d}{2} - \frac{1}{2})} (\frac{d}{2} - \frac{1}{2})^{\frac{d}{2} + \frac{1}{2}}}\\ & = & 1\times\frac{1}{\sqrt{\pi}}\times\frac{1}{\sqrt{2e}} \times\lim\limits_{d\rightarrow\infty}\frac{d}{\sqrt{d - 1}}\left( 1 + \frac{1}{d - 1}\right)^{\frac{d}{2}}\\ & = & 1\times\frac{1}{\sqrt{\pi}}\times\frac{1}{\sqrt{2e}}\times\lim\limits_{d\rightarrow\infty}\frac{d}{\sqrt{d - 1}}\times\lim\limits_{d\rightarrow\infty}\left( 1 + \frac{1}{d - 1}\right)^{\frac{d}{2}}\\ &=&\infty~\left( \text{since}~\displaystyle\lim\limits_{d\rightarrow\infty}\frac{d}{\sqrt{d - 1}} = \infty\right). \end{array} $$

This implies that $\displaystyle \lim _{d\rightarrow \infty } e_{2}(d) = \infty$. Arguing in similar way one can say that $\displaystyle \lim _{d\rightarrow \infty } e_{3}(d) = \infty$. Furthermore, for d-dimensional Spherical distribution with $g(x) = e^{-x^{100}}$, we have $e_{1} (d) = \frac {100 d\gamma {\Gamma }(\frac {d}{2})}{\pi ^{\frac {d}{2}} {\Gamma }(\frac {1}{100}(\frac {d}{2}+ 1))}$, $e_{2}(d) = \frac {d\gamma {\Gamma }(\frac {d}{2}) ({\Gamma }(\frac {1}{200}))^{2}}{400 \pi ^{\frac {d}{2}} {\Gamma }(\frac {1}{100}(\frac {d}{2}+ 1))}$ and $e_{3}(d) = \frac {53188.48 d\gamma {\Gamma }(\frac {d}{2})}{ \pi ^{\frac {d}{2}} {\Gamma }(\frac {1}{100}(\frac {d}{2}+ 1))}$. In order to investigate the limiting properties of e₁(d) the repeated use of Stirling’s approximation formula leads to

$$\begin{array}{@{}rcl@{}} \lim\limits_{d\rightarrow\infty}\frac{e_{1}(d + 1)}{e_{1}(d)} & =& \lim\limits_{d\rightarrow\infty}\frac{d + 1}{d}\times\frac{1}{\sqrt{\pi}}\times\frac{{\Gamma}(\frac{d + 1}{2})}{{\Gamma}(\frac{d}{2})}\times \frac{{\Gamma}(\frac{1}{100}(\frac{d}{2}+ 1))}{{\Gamma}(\frac{1}{100}(\frac{d + 1}{2}+ 1))}\\ & = & 1\times\frac{1}{\sqrt{\pi}} \times\lim\limits_{d\rightarrow\infty}\frac{{\Gamma}(\frac{d}{2} + \frac{1}{2})}{{\Gamma}(\frac{d}{2})} \times \frac{{\Gamma}(\frac{d}{200}+\frac{1}{100})}{{\Gamma}(\frac{d + 1}{200}+\frac{1}{100})}\\ & = & 1\times\frac{1}{\sqrt{\pi}} \times\lim\limits_{d\rightarrow\infty}\frac{\sqrt{2\pi}e^{-(\frac{d}{2} - \frac{1}{2})} (\frac{d}{2} - \frac{1}{2})^{\frac{d}{2}}}{\sqrt{2\pi}e^{-(\frac{d}{2} - 1)} (\frac{d}{2} - 1)^{\frac{d}{2} - \frac{1}{2}}}\\ &&\times \frac{\sqrt{2\pi}e^{-(\frac{d}{200} - \frac{198}{200})} (\frac{d}{200} - \frac{198}{200})^{\frac{d}{200} - \frac{98}{200}}}{\sqrt{2\pi}e^{-(\frac{d}{200} - \frac{197}{200})} (\frac{d}{200} - \frac{197}{200})^{\frac{d}{200} - \frac{97}{200}}} \\ & = & 1\times\frac{1}{\sqrt{\pi}}\times\frac{(200e)^{\frac{1}{200}}}{\sqrt{2e}} \times\lim\limits_{d\rightarrow\infty}{\sqrt{d - 2}}\left( 1 - \frac{1}{d - 2}\right)^{\frac{d}{2}} \\ &&\times \lim\limits_{d\rightarrow\infty}{\frac{1}{(d-197)^{\frac{1}{200}}}}\left( 1 - \frac{1}{d - 197}\right)^{\frac{d}{200}-\frac{98}{200}}\\ & = & 1\times\frac{1}{\sqrt{\pi}}\times\frac{1}{\sqrt{2e}}\times\lim\limits_{d\rightarrow\infty}\frac{\sqrt{d - 2}}{(d-197)^{\frac{1}{200}}}\\ &&\times\lim\limits_{d\rightarrow\infty}\left( 1 - \frac{1}{d - 2}\right)^{\frac{d}{2}} \times \lim\limits_{d\rightarrow\infty}\left( 1 - \frac{1}{d - 197}\right)^{\frac{d}{200}-\frac{98}{200}}\\ &=&\infty~\left( \text{since}~\displaystyle\lim\limits_{d\rightarrow\infty}\frac{\sqrt{d - 2}}{(d-197)^{\frac{1}{200}}} = \infty\right). \end{array} $$

This leads us to $\displaystyle \lim _{d\rightarrow \infty } e_{1}(d) = \infty$. Similarly one can say that $\displaystyle \lim _{d\rightarrow \infty } e_{2}(d) = \infty$ and $\displaystyle \lim _{d\rightarrow \infty } e_{3}(d) = \infty$, and hence the proof is complete. □

Proof of Theorem 3.1.

In order to establish the contiguity of the distributions associated with sequence {H_1n} relative to those of {H_0n}, it is enough to show that Λ_n, the logarithm of the likelihood ratio, converges weakly to a random variable associated with a normal distribution with location parameter $= -\frac {\sigma ^{2}}{2}$ and variance = σ², where σ is a positive constant (see Hajek et al. 1999, p. 254, Corollary to Lecam’s first lemma). Let y₁, …, y_n be i.i.d. random variables with the probability density function f(y;(.),Σ), where (.) denotes the location parameter involved in the distribution, and Σ is the scatter matrix. We now consider

$$\begin{array}{@{}rcl@{}} {\Lambda}_{n} & = & \displaystyle\log\prod\limits_{i = 1}^{n}\frac{f\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}+\frac{{\boldsymbol{\delta}}}{\sqrt{n}}\right)}{f\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}\right)} = \sum \limits_{i = 1}^{n} \left\{\log f\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0} + \frac{{\boldsymbol{\delta}}}{\sqrt{n}}\right) - \log f\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}\right)\right\}\\ & = & \sum\limits_{i = 1}^{n}\left\{h\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}+\frac{{\boldsymbol{\delta}}}{\sqrt{n}}\right) - h\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}\right)\right\}~(\text{denoted as}~h (.) = \log f(.))\\ & = & \sum\limits_{i = 1}^{n} \left[ \frac{{\boldsymbol{\delta}}^{T}}{\sqrt{n}}\bigtriangledown\{h\left( \mathbf{y}_{i},{\boldsymbol{\mu}}_{0}\right)\} + \frac{1}{2n}{\boldsymbol{\delta}}^{T}H\{h\left( \mathbf{y}_{i}, \zeta_{n}\right)\}{\boldsymbol{\delta}}\right], \end{array} $$

where ζ_n is any point lying on the straight line joining μ₀ and ${\boldsymbol {\mu }}_{0}+\frac {{\boldsymbol {\delta }}}{\sqrt {n}}$, ▽ (.) denotes the gradient vector of (.), and H(.) denotes the Hessian matrix of (.). Note that ζ_n →μ₀ as n →∞.

It now follows from the central limit theorem that $\sum \limits _{i = 1}^{n}\frac {{\boldsymbol {\delta }}^{T}}{\sqrt {n}}\bigtriangledown \{h\left (\mathbf {y}_{i},{\boldsymbol {\mu }}_{0}\right )\}$ converges weakly to a random variable associated with normal distribution with mean = E[δ^T ▽{h (y,μ₀)}] = 0 and variance = E[δ^T ▽{h (y,μ₀)}]² when E[δ^T ▽{h (y,μ₀)}]² < ∞ for all δ. Moreover, a direct algebra implies that E[δ^T ▽{h (y,μ₀)}]² = −E[δ^TH{h (y_i,μ₀)}δ] (see, e.g., Shao 2003). Hence, the asymptotic normality of $\sum \limits _{i = 1}^{n}\frac {{\boldsymbol {\delta }}^{T}}{\sqrt {n}}\bigtriangledown \{h\left (\mathbf {y}_{i},{\boldsymbol {\mu }}_{0}\right )\}$ holds when $E\left \{\frac {\partial ^{2}}{\partial \mu _{i}\partial \mu _{j}}\log f(\mathbf {y}, {\boldsymbol {\mu }}) \right \} < \infty$, where μ_i and μ_j are the i-th and the j-th components of μ.

Next, the other term $\frac {1}{2n}\sum \limits _{i = 1}^{n}{\boldsymbol {\delta }}^{T}H\{h\left (\mathbf {y}_{i},\boldsymbol {\zeta }_{n}\right )\}{\boldsymbol {\delta }}\stackrel {p}\rightarrow \frac {1}{2}E [{\boldsymbol {\delta }}^{T}H\{h\left (\mathbf {y},{\boldsymbol {\mu }}_{0}\right )\}{\boldsymbol {\delta }}]$ in view of weak law of large number and ζ_n →μ₀ as n →∞. Finally, using Slutsky’s result, one can conclude that Λ_n converges weakly to a normal distribution with mean $= \frac {1}{2}E [{\boldsymbol {\delta }}^{T}H\{h\left (\mathbf {y}_{i},{\boldsymbol {\mu }}_{0}\right )\}{\boldsymbol {\delta }}]$ and variance = −E[δ^TH{h (y_i,μ₀)}δ], where E[δ^TH{h (y_i,μ₀)}δ] is a negative constant. Hence, the proof is complete. □

Proof of Theorem 3.2.

To establish this result, one first needs to show that the joint distribution of $\{\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0}), {\Lambda }_{n}\}$ is asymptotically Gaussian under H₀, which is asserted in Le Cam’s third lemma (see, e.g., Hajek et al. 1999). Note that Lemma 1 asserts, $\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})$ converges weakly to a d-dimensional Gaussian distribution under H₀, and further, it follows from the proof of Theorem 3.1 that Λ_n converges weakly to a univariate Gaussian distribution with certain parameters under some conditions. Suppose now that $\mathbf {L}\in \mathbb {R}^{d}$ and $m\in \mathbb {R}$ are two arbitrary constants. In view of the linearization of $\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})$ and Λ_n along with a direct application of central limit theorem, one can establish that $\mathbf {L}^{T}.\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})\} + m{\Lambda }_{n}$ converges weakly to a univariate Gaussian distribution under H₀, where (.) denotes the inner product. This fact implies that the joint distribution of $\{\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0}), {\Lambda }_{n}\}$ is asymptotically (d + 1)-dimensional Gaussian under H₀. Note that the i-th component of d-dimensional covariance between $\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})$ and Λ_n is $E\left \{(\dot {{\boldsymbol {\mu }}}_{\gamma ,n, i} - {\boldsymbol {\mu }}_{0, i})\sum \limits _{j = 1}^{d}\delta _{j}\frac {\partial g(\mathbf {y}, {\boldsymbol {\mu }})}{\partial \mu _{j}}|_{{\boldsymbol {\mu }} = {\boldsymbol {\mu }}_{0}}\right \}$, where δ_i, $\dot {{\boldsymbol {\mu }}}_{\gamma , n, i}$, μ_0,i and μ_i are the i-th component of δ, $\dot {{\boldsymbol {\mu }}}_{\gamma , n}$, μ₀ and μ, respectively, and g(y,μ) = log f(y,μ).

Now, using Le Cam’s third lemma, one can directly establish that under H_n, $\{\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0})\}$ converges weakly to d-dimensional Gaussian distribution with mean vector a = (a₁,…,a_d) and variance-covariance matrix $= \left [\frac {1}{d\gamma } \frac {\pi ^{\frac {d}{2}}}{\Gamma (\frac {d}{2})}\int \limits _{0}^{\infty } x^{\frac {d}{2}} g(x)dx\right ]{\Sigma }$, where $a_{i} = E\left \{(\dot {{\boldsymbol {\mu }}}_{\gamma ,n, i} - {\boldsymbol {\mu }}_{0, i})\sum \limits _{j = 1}^{d}\delta _{j}\frac {\partial g(\mathbf {y}, {\boldsymbol {\mu }})}{\partial \mu _{j}}|_{{\boldsymbol {\mu }} = {\boldsymbol {\mu }}_{0}}\right \}$. Hence, under H_n, for any orthogonal matrix A, $\{A\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0} - \mathbf {a})\}$ converges weakly to a d-dimensional Gaussian distribution with mean vector = 0 and variance-covariance matrix = Diag(λ₁,…,λ_d), where λ_i is the i-th eigenvalue of $\left [\frac {1}{d\gamma } \frac {\pi ^{\frac {d}{2}}}{\Gamma (\frac {d}{2})}\int \limits _{0}^{\infty } x^{\frac {d}{2}} g(x)dx\right ]{\Sigma }$, and consequently, as A is an orthogonal matrix, i.e., A^TA = AA^T = I_d, $\{A\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0} - \mathbf {a})\}^{T}\{A\sqrt {n} (\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0} - \mathbf {a})\} = n||\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0} - \mathbf {a}||^{2}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}{Z_{i}^{2}}$, where Z_i’s are i.i.d. standard normal random variables. This fact directly imply that under H_n and condition assumed in Theorem 3.1., ${T_{n}^{1}} = n||\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - {\boldsymbol {\mu }}_{0}||^{2}$ converges weakly to the distribution of $\sum \limits _{i = 1}^{d}\lambda _{i}(Z_{i} + a_{i})^{2}$.

Similarly, based on the Bahadur expansion of the median (see, e.g., Serfling 1980), the Hodges-Lehmann estimator (see, e.g., Lehmann 2006) and the definition of the sample mean vector, one can conclude that under H₀, $\mathbf {L}^{T}.\{\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{CM} - {\boldsymbol {\mu }}_{0})\} + m{\Lambda }_{n}$, $\mathbf {L}^{T}.\{\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{HL} - {\boldsymbol {\mu }}_{0})\} + m{\Lambda }_{n}$ and $\mathbf {L}^{T}.\{\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{SM} - {\boldsymbol {\mu }}_{0})\} + m{\Lambda }_{n}$ converge weakly to a Gaussian random variable with certain parameters for arbitrary constants $\mathbf {L}\in \mathbb {R}^{d}$ and $m\in \mathbb {R}$. This fact along with Le Cam’s third lemma imply that under H_n, both $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{CM} - {\boldsymbol {\mu }}_{0})$, $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{HL} - {\boldsymbol {\mu }}_{0})$ and $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{SM} - {\boldsymbol {\mu }}_{0})$ converge weakly to a d-dimensional Gaussian random vector with non-zero mean vectors and certain variance-covariance matrix. The i-th component of the mean vector for $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{SM} - {\boldsymbol {\mu }}_{0})$, $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{CM} - {\boldsymbol {\mu }}_{0})$ and $\sqrt {n} (\hat {{\boldsymbol {\mu }}}_{HL} - {\boldsymbol {\mu }}_{0})$ are $a_{i}^{*}$, $a_{i}^{**}$ and $a_{i}^{***}$, respectively, where i = 1,…,d. Finally, as we argued earlier, based on the well-known orthogonalization of the multivariate Gaussian distribution, one can conclude that under H_n, ${T_{n}^{2}}$ converges weakly to $\sum \limits _{i = 1}^{d}\lambda ^{*}_{i}(Z_{i}^{*} + a_{i}^{*})^{2}$, ${T_{n}^{3}}$ converges weakly to $\sum \limits _{i = 1}^{d}\lambda ^{**}_{i}(Z_{i}^{**} + a_{i}^{**})^{2}$, and ${T_{n}^{4}}$ converges weakly to $\sum \limits _{i = 1}^{d}\lambda ^{***}_{i}(Z_{i}^{***} + a_{i}^{***})^{2}$. The description of $\lambda _{i}^{*}$, $a_{i}^{*}$, $Z_{i}^{*}$, $\lambda _{i}^{**}$, $a_{i}^{**}$, $Z_{i}^{**}$, $\lambda _{i}^{***}$, $a_{i}^{***}$ and $Z_{i}^{***}$ are provided in the statement of the theorem. The proof is complete now. □

Proof of Corollary 3.1.

It follows from the assertion in Theorem 3.2. that under H_n, the power of the test based on ${T_{n}^{1}}$ is $P_{{\boldsymbol {\delta }}}\left [\sum \limits _{i = 1}^{d}\lambda _{i}(Z_{i} + a_{i})^{2} > c_{\alpha }\right ]$. Here c_α is such that $P_{\boldsymbol {\delta } = \mathbf {0}}\left [\sum \limits _{i = 1}^{d}\lambda _{i}(Z_{i} + a_{i})^{2} > c_{\alpha }\right ]$ = α since H_n coincides with H₀ when δ = 0. Arguing exactly in a similar way, one can establish the asymptotic power of the tests based on ${T_{n}^{2}}$, ${T_{n}^{3}}$ and ${T_{n}^{4}}$ under H_n. □

Appendix B: Properties of Forward Search Estimator

Here, we study a few fundamental properties of the multivariate forward search location estimator along with its performances.

(Property 1) Robustness of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$

The robustness property of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ is described by the finite sample breakdown point, which is defined as follows. For the estimator $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ based on the data ${\mathcal {Y}} = \{\mathbf {y}_{1}, \ldots , \mathbf {y}_{n}\}$, its finite sample breakdown point is defined as $\epsilon (\dot {{\boldsymbol {\mu }}}_{\gamma ,n}, {\mathcal {Y}}) = \displaystyle \min _{m^{\star }\leq n^{\star }\leq n}\left \{\frac {n^{\star }}{n}:\sup _{{\mathcal {Y}^{*}}}\left |\left |\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - \dot {{\boldsymbol {\mu }}}_{\gamma ,n}^{(n^{\star })}\right |\right | = \infty \right \},$ where m^∗ is the cardinality of the initial subset, and $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}^{(n^{\star })}$ is the forward search estimator of μ based on a modified sample ${\mathcal {Y}^{*}} = \{\mathbf {y}^{*}_{1}, \ldots , \mathbf {y}^{*}_{n^{*}}, \mathbf {y}_{n^{*} + 1},$ …,y_n}. The following theorem describes the breakdown point of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$.

Theorem 6.1.

Suppose thaty’s are in general position i.e., one cannot draw a hyperplanepassing through all the observations, in the case of$\dot {{\boldsymbol {\mu }}}_{\gamma , n}$forγ-thstep. Then, we have$\epsilon (\dot {{\boldsymbol {\mu }}}_{\gamma ,n}, {\mathcal {Y}}) = 1-\gamma$.

Proof of Theorem 6.1.

Suppose that the original observations are denoted by ${\mathcal {Y}} = \{\mathbf {y}_{1}, \ldots , \mathbf {y}_{n}\}$, and without loss of generality, first n^∗ < n obervations are corrupted. Let ${\mathcal {Y}}^{*} = \{\mathbf {y}_{1}^{*}, \ldots , \mathbf {y}_{n^{*}}^{*}, \mathbf {y}_{n^{*} + 1}, \ldots , \mathbf {y}_{n}\}$ denote the contaminated sample, where $\mathbf {y}_{i}^{*}$ are corrupted observations, i = 1,…,n^∗. It follows from the construction of the estimator that $\displaystyle \sup _{{\mathcal {Y}}^{*}}\left |\left |\dot {{\boldsymbol {\mu }}}_{\gamma ,n} - \dot {{\boldsymbol {\mu }}}_{\gamma ,n}^{(n^{\star })}\right |\right | = \infty$ if and only if $\left |\left |\mathbf {y}_{i^{\star }}^{\star }\right |\right |=\infty$ for any i = 1,…,n^∗. Without loss of generality, suppose that the aforementioned equivalent relationship holds for some k, and we than have η_k,γ,n = 1 since $\sum \limits _{i = 1}^{n}\eta _{i,\gamma , n} = m>0$. This fact implies that $Md^{2}_{k,n} = \infty$ for those choices of k. Let us further consider that there are $n_{1}^{*} < n^{*}$ many choices of k for which $Md^{2}_{k,n} = \infty$. Now, in view of the definition of η_k,γ,n, we have $\eta _{j,\gamma ,n} = I(Md^{2}_{j,n} \leq \delta ^{2}_{\gamma , n}) = I(Md^{2}_{j, n} \leq Md^{2}_{(m), n}) = 0$ for $j = 1, \ldots , n_{1}^{*}$ when $n_{1}^{*} < n - m$. Further, note that $Md^{2}_{k,n} = \infty$ for any k = 1,…,n^∗ when $n_{1}^{*} = n^{*}$. Hence, $\epsilon (\dot {{\boldsymbol {\mu }}}_{\gamma ,n}, {\mathcal {Y}}) = 1-\gamma ,$ and the proof is complete. □

Remark 6.1.

We would like to discuss on the breakdown point of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$. Note that (1 − γ) is essentially the trimming proportion of the observations in the forward search estimator, and for that reason, it is expected that this estimator cannot be breakingdown even in the presence of (1 − γ) proportion outliers in the data. It indicates that γ controls the breakdown point of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$. For instance, the breakdown point of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ will achieve the highest possible value = 1/2 when γ = 1/2, and on the other hand, when γ = 1 ⇔ m = n, the breakdown point of the forward search estimator will be = 0. Overall, the robustness behaviour of the forward search estimator is similar to any other trimming based estimator, e.g., the trimmed mean (see, e.g., Tukey 1948; Bickel 1965).

(Property 2) Finite Sample Efficiency of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$

We here investigate the finite sample efficiency of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ relative to the sample mean, the co-ordinate wise median and the Hodges-Lehmann estimator for d-dimensional Gaussian distribution, d-dimensional Cauchy distribution with probability density function f_C(x) = (Γ((d + 1)/2)/π^d/2Γ(1/2))(1 + x^Tx)^{−(d+ 1)/2} and d-dimensional Spherical distribution with with $g(x) = e^{-x^{100}}$. In this simulation study, the finite sample efficiency of an estimator T_n relative to ${T}_{n}^{\prime }$ is defined as $\{|COV(T_{n}^{\prime })|/|COV(T_{n})|\}^{1/d}$, where |COV (T_n)| is the determinant of $COV({T}_{n}) = \frac {1}{m}\sum \limits _{i = 1}^{m}({T}_{n}^{i} -\bar {{T}}_{n})({T}_{n}^{i} -\bar {{T}}_{n})^{T}$, and m is the number of Monte-Carlo replications. Here, ${T}_{n}^{i}$ is the estimate of T_n based on the i-th replication, and $\bar {{T}}_{n} = \frac {1}{m}\sum \limits _{i = 1}^{m}{T}_{n}^{i}$. In this simulation study, we consider m = 1000, and n = 10 and 100, and the results are summarized in Table 3.

Table 3 Finite sample efficiencies of the multivariate forward search location estimator relative to the sample mean, the co-ordinate wise median and the Hodges-Lehmann estimator for different values of n, d and various distributions. Here γ = 1/2

Here, $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ performs well in term of finite sample efficiency for Cauchy distribution. The Hodges-Lehmann estimator also has better efficiency than the co-ordinate wise median for like Cauchy distribution. For normal distribution, the sample mean performs best although it fails to perform well for Cauchy distribution. The Hodges-Lehmann estimator is nearly as efficient as the sample mean for the normal distribution. For Spherical distribution with with $g(x) = e^{-x^{100}}$, $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ performs best compared to other three estimators.

(Property 3) Asymptotic Efficiency of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$

In this study, we consider d-dimensional Gaussian, d-dimensional Cauchy distributions and d-dimensional Spherical distribution with $g(x)=e^{-x^{100}}$ to carry out this study. The asymptotic efficiencies of $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ relative to the sample mean, the co-ordinate wise median and the Hodges-Lehmann estimator for the aforementioned distributions are reported in Table 4.

Table 4 Asymptotic efficiencies of the multivariate forward search location estimator relative to the sample mean, the co-ordinate wise median and the Hodges-Lehmann estimator for different values of dimensions. Here γ = 1/2

It is evident from the figures in Table 4 that $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$ performs well in term of asymptotic efficiency for Cauchy distribution. For Spherical distribution with $g(x)=e^{-x^{100}}$, it performs best among all the estimators. As expected, the sample mean performs best for normal distribution since it is the maximum likelihood estimator of location parameter in the normal distribution. The Hodges-Lehmann estimator is nearly as efficient as the sample mean for the normal distribution. On the other hand, the sample mean was outperformed by $\dot {{\boldsymbol {\mu }}}_{\gamma ,n}$, co-ordinate wise median and Hodges-Lehmann estimator for Cauchy distribution since the sample mean does not have finite second moment when data follow Cauchy distribution.