# On the empirical function

M.A. Taha

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M.A. Taha. On the empirical function. Annales de l'ISUP, 1970, XIX (4), pp.77-90. hal-04083521

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# ON THE EMPIRICAL DISTRIBUTION FUNCTION :

## M.A.H. TAHA*

### 0. INTRODUCTION

Let X to be a random variable (RV) which has a certain continuous distribution function (DF) :

$$P(X \le x) = F(x) \qquad 0.1$$

Let $x_1, x_2, \dots, x_N$ be a series of observations made on this RV. These can be arranged in an ascending order to form so called "ordered sample".

$$-\infty < X_1 < X_2 < X_3 < \dots < X_N < \infty$$

Let

$$U_i = F(X_i), \quad i = 1 (1) N \qquad 0.2$$

to be the integral transformation of these sample points and

$$U_i = U_i - U_{i-1}, \quad i = 1 (1) N \qquad 0.3$$

there successive differences.

In many statistical procedures, we have something to do with the empirical distribution function (EDF). For example in tests of goodness of fit like those of Cramervon Mises-Smirnov, Kolmogorov and of Shermann. Other situation are those of constructing probability-papers or estimating the return period of some event (see Gumbel pp 29).

The most acceptable procedure till now is to define the EDF to be:

$$F_N(x) = \frac{i}{N} X_i \le x < X_{i+1}, \quad i = 1 (1) N \qquad 0.4$$

The theoretical reasoning for this definition is established in Glivenko's work [4] where he proved that

$$P \left\{ \sup_X \left| F_N(x) - F(x) \right| \to 0 \right\} = 1 \qquad 0.5$$

<footnote>* University of Alexandria</footnote>
It can be easily shown that this holds not only for the function defined in 0.4, but for a whole class of EDF (C(S<sub>N</sub>) say) which differs from 0.4 with a value in the order of 1/N.

This class of EDF, although all having the same asymptotic properties, can contain some which have some advantages for small samples.

Our purpose is to find some of those EDF and demonstrate its uses.

Here we meet two problems : firstly, how to construct such a function? and secondly how to choose between them. Gumbel [6] , discussing the problem of plotting an EDF, a situation in which the classical function 0.4 has a quite unhappy position (since in probability paper constructed for unlimited variables there are no point of probability 1 which is attained by this function at X = X<sub>N</sub>) suggested the following rules to judge an EDF.

G.1. The function should be capable of being plotted

G.2. It should be universally acceptable

G.3. Plotting points should lie between 1/N, i/N

G.4. The observation should be equally spaced

G.5. The function should have some intuitive meaning.

These conditions, being suitable for the purpose they are meant for, have not the nature of generality. For example, there seems to be no obvious reason - other than simplicity - why these points should be equally spaced.

Also, there seems to be no obvious reason why these points should lie between (i-1)/N, i/N. Any how, these conditions as they are now presented, are not constructive, i.e. they give no method of finding an EDF.

Aggarwal [1] following a suggestion of Birnbaum [2] obtained some functions which minimize certain risk-functions.

The property of minimizing a risk-function seems with condition G 2- to be good criteria for judging an EDF and in the same time a method of constructing such functions. Condition G 1. of Gumbel should also be considered because of its practical value.
On the other hand it seems impossible to find (even to define) an overall best EDF, since it is hardly possible to count all possible loss-functions, and it is also to be expected that an EDF can only minimize the expectation of some loss function and not all of them simultaneously.

Naturally, such a function should be invariant against linear transformation. This has been noticed by Aggarwal and can be also deduced from G 2. This restrict our discussion to functions of the form

$$S_N(x) = c_1, X_1 \le X < X_{1+1}; i = 1 (1) N \qquad 0.6$$

where the c's are constants.

Aggarwal discussed the loss function

$$W_N = \int |S_N(x) - F(x)|^r Q[F(x)] dF(x) \\ \text{for } Q(u) = \begin{cases} Q_1(u) = 1 \\ Q_2(u) = (u(1-u))^{-1} \end{cases}$$

The interesting case r = 2 gave

$$S_N(x) = \frac{i+1}{N+2} \quad \text{for } Q_1$$

$$\text{and } S_N(x) = \frac{i}{N} \quad \text{for } Q_2$$

These results were obtained also in [9].

In the next 2 sections other 2 minimax procedures are used to obtain an estimate for $S_N(x)$. Fortunately both the two procedures led to the same estimate some of whose properties are then discussed in section 3.

# 1. SHERMANN'S STATISTICS AS A LOSS FUNCTION.

An estimate can be obtained by finding the spaces between sample points which have some optimal properties. Consider the loss-function.

$$L_1 = \frac{1}{2} \sum_{i=1}^{N+1} |F(X_i) - F(X_{i-1}) - d_i| \qquad (1.1)$$
$$ \text{where } d_i = C_i - C_{i-1}, F(X_0) = U_0 = 0 $$

$$ d_{N+1} = C_0 + (1 - C_N) $$

$$ F_{X_{N+1}} = U_{N+1} = 1 $$

The risk with this loss-function i.e. its expectation is given by.

$$ R_1 = E(L_1) = \frac{N!}{2} \iint_{D_0} \dots \iint_{D_{N+1}} \sum_{i=1}^{N+1} |U_i - U_{i-1} - d_i| \quad dU_1 \, dU_2 \, \dots \, dU_N \quad (1.2) $$

$$ \text{where } D_0 \text{ is the domain} \quad 0 \le U_1 < U_2 < \dots < U_N < 1 $$

This is equivalent to

$$ R_1 = \frac{1}{2} N! \iint_{D_1} \dots \iint_{D_{N+1}} \sum_{i=1}^{N+1} |u_i - d_i| \, du_1 \dots du_N \quad (1.3) $$

$$ \text{where } D_1 \equiv \begin{cases} u_1 > 0, & i=1(1)N \\ \sum_{i=1}^{N} u_i < 1 \end{cases} \quad (1.4) $$

Since $u_i$ are all symmetrical and can be permutated between themselves we can write 1-3 as follows.

$$ R_1 = \frac{N!}{2} \sum_{i=1}^{N+1} \iint_{D_1} \dots \iint_{D_{N+1}} |u_i - d_i| \, du_1 \dots du_N \quad (1.5) $$

Evaluating the integral we get

$$ R_1 = \frac{N!}{2} \sum_{i=1}^{N+1} \left\{ \frac{2(1-d_i)^{N+1}}{N+1} + \frac{d_i}{N!} - \frac{1}{(N+1)!} \right\} \quad (1.6) $$
Noticing that the sum of the spaces should be 1, we get.

$$
\begin{align*}
R_1 &= \frac{1}{N+1} \sum_{i=1}^{N+1} (1 - d_i)^{N+1} \\
&= \frac{1}{N+1} \sum_{i=1}^{N} (1 - d_i)^{N+1} + (1 - d_{N+1})^{N+1} \\
&= \frac{1}{N+1} \sum_{i=1}^{N} (1 - d_i)^{N+1} + \left( \sum_{i=1}^{N} d_i \right)^{N+1} \quad (1.7)
\end{align*}
$$

differentiating with respect to every $d_i$ and equating to zero we get
the system of equations

$$
d_k = 1 - \sum_{i=1}^{N} d_i = d_{N+1}, \quad k = 1(1)N \quad (1.8)
$$

which has the unique solution

$$
d_k = \frac{1}{N+1}, \quad k = 1(1)N \tag{1.9}
$$

This value for the d's gives the minimum risk

$$
\min (R_1) = \left( \frac{N}{N+1} \right)^{N+1} \qquad (1.9)
$$

which tends to $e^{-1}$ as $N \to \infty$

It also leads to the estimate

$$
S_N(x) = \frac{i + \epsilon}{N + 1} \qquad X_i \le X < X_{i+1} \tag{1.10}
$$

where $0 \le \epsilon \le 1$

$\epsilon$ can then be chosen to fit practical requirements since it does not
affect the loss function.
## II. THE KOLMOGOROV DISTANCE

Birnbaum and Pyke [3] discussed some statistics related to the Kolmogorov's distance

$$D_N^+ = \sup_x F_N(x) - F(x) \quad (2.1)$$

They proved that in spite of the fact that the maximum deviation is equally probable all over the interval (0.1), it is more likely to occur at the sample point $X_{i+1}$ rather than $X_i$ for all $i = 1, 2... N$.

Since this maximum is bound to occur at one of the sample points, it is then interesting to find the EDF, if there is any, for which the maximum deviation is equally probable at all sample points.

Pyke [7] gave a partial answer to this question. He showed that the EDF

$$S_N(x) = \frac{i}{N+1} \quad (2.2)$$

has this property. There is still the question if this is the only function which satisfies this requirement.

Let

$$D_N^+ = \max_{1 \le i \le N} (C_i - U_i) \quad (2.3)$$

If the maximum deviation occurs at the m-th sample point then

$$D_N^+ = C_m - U_m \quad (2.4)$$

We have now to determine the C's so that

$$P(m=k) = \text{const.} \quad k = 0, 1... N \quad (2.5)$$

This probability is given by

$$\begin{align}
P(m=k) &= P \left\{ (C_k - U_k) \ge C_j - U_j, j \ne k \right\} \nonumber \\
&= N! \iiint_{G_k} \dots \int dU_1 \dots dU_N \quad (2.6)
\end{align}$$
where $G_k$ is the domain :

$$0 < U_1 < U_2 < \dots < U_N < 1 \quad (2.7a)$$

$$U_k - U_j \le C_k - C_j \quad j = 0,1,2,\dots, k-1 \quad (2.7b)$$

$$U_j - U_k \ge C_j - C_k \quad j = k+1,\dots,N \quad (2.7c)$$

2.7a can be written as follows :

$$-U_k \le U_1 - U_k \le U_2 - U_k \le \dots \le U_{k-1} - U_k \le 0 \le U_{k+1} - U_k \le \dots \le 1 - U_k$$

adding 1 to the negative values we get :

$$1 - U_k \le 1 - (U_k - U_1) \le 1 - (U_k - U_2) \le \dots \le 1 - (U_k - U_{k-1}) \le 1$$

that is

$$0 \le U_{k+1} - U_k \le \dots \le U_N - U_k \le 1 - U_k \le 1 - (U_{k-1} - U_1) \le \dots \le 1 - (U_k - U_{k-1}) \le 1 \quad \dots \quad (2.8)$$

using the transformation

$$U_{k+j} - U_k = Z_j, \quad = 1,2,\dots,N-k \quad (2.9) \\ 1 - U_k = Z_{N-k+1}$$

$$1 - (U_k - U_j) = Z_{N-k+1+j}, \quad = 1,2,\dots,k-1$$

The domain $G_k$ will be then

$$0 \le Z_1 \le \dots \le Z_N \le 1 \\ Z_1 \le l_i \quad i = 1,2,\dots,N$$

where $l_i = C_{k+1} - C_k, i = 1,2,\dots,N-k$

$$l_{N-k+j} = 1 - (C_k - C_j) \quad j = 0,1,\dots,k-1 \quad (2.10)$$

The Jacobian of the transformation is $= 1$ and the integral over $G_k$ can then be evaluated with techniques similar to that of Wald and Wolfowitz [10] to give
$$
P (m=k) = \begin{vmatrix}
\binom{N}{0} & \binom{N}{1} & \binom{N}{2} & \cdots & \binom{N}{N} \\
l_1 & 1 & 0 & \cdots & 0 & 0 \\
l_2 & l_2 & 1 & 0 & \cdots & 0 \\
l_3 & l_3 & l_3 & 1 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
l_N & l_N & \binom{N}{1} & \binom{N}{2} & \cdots & \binom{N}{N-1} \\
\end{vmatrix} = P(N,k) \tag{2.11}
$$

we have then to solve the equations

$$
P(N,k) = \text{const.} \quad k = 0,1,\dots N \quad (2.12)
$$

in the variable $d_i = C_i - C_{i-1}$

Because of the symmetry of these events it is easy to show that if
there is a solution then it should be such that all the d's are equal say = d.

In this case the determinant 2.11 can be easily evaluated to give.

$$
\begin{align*}
P(m=k) &= 1 - N + d - \left\{1-(N+1)d\right\} \sum_{r=0}^{k-1} \binom{N}{r} d^r (r+1)^{r-1} \, 1-d(r+1)^{N-(r+1)} \\
&= 1 - dN + \left\{(N+1)d - 1\right\} \sum_{j=1}^{k} \binom{N}{j} \frac{1}{N-j+1} (jd)^{j-1} (1-jd)^{N-j} \tag{2.13}
\end{align*}
$$

Since the d's are none negative this probability will be independant
of k only if the 1st. factor of the second term is zero which means.

$$
d = \frac{1}{N+1} \tag{2.14}
$$

It may be interesting to notice that for $d > \frac{1}{N+1}$ this probability will in-
crease with k and if $d < \frac{1}{N+1}$ it will decrease with k
It is important here to notice that the general form 2.11 is independent of the C's and depends only on their successive difference $d_1$:

Once more we are led to the EDF

$$S_N(x) = \frac{i+\epsilon}{N+1} \quad X_1 \le X < X_{i+1} \quad i = 1 (1) N \quad (2.15)$$

where $0 \le \epsilon \le 1$

It can be proved that this is the case also if we use the negative distance i.e.

$$D_N = \sup_x \{F(x) - F_N(x)\}$$

### III. DISCUSSION.

For choosing a value for $\epsilon$ we meet some problems. Firstly for any value $0 < \epsilon < 1$, the resulting EDF is no more a proper one since it gives positive mass for either $-\infty$ or $+\infty$ or both. If we restrict ourselves to estimators which are proper DF (and this sound quite reasonable) then we have to give $S_N(x)$ the value $\theta$ at some point between 0 and $U_1$ and the value 1 at some point between $U_N$ and 1.

For reasons of symmetry the value $\epsilon = \frac{1}{2}$ seems to be reasonable any how the question will stay open. May be more studies on the efficiency of the above test statistics can help in finding an answer.

The following definition of the EDF is a one which satisfies most of the mentioned demands:
$$
S_N(x) = \begin{cases}
\frac{0_1}{2(N+1)} & \dots\dots\dots \quad x < \frac{1}{2} X_1 \\
\frac{i+\frac{1}{2}}{N+1} & \dots\dots\dots \quad \frac{1}{2} X_1 \le X < X_1 \\
\frac{N+\frac{1}{2}}{N+1} & \dots\dots\dots \quad X_N \le X < X_N + \frac{1}{2} (1-X_N) \\
1 & \dots\dots\dots \quad X_N + \frac{1}{2} (1-X_N) \le X < 1
\end{cases} \quad i = 1, 2, \dots, N \quad (3.1)
$$

This- or the general one 2.15- should be used in the 2 test statics
mentioned. This in fact is one of its merits and fortunately this is quite
possible. Shermann studied his test intensively using this EDF from the
beginning.

For the Kolmogorov statistics this is also possible. The distribu-
tion of the criterium with the EDF can be obtained from a general formula
obtained by Dwass [4] and also by Pyke [7].

Using the general form 2.15 with $\epsilon = \frac{1}{2}$ we get

$$
P(D_N^+ \le \lambda) = 1 - \left\{ \lambda + \frac{1}{2(N+1)} \right\} \sum_{j=0}^{K} \left( \binom{N}{j} \frac{2j+1}{2N+1} + \lambda \right) \stackrel{J=1}{=} \left( \frac{2N-2j+1}{2N+2} - \right) \stackrel{N-J}{=} \\
\text{where } k = \left[ (N+1)(1-\lambda) - \frac{1}{2} \right] \quad \text{for} \quad \frac{1}{2(N+1)} < \lambda \le \frac{2}{N+2} \qquad (3.2)
$$

The critical values of this test are given in table 1 for sample of
5 up to 100 - for more than this the asymptotic formula can be used conservatively.

The formulae

$$
\lim_{N \to \infty} P \left( D_N^+ \le \lambda \right) = 1 - e^{-2(N+1) \left( t^2 - \frac{1}{4N^2} \right)}
$$
$$ \text{where} \qquad t = \frac{N+1}{N} \lambda \qquad (3.3) $$

gives a good approximation even for moderate N (about 30)

### ACKNOWLEDGMENT.

This work was partially supported by the Swiss Federal Institute for Forest Research. I would like also to thank Paul Schmid for many valuable discussions and for reading the manuscript.

## LITERATURE

[1] AGGARWAL (OM.P.) - (1955) Some minimax invariant procedures for estimating a cumulative distribution function. *Ann. Math. Statistics* 26 (450 - 463)

[2] BIRNBAUM (Z.W.) - (1953) Distribution-free tests of fit for continuous functions. *Ann. Math. Statistics* 24 (1 - 8)

[3] BIRNBAUM (Z.W.) and PYKE (R.) - (1958) On some distribution related to the statistic $D_N^+$. *Ann. Math. Statistics* 29 (179 - 187)

[4] DWASS (M.) - (1959) The distribution of a generalized $D_N^+$ statistic. *Ann. Math. Statistics* 30 (1024 - 1028)

[5] GLIVENKO (V.) - (1933) Sulla determinazione emperica delle leggi di probabilità. *Giorn. dell'Istit. degli att., Vol.* 4 (1933), pp. 92-99

[6] GUMBEL (E.J.) - Statistic of extremes

[7] PYKE (R.) - (1959) The supremum and infimum of the poisson process. *Ann. Math. Statistics* 30 (568)
[8] - SHERMAN (B.) - (1950) A random variable related to the spacing of sample values. *Ann. Math. Statistics* 21 (339 - 361)

[9] - STEINHAUS (H.) (1957) The problem of estimation. *Ann. Math. Statistics* 28 (633 - 648)

[10] - WALD (A.) - WOLFOWITZ (J.) - (1939) Confidence limits for continuous distribution functions. *Ann. Math. Statistics* 10 (105)

15 Moustafa Abady Street.

Moharram Bey

ALEXANDRIE

République Arabe d'Egypte
On the empirical distribution function.

<table><caption>Table 1 — Critical values of $D_N^+$ with $F_N(x) = \frac{2i+1}{2N+2}$</caption><thead><tr><th>N/P</th><th>0.001</th><th>0.005</th><th>0.01</th><th>0.025</th><th>0.05</th><th>0.1</th></tr></thead><tbody><tr><td>5</td><td>0.67269</td><td>0.59910</td><td>0.56026</td><td>0.50508</td><td>0.45693</td><td>0.39890</td></tr><tr><td>6</td><td>0.63375</td><td>0.56133</td><td>0.52584</td><td>0.47190</td><td>0.42666</td><td>0.37388</td></tr><tr><td>7</td><td>0.59963</td><td>0.53011</td><td>0.49586</td><td>0.44544</td><td>0.40155</td><td>0.35241</td></tr><tr><td>8</td><td>0.57055</td><td>0.50375</td><td>0.47078</td><td>0.42253</td><td>0.38094</td><td>0.33384</td></tr><tr><td>9</td><td>0.54526</td><td>0.48072</td><td>0.40278</td><td>0.36326</td><td>0.36326</td><td>0.31794</td></tr><tr><td>10</td><td>0.52292</td><td>0.46063</td><td>0.43026</td><td>0.38567</td><td>0.34767</td><td>0.30430</td></tr><tr><td>11</td><td>0.50315</td><td>0.44289</td><td>0.51354</td><td>0.37062</td><td>0.33394</td><td>0.29238</td></tr><tr><td>12</td><td>0.48543</td><td>0.42703</td><td>0.39866</td><td>0.35716</td><td>0.32178</td><td>0.28172</td></tr><tr><td>13</td><td>0.46944</td><td>0.41275</td><td>0.38527</td><td>0.34507</td><td>0.31088</td><td>0.27212</td></tr><tr><td>14</td><td>0.45494</td><td>0.39984</td><td>0.37314</td><td>0.33416</td><td>0.30102</td><td>0.26344</td></tr><tr><td>15</td><td>0.44168</td><td>0.38804</td><td>0.36208</td><td>0.32421</td><td>0.29201</td><td>0.25555</td></tr><tr><td>16</td><td>0.42955</td><td>0.37726</td><td>0.35199</td><td>0.31513</td><td>0.28381</td><td>0.24837</td></tr><tr><td>17</td><td>0.41834</td><td>0.36732</td><td>0.34268</td><td>0.30676</td><td>0.27626</td><td>0.24176</td></tr><tr><td>18</td><td>0.40797</td><td>0.35813</td><td>0.33408</td><td>0.29903</td><td>0.26929</td><td>0.23566</td></tr><tr><td>19</td><td>0.39833</td><td>0.34960</td><td>0.32609</td><td>0.29186</td><td>0.26282</td><td>0.22999</td></tr><tr><td>20</td><td>0.38935</td><td>0.34166</td><td>0.31866</td><td>0.28519</td><td>0.25680</td><td>0.22472</td></tr><tr><td>21</td><td>0.38095</td><td>0.33423</td><td>0.31172</td><td>0.27895</td><td>0.25118</td><td>0.21979</td></tr><tr><td>22</td><td>0.37308</td><td>0.32727</td><td>0.30521</td><td>0.27311</td><td>0.24591</td><td>0.21518</td></tr><tr><td>23</td><td>0.36566</td><td>0.32073</td><td>0.29909</td><td>0.26763</td><td>0.24096</td><td>0.21085</td></tr><tr><td>24</td><td>0.35869</td><td>0.31457</td><td>0.29334</td><td>0.26247</td><td>0.23631</td><td>0.20678</td></tr><tr><td>25</td><td>0.35209</td><td>0.30875</td><td>0.28790</td><td>0.25759</td><td>0.23192</td><td>0.20294</td></tr><tr><td>26</td><td>0.34585</td><td>0.30325</td><td>0.28276</td><td>0.25298</td><td>0.22777</td><td>0.19930</td></tr><tr><td>27</td><td>0.33993</td><td>0.29803</td><td>0.27788</td><td>0.24861</td><td>0.22383</td><td>0.19586</td></tr><tr><td>28</td><td>0.33430</td><td>0.29307</td><td>0.27325</td><td>0.24446</td><td>0.22009</td><td>0.19259</td></tr><tr><td>29</td><td>0.32895</td><td>0.28835</td><td>0.26885</td><td>0.24051</td><td>0.21653</td><td>0.18947</td></tr><tr><td>30</td><td>0.32385</td><td>0.28387</td><td>0.26465</td><td>0.23675</td><td>0.21315</td><td>0.18652</td></tr><tr><td>31</td><td>0.31895</td><td>0.27955</td><td>0.26063</td><td>0.23314</td><td>0.20989</td><td>0.18367</td></tr><tr><td>32</td><td>0.31430</td><td>0.27546</td><td>0.25681</td><td>0.22972</td><td>0.20682</td><td>0.18098</td></tr><tr><td>33</td><td>0.30985</td><td>0.27154</td><td>0.25315</td><td>0.22645</td><td>0.20386</td><td>0.17840</td></tr><tr><td>34</td><td>0.30557</td><td>0.26778</td><td>0.24964</td><td>0.22330</td><td>0.20103</td><td>0.17592</td></tr><tr><td>35</td><td>0.30147</td><td>0.26417</td><td>0.24627</td><td>0.22029</td><td>0.19832</td><td>0.17354</td></tr><tr><td>36</td><td>0.29753</td><td>0.26071</td><td>0.24304</td><td>0.21739</td><td>0.19571</td><td>0.17126</td></tr><tr><td>37</td><td>0.29374</td><td>0.25737</td><td>0.23993</td><td>0.21461</td><td>0.19320</td><td>0.16907</td></tr><tr><td>38</td><td>0.29009</td><td>0.25417</td><td>0.23693</td><td>0.21193</td><td>0.19079</td><td>0.16696</td></tr><tr><td>39</td><td>0.28657</td><td>0.25108</td><td>0.23405</td><td>0.20934</td><td>0.18846</td><td>0.16492</td></tr><tr><td>40</td><td>0.28319</td><td>0.24810</td><td>0.23127</td><td>0.20686</td><td>0.18622</td><td>0.16297</td></tr><tr><td>41</td><td>0.27991</td><td>0.24522</td><td>0.22859</td><td>0.20446</td><td>0.18406</td><td>0.16108</td></tr><tr><td>42</td><td>0.27675</td><td>0.24245</td><td>0.22600</td><td>0.20214</td><td>0.18197</td><td>0.15925</td></tr><tr><td>43</td><td>0.27369</td><td>0.23976</td><td>0.22349</td><td>0.19990</td><td>0.17996</td><td>0.15749</td></tr><tr><td>44</td><td>0.27074</td><td>0.23717</td><td>0.22107</td><td>0.19773</td><td>0.17801</td><td>0.15578</td></tr><tr><td>45</td><td>0.26788</td><td>0.23465</td><td>0.21873</td><td>0.19563</td><td>0.17612</td><td>0.15413</td></tr><tr><td>46</td><td>0.26510</td><td>0.23222</td><td>0.21645</td><td>0.19360</td><td>0.17429</td><td>0.15253</td></tr><tr><td>47</td><td>0.26241</td><td>0.22985</td><td>0.21425</td><td>0.19162</td><td>0.17251</td><td>0.15097</td></tr><tr><td>48</td><td>0.25981</td><td>0.22757</td><td>0.21212</td><td>0.18972</td><td>0.17079</td><td>0.14947</td></tr><tr><td>49</td><td>0.25728</td><td>0.22535</td><td>0.21004</td><td>0.18786</td><td>0.16912</td><td>0.14801</td></tr><tr><td>50</td><td>0.25482</td><td>0.22319</td><td>0.20803</td><td>0.18606</td><td>0.16750</td><td>0.14660</td></tr></tbody></table>
<table><caption>Table 1 (cont.)</caption><thead><tr><th>N/P</th><th>0.001</th><th>0.005</th><th>0.01</th><th>0.025</th><th>0.05</th><th>0.1</th></tr></thead><tbody><tr><td>51</td><td>0.25243</td><td>0.22109</td><td>0.20607</td><td>0.18431</td><td>0.16593</td><td>0.14522</td></tr><tr><td>52</td><td>0.25011</td><td>0.21905</td><td>0.20417</td><td>0.18261</td><td>0.16440</td><td>0.14388</td></tr><tr><td>53</td><td>0.24785</td><td>0.21707</td><td>0.20232</td><td>0.18095</td><td>0.16291</td><td>0.14258</td></tr><tr><td>54</td><td>0.24565</td><td>0.21514</td><td>0.20052</td><td>0.17934</td><td>0.16146</td><td>0.14131</td></tr><tr><td>55</td><td>0.24350</td><td>0.21326</td><td>0.19877</td><td>0.17777</td><td>0.16004</td><td>0.14007</td></tr><tr><td>56</td><td>0.24142</td><td>0.21143</td><td>0.19707</td><td>0.17625</td><td>0.15867</td><td>0.13887</td></tr><tr><td>57</td><td>0.23939</td><td>0.20965</td><td>0.19540</td><td>0.17476</td><td>0.15733</td><td>0.13770</td></tr><tr><td>58</td><td>0.23740</td><td>0.20791</td><td>0.19378</td><td>0.17331</td><td>0.15602</td><td>0.13656</td></tr><tr><td>59</td><td>0.23547</td><td>0.20621</td><td>0.19220</td><td>0.17189</td><td>0.15475</td><td>0.13544</td></tr><tr><td>60</td><td>0.23358</td><td>0.20455</td><td>0.19065</td><td>0.17051</td><td>0.15351</td><td>0.13436</td></tr><tr><td>61</td><td>0.23174</td><td>0.20294</td><td>0.18915</td><td>0.16916</td><td>0.15229</td><td>0.13330</td></tr><tr><td>62</td><td>0.22994</td><td>0.20136</td><td>0.18767</td><td>0.16785</td><td>0.15111</td><td>0.13226</td></tr><tr><td>63</td><td>0.22816</td><td>0.19980</td><td>0.18622</td><td>0.16654</td><td>0.14993</td><td>0.13123</td></tr><tr><td>64</td><td>0.22646</td><td>0.19831</td><td>0.18483</td><td>0.16530</td><td>0.14882</td><td>0.13026</td></tr><tr><td>65</td><td>0.22478</td><td>0.19684</td><td>0.18346</td><td>0.16408</td><td>0.14771</td><td>0.12929</td></tr><tr><td>66</td><td>0.22314</td><td>0.19539</td><td>0.18211</td><td>0.16287</td><td>0.14663</td><td>0.12835</td></tr><tr><td>67</td><td>0.22153</td><td>0.19398</td><td>0.18080</td><td>0.16170</td><td>0.14557</td><td>0.12742</td></tr><tr><td>68</td><td>0.21996</td><td>0.19261</td><td>0.17951</td><td>0.16055</td><td>0.14454</td><td>0.12652</td></tr><tr><td>69</td><td>0.21842</td><td>0.19126</td><td>0.17826</td><td>0.15942</td><td>0.14353</td><td>0.12563</td></tr><tr><td>70</td><td>0.21691</td><td>0.18993</td><td>0.17702</td><td>0.15832</td><td>0.14253</td><td>0.12476</td></tr><tr><td>71</td><td>0.21543</td><td>0.18864</td><td>0.17581</td><td>0.15724</td><td>0.14156</td><td>0.12391</td></tr><tr><td>72</td><td>0.21398</td><td>0.18737</td><td>0.17463</td><td>0.15618</td><td>0.14061</td><td>0.12308</td></tr><tr><td>73</td><td>0.21257</td><td>0.18613</td><td>0.17347</td><td>0.15515</td><td>0.13968</td><td>0.12226</td></tr><tr><td>74</td><td>0.21118</td><td>0.18491</td><td>0.17234</td><td>0.15413</td><td>0.13876</td><td>0.12147</td></tr><tr><td>75</td><td>0.20981</td><td>0.18371</td><td>0.17122</td><td>0.15313</td><td>0.13787</td><td>0.12068</td></tr><tr><td>76</td><td>0.20847</td><td>0.18254</td><td>0.17013</td><td>0.15216</td><td>0.13699</td><td>0.11991</td></tr><tr><td>77</td><td>0.20716</td><td>0.18139</td><td>0.16906</td><td>0.15120</td><td>0.13612</td><td>0.11916</td></tr><tr><td>78</td><td>0.20587</td><td>0.18026</td><td>0.16801</td><td>0.15026</td><td>0.13528</td><td>0.11842</td></tr><tr><td>79</td><td>0.20461</td><td>0.17915</td><td>0.16697</td><td>0.14933</td><td>0.13444</td><td>0.11769</td></tr><tr><td>80</td><td>0.20337</td><td>0.17807</td><td>0.16596</td><td>0.14843</td><td>0.13363</td><td>0.11698</td></tr><tr><td>81</td><td>0.20215</td><td>0.17700</td><td>0.16497</td><td>0.14754</td><td>0.13283</td><td>0.11628</td></tr><tr><td>82</td><td>0.20096</td><td>0.17595</td><td>0.16399</td><td>0.14666</td><td>0.13204</td><td>0.11559</td></tr><tr><td>83</td><td>0.19978</td><td>0.17492</td><td>0.16303</td><td>0.14580</td><td>0.13127</td><td>0.11491</td></tr><tr><td>84</td><td>0.19863</td><td>0.17391</td><td>0.16209</td><td>0.14496</td><td>0.13051</td><td>0.11425</td></tr><tr><td>85</td><td>0.19749</td><td>0.17291</td><td>0.16116</td><td>0.14413</td><td>0.12977</td><td>0.11360</td></tr><tr><td>86</td><td>0.19638</td><td>0.17194</td><td>0.16025</td><td>0.14332</td><td>0.12903</td><td>0.11295</td></tr><tr><td>87</td><td>0.19528</td><td>0.17097</td><td>0.15935</td><td>0.14251</td><td>0.12831</td><td>0.11232</td></tr><tr><td>88</td><td>0.19420</td><td>0.17003</td><td>0.15847</td><td>0.14173</td><td>0.12760</td><td>0.11170</td></tr><tr><td>89</td><td>0.19314</td><td>0.16910</td><td>0.15760</td><td>0.14095</td><td>0.12690</td><td>0.11109</td></tr><tr><td>90</td><td>0.19209</td><td>0.16819</td><td>0.15675</td><td>0.14019</td><td>0.12622</td><td>0.11049</td></tr><tr><td>91</td><td>0.19107</td><td>0.16729</td><td>0.15591</td><td>0.13944</td><td>0.12554</td><td>0.10990</td></tr><tr><td>92</td><td>0.19006</td><td>0.16640</td><td>0.15508</td><td>0.13870</td><td>0.12488</td><td>0.10932</td></tr><tr><td>93</td><td>0.18906</td><td>0.16553</td><td>0.15427</td><td>0.13797</td><td>0.12422</td><td>0.10875</td></tr><tr><td>94</td><td>0.18808</td><td>0.16467</td><td>0.15347</td><td>0.13726</td><td>0.12358</td><td>0.10819</td></tr><tr><td>95</td><td>0.18711</td><td>0.16382</td><td>0.15268</td><td>0.13655</td><td>0.12294</td><td>0.10763</td></tr><tr><td>96</td><td>0.18617</td><td>0.16299</td><td>0.15191</td><td>0.13586</td><td>0.12232</td><td>0.10709</td></tr><tr><td>97</td><td>0.18523</td><td>0.16217</td><td>0.15115</td><td>0.13518</td><td>0.12171</td><td>0.10655</td></tr><tr><td>98</td><td>0.18431</td><td>0.16136</td><td>0.15039</td><td>0.13450</td><td>0.12110</td><td>0.10602</td></tr><tr><td>99</td><td>0.18340</td><td>0.16057</td><td>0.14965</td><td>0.13384</td><td>0.12051</td><td>0.10550</td></tr><tr><td>100</td><td>0.18251</td><td>0.15979</td><td>0.14892</td><td>0.13319</td><td>0.11992</td><td>0.10499</td></tr></tbody></table>