A class of percentile modified Lepage-type tests

Abstract

The two-sample problem usually tests for a difference in location. However, there are many situations, for example in biomedicine, where jointly testing for difference in location and variability may be more appropriate. Moreover, heavy-tailed data, outliers and small-sample sizes are common in biomedicine and in other fields. These considerations make the use of nonparametric methods more appealing than parametric ones. The aim of the paper is to contribute to the literature about nonparametric simultaneous location and scale testing. More precisely, several existing tests are generalized and unified, and a new class of tests based on the Mahalanobis distance between the percentile modified test statistics for location and scale differences is introduced. The asymptotic distributions of the test statistics are obtained, and small-sample size behaviour of the tests is studied and compared to other tests via Monte Carlo simulations. It is shown that the proposed class of tests performs well when there are differences in both location and variability. A practical application is presented.

Appendix 1

Lemma 1

Under$H_{0}$, first and second moments of$T_{p}$and$B_{r}$are

• (i)
  $$ E\left( {T_{p} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{mP\left( {P + 1} \right)}}{2N}} \hfill &\quad {{\text{if}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{mP^{2} }}{2N}} \hfill &\quad {{\text{if}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right.\quad {\text{and}}\quad E\left( {B_{r} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{mR\left( {R + 1} \right)}}{2N}} \hfill &\quad {{\text{if}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{mR^{2} }}{2N}} \hfill &\quad {{\text{if}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} ;} \right. $$
• (ii)
  $$ Var\left( {T_{p} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{mnP\left( {P + 1} \right)}}{{12N^{2} \left( {N - 1} \right)}}\left[ {4NP + 2N - 3P\left( {P + 1} \right)} \right]} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{mnP}{{12N^{2} \left( {N - 1} \right)}}\left[ {4NP^{2} - 3P^{3} - N} \right]} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right.; $$
• (iii)
  $$ Var\left( {B_{r} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{mnR\left( {R + 1} \right)}}{{12N^{2} \left( {N - 1} \right)}}\left[ {4NR + 2N - 3R\left( {R + 1} \right)} \right]} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{mnR}{{12N^{2} \left( {N - 1} \right)}}\left[ {4NR^{2} - 3R^{3} - N} \right]} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right.; $$

$$ COV\left( {T_{p} ,B_{r} } \right) = \left\{ {\begin{array}{*{20}l} { - \frac{{mnPR\left( {P + 1} \right)\left( {R + 1} \right)}}{{4N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ { - \frac{{mnP^{2} R^{2} }}{{4N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{if}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right.. $$

Proof

• (i) See Gastwirth (1965).

Proofs of parts (ii) and (iii) are not given by Gastwirth (1965), they are reported below.

• (ii) We prove that under ${H_{0} ,\; V\left( {T_{p} } \right) = \frac{{mnP\left( {P + 1} \right)}}{{12N^{2} \left( {N - 1} \right)}}\left( {4NP + 2N - 3P\left( {P + 1} \right)} \right)^{2} \; {\text{if}}}\; N$ is odd. $E\left( {T_{p}^{2} } \right)$ is needed. Note that

  $$ \begin{aligned} T_{p}^{2} & = \left( {\mathop \sum \limits_{i = N - P + 1}^{N} \left( {i - \left( {N - P} \right)} \right)\lambda_{i} } \right)^{2} \\ & = \mathop \sum \limits_{i = N - P + 1}^{N} \left( {i - \left( {N - P} \right)} \right)^{2} \lambda_{i}^{2} + \mathop \sum \limits_{i = N - P + 1}^{N} \mathop \sum \limits_{i = N - P + 1,i \ne j}^{N} \left( {i - \left( {N - P} \right)} \right)\left( {j - \left( {N - P} \right)} \right)\lambda_{i} \lambda_{j}. \\ \end{aligned} $$

  Therefore

  $$ \begin{aligned} E\left( {T_{p}^{2} } \right) & = \mathop \sum \limits_{i = N - P + 1}^{N} \left( {i - \left( {N - P} \right)} \right)^{2} E\left( {\lambda_{i}^{2} } \right) + \mathop \sum \limits_{i = N - P + 1}^{N} \mathop \sum \limits_{i = N - P + 1,i \ne j}^{N} \left( {i - \left( {N - P} \right)} \right)\left( {j - \left( {N - P} \right)} \right)E\left( {\lambda_{i} \lambda_{j} } \right) \\ & = \frac{{P\left( {P + 1} \right)\left( {2P + 1} \right)m}}{6N} + \frac{{m\left( {m - 1} \right)}}{{N\left( {N - 1} \right)}}\frac{{P\left( {P + 1} \right)\left( {3P^{2} - P - 2} \right)}}{12}. \\ \end{aligned} $$

  Since under H₀, $E\left( {\lambda_{i}^{2} } \right) = \frac{m}{N} \,{\text{and}}\, E\left( {\lambda_{i} \lambda_{j} } \right) = \frac{{m\left( {m - 1} \right)}}{{N\left( {N - 1} \right)}}$ then the result follows. The proofs for $N$ even is very similar. Also the proofs for $V\left( {B_{r} } \right)$ (for odd and even $N$) under $H_{0}$ are very similar and therefore omitted.

• (iii) We prove that under $H_{0} ,\quad COV\left( {T_{p} ,B_{r} } \right) = - \frac{{mnPR\left( {P + 1} \right)\left( {R + 1} \right)}}{{4N^{2} \left( {N - 1} \right)}}\quad {\text{if}}\;N\;{\text{is}}\;{\text{odd}}$. $E\left( {T_{p} B_{r} } \right)$ is needed. Note that

  $$ \begin{aligned} T_{p} B_{r} & = \mathop \sum \limits_{i = N - P + 1}^{N} \left( {i - \left( {N - P} \right)} \right)\lambda_{i} \mathop \sum \limits_{J = 1}^{R} \left( {R - j + 1} \right)\lambda_{j} \\ & = \mathop \sum \limits_{i = N - P + 1}^{N} \mathop \sum \limits_{J = 1}^{R} \left( {\left( {R + 1} \right)i + \left( {N - P} \right)j - ij - \left( {N - P} \right)\left( {R + 1} \right)} \right)\lambda_{i} \lambda_{j} . \\ \end{aligned} $$

  Therefore

  $$ E\left( {T_{p} B_{r} } \right) = \mathop \sum \limits_{i = N - P + 1}^{N} \mathop \sum \limits_{J = 1}^{R} \left( {\left( {R + 1} \right)i + \left( {N - P} \right)j - ij - \left( {N - P} \right)\left( {R + 1} \right)} \right)E\left( {\lambda_{i} \lambda_{j} } \right). $$

  Since under $H_{0} \;E\left( {\lambda_{i} \lambda_{j} } \right) = \frac{{m\left( {m - 1} \right)}}{{N\left( {N - 1} \right)}}$ then

  $$ E\left( {T_{p} B_{r} } \right) = \frac{{m\left( {m - 1} \right)PR\left( {P + 1} \right)\left( {R + 1} \right)}}{{4N\left( {N - 1} \right)}}, $$

  and consequently,

  $$ COV\left( {T_{p} ,B_{r} } \right) = - \frac{{mnPR\left( {P + 1} \right)\left( {R + 1} \right)}}{{4N^{2} \left( {N - 1} \right)}}. $$

  The proof for $N$ even is very similar and therefore omitted.

Lemma 2

Under $H_{0}$ first and second moments of $W$ and $S$ are given by

• (i)
  $$ \mu_{W} = E\left[ W \right] = E\left[ {T_{p} } \right] - E\left[ {B_{r} } \right] = \left\{ {\begin{array}{*{20}l} {\frac{{m\left( {P - R} \right)\left( {P + R + 1} \right)}}{2N}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{m\left( {P - R} \right)\left( {P + R} \right)}}{2N}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right. $$
  $$ \mu_{S} = E\left[ S \right] = E\left[ {T_{p} } \right] + E\left[ {B_{r} } \right] = \left\{ {\begin{array}{*{20}l} {\frac{{m\left( {P^{2} + R^{2} + P + R} \right)}}{2N}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{m\left( {P^{2} + R^{2} } \right)}}{2N}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right. $$
• (ii)
  $$ \begin{aligned} \sigma_{W}^{2} & = Var\left[ W \right] = Var\left[ {T_{p} } \right] + Var\left[ {B_{r} } \right] - 2Cov\left( {T_{p} ,B_{r} } \right) \\ & = \left\{ {\begin{array}{*{20}l} {\frac{{mn\left\{ {\left( {4N - 6} \right)\left( {P^{3} + R^{3} } \right) + \left( {6N - 3} \right)\left( {P^{2} + R^{2} } \right) + 2N\left( {P + R} \right) - 3\left( {P^{4} + R^{4} } \right) + 6PR\left( {P + 1} \right)\left( {R + 1} \right)} \right\}}}{{12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{mn\left\{ {\left( {4N - 6} \right)\left( {P^{3} + R^{3} } \right) - 3\left( {P^{4} + R^{4} } \right) - N\left( {P + R} \right) + 6P^{2} R^{2} } \right\}}}{{12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$
  $$ \begin{aligned} \sigma_{S}^{2} & = Var\left[ S \right] = Var\left[ {T_{p} } \right] + Var\left[ {B_{r} } \right] + 2Cov\left( {T_{p} ,B_{r} } \right) \\ & = \left\{ {\begin{array}{*{20}l} {\frac{{mn\left\{ {\left( {4N - 6} \right)\left( {P^{3} + R^{3} } \right) + \left( {6N - 3} \right)\left( {P^{2} + R^{2} } \right) + 2N\left( {P + R} \right) - 3\left( {P^{4} + R^{4} } \right) - 6PR\left( {P + 1} \right)\left( {R + 1} \right)} \right\}}}{{12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{mn\left\{ {\left( {4N - 6} \right)\left( {P^{3} + R^{3} } \right) - 3\left( {P^{4} + R^{4} } \right) - N\left( {P + R} \right) - 6P^{2} R^{2} } \right\}}}{{12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$
• (iii)
  $$ \begin{aligned} Cov\left( {W,S} \right) & = \sigma_{WS} = Var\left[ {T_{p} } \right] - Var\left[ {B_{r} } \right] \\ & = \left\{ {\begin{array}{*{20}l} {\frac{{mn\left\{ {\left( {4N - 6} \right)\left( {P^{3} - R^{3} } \right) + \left( {6N - 3} \right)\left( {P^{2} - R^{2} } \right) + 2N\left( {P - R} \right) - 3\left( {P^{4} - R^{4} } \right)} \right\}}}{{12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{odd}}} \hfill \\ {\frac{{mn\left\{ {4N\left( {P^{3} - R^{3} } \right) - N\left( {P - R} \right) - 3\left( {P^{4} - R^{4} } \right)} \right\}}}{{ 12N^{2} \left( {N - 1} \right)}}} \hfill & {{\text{when}}\;N\;{\text{is}}\;{\text{even}}} \hfill \\ \end{array} } \right.. \\ \end{aligned} $$

Proof

Straightforward using results in Lemma 1.

To prove Theorem 1 we use the following Lemma 3.

Lemma 3

As$N \to \infty$, under H₀

$$ W^{*} = \frac{{\left( {W - \mu_{W} } \right)}}{{\sigma_{W} }}\mathop \to \limits^{d}N\left( {0,1} \right)\quad {\text{and}}\quad S^{*} = \frac{{\left( {S - \mu_{S} } \right)}}{{\sigma_{S} }}\mathop \to \limits^{d}N\left( {0,1} \right) $$

where$\mathop \to \limits^{d}$stands for convergence in distribution.

Proof

Note that in either case the score functions $U_{W} \left( u \right)$ and $U_{S} \left( u \right)$ corresponding to W and S respectively are continuous, differentiable (except at two points) in the interval [0,1], square integrable and are bounded. Therefore we can apply the Chernoff-Savage (1958) theorem and prove the lemma. Two special cases, one with $p = r = 0.25$ and the other with $p = r = 0.5$ are discussed in Büning and Thadewald (2000).

Proof of Theorem 1

Consider the linear combination $a_{1} W^{*} + a_{2} S^{*}$ where $a_{1}$ and $a_{2}$ are two real numbers. It is $a_{1} W^{*} + a_{2} S^{*} = \left( {\frac{{a_{1} }}{{\sigma_{W} }}W + \frac{{a_{2} }}{{\sigma_{S} }}S} \right) - \left( {\frac{{a_{1} \mu_{W} }}{{\sigma_{W} }} + \frac{{a_{2} \mu_{S} }}{{\sigma_{S} }}} \right) = \left( {\frac{{a_{1} }}{{\sigma_{W} }} + \frac{{a_{2} }}{{\sigma_{S} }}} \right)T_{p} + \left( {\frac{{a_{2} }}{{\sigma_{S} }} - \frac{{a_{1} }}{{\sigma_{W} }}} \right)B_{r} - \left( {\frac{{a_{1} \mu_{W} }}{{\sigma_{W} }} + \frac{{a_{2} \mu_{S} }}{{\sigma_{S} }}} \right),$ and the corresponding score function can be expressed as

$$ U_{{a_{1} W^{*} + a_{2} S^{*} }} \left( u \right) = \left\{ {\begin{array}{*{20}l} {\left( {\frac{{a_{2} }}{{\sigma_{S} }} - \frac{{a_{1} }}{{\sigma_{W} }}} \right)\left[ {u - r} \right]} \hfill &\quad {{\text{if}}\quad 0 < u < r} \hfill \\ 0 \hfill &\quad {{\text{if}}\quad r \le u \le \left( {1 - p} \right)} \hfill \\ {\left( {\frac{{a_{1} }}{{\sigma_{W} }} + \frac{{a_{2} }}{{\sigma_{S} }}} \right)\left[ {u - \left( {1 - p} \right)} \right]} \hfill &\quad {{\text{if}}\quad \left( {1 - p} \right) < u < 1} \hfill \\ \end{array} } \right.. $$

Here also, the score function is continuous, differentiable (except at two points) in the interval [0,1], square integrable and bounded and therefore the Chernoff-Savage theorem can be applied to conclude that as $N \to \infty$, for any arbitrary real numbers $a_{1}$ and $a_{2}$, under H₀

$$ a_{1} W^{*} + a_{2} S^{*} \mathop \to \limits^{d} N\left( {0, a_{1}^{2} + a_{2}^{2} + 2a_{1} a_{2} \rho_{WS} } \right) $$

where $\rho_{WS} = \frac{{\sigma_{WS} }}{{\sigma_{W} \sigma_{S} }}.$ Since an arbitrary linear combination of $W^{*}$ and $S^{*}$ converges in distribution to the normal, then using the Cramér-Wold device we conclude that the joint distribution of $W^{*}$ and $S^{*}$ is a bivariate normal with mean vector ${\mathbf{0}} = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right)$ and dispersion matrix $\varSigma = \left( {\begin{array}{*{20}c} 1 & {\rho_{WS} } \\ {\rho_{WS} } & 1 \\ \end{array} } \right)$. Noting that

$$ \mathop {\lim }\limits_{n \to \infty } f\left( {W^{*} , S^{*} } \right) = \frac{1}{{2\pi \sqrt {1 - \rho_{WS}^{2} } }}e^{{ - \frac{{PML\left( {p,r} \right)}}{2}}} $$

where $f(W^*,S^*)$ denotes the density function of $\left( {W^{*} , S^{*} } \right)$, we conclude that $PML\left( {p,r} \right)$ is asymptotically distributed as a Chi squared with 2 d.f.

Proof of Theorem 2

We follow Kössler (2006). Suppose $\theta = \left( {\mu ,\vartheta } \right)$ and the c.d.f. $F$ twice continuously differentiable. Using Taylor expansion, and neglecting higher order terms in $\left\| {\theta } \right\|$; we have

$$ \begin{aligned} F\left( {x;\mu ,\vartheta } \right) \approx F\left( {\frac{x - \mu }{{e^{\vartheta } }}} \right) & = F\left( {x; 0, 0} \right) + \left. {\mu \frac{\partial }{\partial \mu }F\left( {x;\mu ,\vartheta } \right)} \right|_{\theta = 0} + \left. {\vartheta \frac{\partial }{\partial \vartheta }F\left( {x;\mu ,\vartheta } \right)} \right|_{\theta = 0} \\ & = F\left( {x; 0, 0} \right) - \mu f\left( x \right) - \vartheta xf\left( x \right) \\ \end{aligned} $$

Now, asymptotic normality of W and S follows from the Chernoff-Savage theorem (e.g. Puri and Sen (1971, Sect. 3.6)) and the expectations of $W$ and $S$, are given by:

$$ \begin{aligned} E\left( W \right) & = \mu_{W} + m\mathop \int \limits_{ - \infty }^{\infty } U_{W} \left( {\frac{m}{N}F\left( x \right) + \frac{n}{N}F\left( {\frac{{x - \mu_{N} }}{{e^{{\vartheta_{N} }} }}} \right)} \right)dF\left( x \right) \\ & \approx \mu_{W} + m\left[ {\mathop \int \limits_{0}^{1} U_{W} \left( u \right)du - \frac{n}{N}\mathop \int \limits_{ - \infty }^{\infty } U_{W}^{'} \left( {F\left( x \right)) f\left( x \right)\left( {\mu_{N} + \vartheta_{N} x} \right)} \right)dF\left( x \right)} \right] \\ & = \mu_{W} - \frac{mn}{N}\mathop \int \limits_{0}^{1} U_{W}^{'} \left( u\right) f\left( {F^{ - 1} \left( u \right)} \right)\left(\mu_{N} + \vartheta_{N} F^{ - 1} \left( u \right) \right)du \\ & = \mu_{W} - \frac{mn}{N}\left[ {\mu_{N} d_{W} \left( {f,g_{1} } \right) + \vartheta_{N} d_{WS} \left( {f,g_{1} } \right)} \right], {\text{because }} {U_{W}^{'} \left( u \right) = 1.} \\ \end{aligned} $$

Similarly, we can show that

$$ E\left( S \right) = \mu_{S} - \frac{mn}{N}\left[ {\mu_{N} d_{SW} \left( {f,g_{2} } \right) + \vartheta_{N} d_{S} \left( {f,g_{2} } \right)} \right]. $$

Further, the variance of $W$ and $S$ are given by

$$ \sigma_{W}^{2} = \frac{mn}{N}I_{W} = \frac{mn}{N}\frac{{2\eta^{3} }}{3}\quad {\text{and}}\quad \sigma_{S}^{2} = \frac{mn}{N}I_{S} = \frac{mn}{N}\left( {\frac{{2\eta^{3} }}{3} - \eta^{4} } \right). $$

Since the score functions are orthogonal for $p = r = \eta ;$ the proof follows.