# Rank test for scale parameter for asymmetrical one-sided distributions

Mohamed A. H. Taha

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## To cite this version:

Mohamed A. H. Taha. Rank test for scale parameter for asymmetrical one-sided distributions. Annales de l'ISUP, 1964, XIII (3), pp.169-180. hal-04085215

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Distributed under a Creative Commons Attribution 4.0 International License
# RANK TEST FOR SCALE PARAMETER FOR ASYMMETRICAL ONE-SIDED DISTRIBUTIONS

Mohamed A.H. TAHA

## 0. Introduction

Much attention has been paid to the problem of testing the variability of dispersion in the two-sample problem. Besides the standard F-test, a number of rank tests were suggested to overcome its sensitivity for asymmetry and skewness like those of Mood, Freund and Ansary and that of Siegel and Tukey... etc. Unfortunately, most of these tests, if not all of them, are in turn sensitive to asymmetry and skewness. Another lack was the necessity of some knowledge about the relative location of the parent distributions. However Moses [4] pointed out that it is of no use to try to construct tests for dispersion without such restrictions. At the same time Tamura [6] showed that we can derive more powerful tests with some simple modifications in the known tests.

It is natural then to ask if it is not possible to construct better tests for dispersion if we restrict ourselves to some class of distributions. The class of one-sided asymmetrical distribution functions which is of considerable importance is very easy to identify in the praxis. A typical example of it is the Γ-distribution with small values of shape parameter, a special case of which is the exponential distribution.

Against scale variability of this class of distribution functions we can use rank tests of location such as Wilcoxon's test, because with a transformation of the form

$$u = \log x \qquad (0,1)$$

which leaves the distribution of the ranks unaltered, the scale parameter will be transformed into a location parameter. It seems more reasonable to try here to construct more powerful tests than those we know for dispersion.
Here we suggest a test, which can be considered as a modification of Wilcoxon's test as well as that of Mood and which shows a considerable power as compared to both of the previous ones for this special alternative.

## 1. The L-Test

Let

$X_1, X_2, X_3, \dots, X_m$

$Y_1, Y_2, Y_3, \dots, Y_n$

be two samples of sizes m, n (m+n = s) which are known to be coming from populations of continuous one-sided distributions. (we will restrict ourselves to non-negative distributions, without any loss of generality).

Let the parent distributions of the X's and the Y's be F(x), and G(x) respectively. The problem is to test the Null-Hypothesis

$$H_0: G(x) = F(x) \qquad (1.1)$$

against the alternative

$$H_1: G(x) = F(\beta x), \quad \beta < 1 \qquad (1.2)$$

The two samples will then be rearranged in one sequence :

$$Z_1 \le Z_2 \le Z_3 \le \dots \le Z_s \qquad (1.3)$$

Let R(yᵢ) be the rank of the i-th Y in the sequence 1, 3 of the ordered combined sample and let

$$L = \sum_{i=1}^{n} R^{2}(y_{i}) \qquad (1.4)$$

The hypothesis 1.2 is to be accepted if L is too great.

## 2. The Distribution of the L-Statistic

The distribution of this criterion for finite n, m is not very easy to be written explicitly. However its frequency generating function which is
$$ \theta(x, y; t) = \prod_{i=1}^{s} (y_i - t_i^2 + x_i) \quad (2.1) $$

enables us to get many useful relations concerning its distribution. Let $f_{n,m}(L)$ be the coefficient of $t^l$ in the coefficient of $y^n \cdot x^m$ in the expansion of 2.1, then

$$ P_{n,m}(L) = f_{n,m}(L) / \binom{s}{n} \quad (2.2) $$

and it can be easily shown that

$$ f_{n,m}(L) = f_{n-1,m}(L-s^2) + f_{n,m-1}(L) \quad (2.3) $$

where

$$ f_{o,m}(O) = 1; $$

$$ f_{n,o}(O) = 0 \text{ for } n = 0; $$

$$ f_{n,m}(L) = 0 \text{ for } L < \sum_{i=1}^{n} j_i^2 \text{ or } L > \sum_{j=m+1}^{s} j_j^2. $$

Differentiating 2.1 with respect to $t$, putting $t = 1$ picking out the coefficient of $y^n \cdot x^m$, we can show that

$$ E(L) = n(s+1) (2s+1)/6 \quad (2.4) $$

Similarly we can show that

$$ V(L) = nm(s+1) (2s+1) (8s+11)/180 \quad (2.5) $$

$$ E\{(L-E(L))^3\} = nm(m-n) (s+1) (s+2) (2s+1) (32s+47)/3780 \quad (3.6) $$

$$ E\{(L-E(L))^4\} = nm(s+1) (2s+1) \left\{ \begin{aligned} & nm(896s^4+3552s^3+3014s^2-3327s-4315) - 2s(s+1) (128s^3+250s^2-104s-289) \end{aligned} \right\} / 75600 \quad (2.7) $$

which give

$$ \gamma_1 = \mu_3 / \mu_2^{3/2} = \frac{2 \sqrt{5} (m-n) (s+2) (32s+47)}{7(8s+11) \sqrt{nm(s+1)(2s+1)(8s+11)}} \quad (2.8) $$

and ;

$$ \gamma_2 = \mu_4 / \mu_2^2 - 3 = -6 \left\{ \frac{(128s^3+1412s^2+3550s+2581)}{(s+1)} + s \frac{(128s^3+250s^2-104s-289)}{(nm)} \right\} / 7(2s+1)(8s+11)^2 \quad (2.9) $$
About the asymptotic behaviour of L we can see by applying theorem (1) of Chernoff and Savage [1], that L is asymptotically normally distributed, when we assume that m/n tends to a finite constant different from zero.

In the notation of Chernoff and Savage (which is slightly modified to meet our own notation) let $F_m(x)$ and $G_n(x)$ be the empirical distribution functions of the X's and the Y's respectively and

$$H_s(x) = \lambda_s F_m(x) + (1 - \lambda_s) G_n(x) \quad (2.10)$$

$$\text{where} \qquad \lambda_s = m/(n+m)$$

$$\text{and} \qquad H(x) = \lambda_s F(x) + (1 - \lambda_s) G(x). \quad (2.11)$$

Let there exist $\lambda_o$ such that

$$0 < \lambda_o \le \lambda_s \le (1 - \lambda_o) < 1 \quad (2.11)$$

Theorem (1) of Chernoff and Savage states that the statistic

$$T_s = \int J_s(H_s(x)) dG_m(x) \quad (2.12)$$

where $J_s$ need be defined only at $1/s, 2/s, \dots, (s-1)/s$, but may have its domain of definition extended to $[0, 1]$ by a suitable convention and $I_s$ is the interval $\{H_s(x) : 0 < H_s(x) < 1\}$, this statistic is asymptotically distributed if

(i) $J(H) = \lim J_s(H)$ exists for $0 < H < 1$ and is not constant,

$$(ii) \int [J_s(H_s) - J(H)] dG_m(x) = o_p(s^{-1/2})$$

$$(iii) J_s(1) = o(s^{1/2})$$

$$(iv) |J^{(i)}(H)| = |d^i J/dH^i| < K [H(1-H)]^{-i-\frac{1}{2}\delta} \quad \text{for } i = 0, 1, 2 \text{ and some } \delta > 0$$

$$\text{If we define} \qquad T_s = L/ns^2 \quad (2.15)$$

$$\text{then we see that} \qquad T_s = \int H_s^2(x) dG_n(x) \quad (2.16)$$

$$\text{that is} \qquad J_s(H) = H_s^2 \quad (2.17)$$
The condition 2.12 is satisfied by the assumption that m/n tends to a finite limit different from zero, and for such a simple function 2.17 we can easily show that all the four conditions of Chernoff and Savage are satisfied, from which we can deduce the asymptotic normality of the L-statistic.

### 3. Relation to Wilcoxon's and Mood's Tests

As we have pointed out, this test can be considered as a modified Wilcoxon's test as well as a modified one of Mood. Indeed, it is a linear combination of both of them.

Let

$$M_c = M - E(M) \quad (3.1a)$$

$$W_c = W - E(W) \quad (3.1b)$$

$$L_c = L - E(L) \quad (3.1c)$$

where M, W and L are Mood's, Wilcoxon's and our statistics respectively. Then we find that

$$L_c = M_c + (s+1) W_c \quad (3.2)$$

It can be also shown that

$$R(L, W) = \sqrt{15(s+1)/(2s+1)} \quad (8s+11) \quad \text{which tends to } \sqrt{15}/4 \quad (3.3)$$

$$R(L, M) = \sqrt{(s+2)(s-2)/(2s+1)} \quad (8s+11) \quad \text{which tends to } 1/4 \quad (3.5)$$

while

$$R(W, M) = 0 \quad (3.5)$$

### 4. Comparison with other tests

As a criterion for comparison with other tests we adopt the Pitman's asymptotic efficiency $e_{x,y}$, which is given by

$$e_{x,y} = \lim P_x / P_y \quad (4.1)$$

where

$$P = \frac{\partial E(x, \theta)}{\partial \theta} \Big|_{\theta=\theta_o} \quad \pm \quad \sqrt{V(x, \theta_o)} \quad (4.2)$$

Now we can write the L-statistic in the form
$$L = \sum_{i=1}^{n} \left\{ \sum_{j=1}^{m} (z_{ij} + i) \right\}^2 \quad (4.3)$$

$$\text{where} \quad z_{ij} = \begin{cases} 1 & \text{if } Y_i > X_j \\ 0 & \text{otherwise} \end{cases}$$

and i is the rank of the Y in its own sample.

Let g(y) be the density function of the Y's and f(x) that of the X's, we have

$$\begin{aligned} E(z_i^2) &= \iint \int z_{ij}^2 g(y_i) f(x_j) \, dy \, dx \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{x} f(x_j) g(y_i) \, dy_i \, dx_i = \int F \, dG \end{aligned} \quad (4.4)$$

$$\begin{aligned} E(z_{ij} z_{ik}, j \neq k) &= \iint \int z_{ij} z_{ik} g(y_i) f(x_k) \, dy_i \, dx_j \, dx_k \\ &= \int F^2 \, dG. \end{aligned} \quad (4.5)$$

To calculate the expected value of $\sum_i \sum_j z_{ij}$, we have to notice that it depends upon the order of the $y_i$, that is

$$g(y_i) = i \binom{n}{i} G^{i-1} (1-G)^{n-i} \, dG \quad (4.6)$$

This gives

$$E(iz_{ij}) = \iint z_{ij} f(x_j) i^2 \binom{n}{i} G^{i-1} (1-G)^{n-i} \, dG \, dx_j$$

$$\begin{aligned} \sum_i \sum_j E(iz_{ij}) &= m \sum_i i^2 \binom{n}{i} \int F G^{i-1} (1-G)^{n-i} \, dG \\ &= m \int F \left\{ \sum_i i^2 \binom{n}{i} G^{i-1} (1-G)^{n-i} \right\} \, dG \\ &= m \int F \{n + n(n-1)H\} \, dG \\ &= mn \int F \, dG + mn(n-1) \int FG \, dG \end{aligned} \quad (4.7)$$
From 4.4 to 4.7 we can conclude that

$$E(L/H_1) = n(n+1)(2n+1)/6 + nm \left\{ (m-1) \int F^2 dG + 2(n-1) \int FG dG + 3 \int F dG \right\} \quad (4.8)$$

when $F = G$, we get directly 2.4.

For the W-statistic we have

$$E(W/H_1) = n(n+1)/2 + nm \int F dG \quad (4.9)$$

and for the M-statistic, we have

$$E(M/H_1) = k + nm \left\{ (m-1) \int F^2 dG + 2(n-1) \int FG dG + (s-2) \int F dG \right\} \quad (4.10)$$

Considering the alternative 1.2 and letting N stand for either n or m, we see that

$$P_L = \left\{ \sqrt{45N} \int xFf^2 dx \right\} (1 + O(1/N))$$

$$P_W = \left\{ \sqrt{12N} \int xf^2 dx \right\} (1 + O(1/N))$$

$$P_M = \left\{ \sqrt{45N}/2 \left( 2 \int xFf^2 dx - \int xf^2 dx \right) \right\} (1 + O(1/N))$$

For non-negative distributions, it is quite evident that $P_M < P_L$ and hence we can conclude directly that Mood's test is less efficient than the L-test, so we have only to compare the L-test with that of Wilcoxon.

For the $\Gamma$-distribution with shape parameter $p$ we have

$$e_{L,W} = \sqrt{15/16} \left\{ 2^{1-p} / \Gamma(p) \Gamma(2p) \right\} \sum_{r=0}^{\infty} \left( -1 \right)^r \frac{\Gamma(3p+r)}{2^r r!(p+r)}$$

or for integer $p$;

$$e_{L,W} = 2 \sqrt{15/16} \left\{ 1 - (2/3)^{2p} \sum_{r=0}^{p-1} \frac{\Gamma(2p+r)}{\Gamma(2p)3^r r!} \right\}$$

This is calculated and plotted in figure 1 for $0 < p < 11$.
<figure><img src="image_3.png" /><figcaption>Fig. 1 - The relative efficiency of the L-Test compared to the Wilcoxon-Test for $\Gamma$-distributions with shape-parameter $p$.</figcaption></figure>
As an example we bring here all the numerical results for the case $p = 1$ (exponential distribution).

$$e_{L,W} = 1,076$$

$$e_{L,M} = 2,5$$

$$e_{L,F} = 1,318$$

It should be noticed that for this distribution, Mood's test is considerably weaker than the standard F-test; indeed

$$e_{M,F} = 0,527$$

## 5. Application of the Test

As we said before, the distribution of the statistic L tends to the normal distribution as n, m grow, provided that m/n tends to a finite limit. However while $\gamma_1$ vanishes rapidly, $\gamma_2$ tends slowly to zero with $(1/n + 1/m)$.

Here we have given the critical values of the statistic for $3 < n, m < 8$. For great values of n, m we can apply the normal approximation, for moderate values of n, m it would be better to use an Edgworth-series of the form :

$$F(x) = \Phi(x) - \frac{\gamma_1}{6} \Phi^{(3)}(x) + \frac{\gamma_2}{24} \Phi^{(4)}(x) + \dots$$

where $\Phi^{(i)}(x)$ is the i-th differential for the normal function.

$$\Phi(x) = 1/\sqrt{2\pi} \int_{-\infty}^{x} e^{-\frac{x^2}{2}} dx$$

with two or three terms. To give an idea about how good such an approximation can be, we give here in table II some critical values of the statistic for

$$n=m=8; n=8, m=7; n=m=7$$

with their real probabilities and under I the normal approximation under II the Edgworth approximation with two terms and with three terms under III.

It is clear from the table that the normal approximation is reasonable even for such small values of n, m especially for values of n which are nearly equal to m.

**Acknowledgement.** The author gratefully acknowledges the valuable suggestions and criticisms of the referee.
<table><caption>Table I
Critical values of the L-Test</caption><thead><tr><th>n</th><th>m</th><th>0,010</th><th>0,025</th><th>0,050</th><th>0,100</th><th>0,900</th><th>0,950</th><th>0,975</th><th>0,990</th></tr></thead><tbody><tr><td rowspan="5">3</td><td>3</td><td>---</td><td>---</td><td>14</td><td>21</td><td>70</td><td>77</td><td>---</td><td>---</td></tr><tr><td>4</td><td>---</td><td>---</td><td>14</td><td>26</td><td>94</td><td>110</td><td>---</td><td>---</td></tr><tr><td>5</td><td>---</td><td>14</td><td>21</td><td>30</td><td>122</td><td>138</td><td>149</td><td>---</td></tr><tr><td>6</td><td>---</td><td>21</td><td>29</td><td>41</td><td>154</td><td>166</td><td>181</td><td>---</td></tr><tr><td>7</td><td>14</td><td>26</td><td>35</td><td>49</td><td>185</td><td>200</td><td>217</td><td>245</td></tr><tr><td rowspan="5">4</td><td>8</td><td>14</td><td>29</td><td>41</td><td>56</td><td>222</td><td>245</td><td>266</td><td>302</td></tr><tr><td>3</td><td>---</td><td>---</td><td>30</td><td>46</td><td>114</td><td>126</td><td>---</td><td>---</td></tr><tr><td>4</td><td>---</td><td>---</td><td>30</td><td>46</td><td>147</td><td>158</td><td>174</td><td>---</td></tr><tr><td>5</td><td>30</td><td>46</td><td>54</td><td>70</td><td>185</td><td>198</td><td>210</td><td>230</td></tr><tr><td>6</td><td>39</td><td>51</td><td>65</td><td>84</td><td>225</td><td>249</td><td>261</td><td>281</td></tr><tr><td rowspan="5">5</td><td>7</td><td>46</td><td>62</td><td>75</td><td>98</td><td>271</td><td>295</td><td>315</td><td>338</td></tr><tr><td>8</td><td>50</td><td>70</td><td>87</td><td>113</td><td>323</td><td>351</td><td>378</td><td>410</td></tr><tr><td>3</td><td>---</td><td>---</td><td>55</td><td>66</td><td>82</td><td>174</td><td>183</td><td>190</td><td>---</td></tr><tr><td>4</td><td>55</td><td>75</td><td>87</td><td>100</td><td>120</td><td>215</td><td>231</td><td>239</td><td>255</td></tr><tr><td>5</td><td>66</td><td>87</td><td>100</td><td>120</td><td>265</td><td>285</td><td>298</td><td>319</td></tr><tr><td rowspan="5">6</td><td>6</td><td>79</td><td>99</td><td>118</td><td>139</td><td>322</td><td>343</td><td>367</td><td>387</td></tr><tr><td>7</td><td>90</td><td>111</td><td>134</td><td>162</td><td>380</td><td>411</td><td>435</td><td>462</td></tr><tr><td>8</td><td>100</td><td>129</td><td>154</td><td>186</td><td>446</td><td>480</td><td>510</td><td>547</td></tr><tr><td>3</td><td>---</td><td>---</td><td>104</td><td>119</td><td>131</td><td>244</td><td>256</td><td>264</td><td>---</td></tr><tr><td>4</td><td>104</td><td>124</td><td>136</td><td>160</td><td>301</td><td>320</td><td>334</td><td>346</td></tr><tr><td rowspan="5">7</td><td>5</td><td>119</td><td>139</td><td>163</td><td>184</td><td>367</td><td>388</td><td>407</td><td>427</td></tr><tr><td>6</td><td>136</td><td>163</td><td>184</td><td>214</td><td>436</td><td>466</td><td>487</td><td>514</td></tr><tr><td>7</td><td>154</td><td>182</td><td>210</td><td>246</td><td>512</td><td>546</td><td>575</td><td>608</td></tr><tr><td>8</td><td>171</td><td>207</td><td>238</td><td>279</td><td>593</td><td>635</td><td>671</td><td>709</td></tr><tr><td>3</td><td>140</td><td>168</td><td>185</td><td>200</td><td>336</td><td>350</td><td>359</td><td>371</td></tr><tr><td rowspan="5">8</td><td>4</td><td>168</td><td>191</td><td>211</td><td>235</td><td>408</td><td>431</td><td>444</td><td>460</td></tr><tr><td>5</td><td>188</td><td>215</td><td>239</td><td>270</td><td>488</td><td>516</td><td>539</td><td>560</td></tr><tr><td>6</td><td>211</td><td>244</td><td>273</td><td>307</td><td>573</td><td>609</td><td>637</td><td>665</td></tr><tr><td>7</td><td>235</td><td>273</td><td>307</td><td>349</td><td>666</td><td>708</td><td>742</td><td>780</td></tr><tr><td>8</td><td>259</td><td>304</td><td>343</td><td>392</td><td>765</td><td>815</td><td>857</td><td>901</td></tr><tr><td rowspan="5">8</td><td>3</td><td>204</td><td>240</td><td>261</td><td>284</td><td>450</td><td>465</td><td>477</td><td>493</td></tr><tr><td>4</td><td>240</td><td>272</td><td>299</td><td>327</td><td>537</td><td>563</td><td>580</td><td>600</td></tr><tr><td>5</td><td>272</td><td>309</td><td>339</td><td>373</td><td>633</td><td>665</td><td>690</td><td>719</td></tr><tr><td>6</td><td>306</td><td>344</td><td>380</td><td>422</td><td>736</td><td>777</td><td>808</td><td>844</td></tr><tr><td>7</td><td>339</td><td>380</td><td>425</td><td>475</td><td>848</td><td>897</td><td>936</td><td>981</td></tr><tr><td>8</td><td>371</td><td>424</td><td>472</td><td>530</td><td>966</td><td>1024</td><td>1072</td><td>1125</td></tr></tbody></table>
<table><caption>Table II</caption><thead><tr><th>n</th><th>m</th><th>L</th><th>P(L)</th><th>I</th><th>II</th><th>III</th></tr></thead><tbody><tr><td rowspan="4">8</td><td rowspan="4">8</td><td>966</td><td>0,9008</td><td>0,9047</td><td>0,9047</td><td>0,9015</td></tr><tr><td>1024</td><td>0,9503</td><td>0,9515</td><td>0,9515</td><td>0,9495</td></tr><tr><td>1072</td><td>0,9754</td><td>0,9743</td><td>0,9743</td><td>0,9753</td></tr><tr><td>1125</td><td>0,9901</td><td>0,9883</td><td>0,9883</td><td>0,9900</td></tr><tr><td rowspan="4">8</td><td rowspan="4">7</td><td>848</td><td>0,8991</td><td>0,9059</td><td>0,9061</td><td>0,9026</td></tr><tr><td>897</td><td>0,9514</td><td>0,9516</td><td>0,9519</td><td>0,9514</td></tr><tr><td>936</td><td>0,9753</td><td>0,9735</td><td>0,9737</td><td>0,9748</td></tr><tr><td>981</td><td>0,9905</td><td>0,9878</td><td>0,9880</td><td>0,9900</td></tr><tr><td rowspan="4">7</td><td rowspan="4">7</td><td>666</td><td>0,9009</td><td>0,9061</td><td>0,9061</td><td>0,9044</td></tr><tr><td>708</td><td>0,9513</td><td>0,9520</td><td>0,9520</td><td>0,9516</td></tr><tr><td>742</td><td>0,9752</td><td>0,9742</td><td>0,9742</td><td>0,9754</td></tr><tr><td>780</td><td>0,9904</td><td>0,9881</td><td>0,9881</td><td>0,9902</td></tr></tbody></table>

REFERENCES

[1] CHERNOFF (H.) and SAVAGE (I.R.) : Asymptotic normality and efficiency of certain nonparametric test-statistics, *Ann. Math. Statist.* 29 (1958), 972-994.

[2] CRAMER (H.) : *Mathematical Methods of Statistics*. Princeton University Press (1951).

[3] MOOD (A.) : On the asymptotic efficiency of certain nonparametric two sample tests, *Ann. Math. Statist.* 25 (1954), 514-522.

[4] MOSES (L.E.) : Rank Tests of Dispersion, *Ann. Math. Statist.* 34 (1963), 973-983.

[5] SCHMETTERER (L.) : *Einfuehrung in die mathematische Statistik*. Wien, Springer Verlag (1956).

[6] ROYJI TAMURA : On a modification of certain Rank Tests, *Ann. Math. Statist.* 34 (1963), 1101-1103.

[7] WILCOXON (F.) : Individual comparisons by ranking methods, *Biometrics* 1 (1954), 80-83.

Faculty of Science

Alexandria University

Alexandria

U.A.R.

Reçu le 28 février 1964

Révisé le 26 juillet 1964

<figure><img src="image_4.png" /></figure>
IMP. LOUIS-JEAN - GAP

Dépôt légal n° 80 - 1964