FBar Class Reference

This class is similar to FDist, except that it provides static methods to compute or approximate the complementary distribution function of \(X\), which we define as \(\bar{F} (x) = P[X\ge x]\), instead of \(F (x)=P[X\le x]\). More...

Static Public Member Functions
static double	scan (int n, double d, int m)
	Return \(P[S_N (d) \ge m]\), where \(S_N (d)\) is the scan statistic(see [71], [70]  and GofStat.scan ), defined as. More...

Detailed Description

This class is similar to FDist, except that it provides static methods to compute or approximate the complementary distribution function of \(X\), which we define as \(\bar{F} (x) = P[X\ge x]\), instead of \(F (x)=P[X\le x]\).

Note that with our definition of \(\bar{F}\), one has \(\bar{F} (x) = 1 - F (x)\) for continuous distributions and \(\bar{F} (x) = 1 - F (x-1)\) for discrete distributions over the integers.

Member Function Documentation

scan()

static double scan ( int  n, double  d, int  m  )

static

Return \(P[S_N (d) \ge m]\), where \(S_N (d)\) is the scan statistic(see [71], [70]  and GofStat.scan ), defined as.

\[ S_N (d) = \sup_{0\le y\le1-d} \eta[y, y+d], \tag{scan} \]

where \(d\) is a constant in \((0, 1)\), \(\eta[y, y+d]\) is the number of observations falling inside the interval \([y, y+d]\), from a sample of \(N\) i.i.d. \(U (0,1)\) random variables. One has (see [7] ),

\begin{align} P[S_N (d) \ge m] & \approx \left(\frac{m}{d}-N-1\right) b (m) + 2 \sum_{i=m}^N b (i) \tag{DistScan1} \\ & \approx 2 (1-\Phi(\theta\kappa)) + \theta\kappa\frac{\exp(-\theta^2\kappa^2 /2)}{d \sqrt{2\pi}} \tag{DistScan2} \end{align}

where \(\Phi\) is the standard normal distribution function.

\begin{align*} b (i) & = \binom{N}{i} d^i (1-d)^{N-i}, \\ \theta & = \sqrt{\frac{d}{1-d}}, \\ \kappa & = \frac{m}{d \sqrt{N}} - \sqrt{N}. \end{align*}

For \(d \le1/2\), ( DistScan1 ) is exact for \(m > N/2\), but only an approximation otherwise. The approximation ( DistScan2 ) is good when \(N d^2\) is large or when \(d > 0.3\) and \(N>50\). In other cases, this implementation sometimes use the approximation proposed by Glaz [71] . For more information, see [7], [71], [234] . The approximation returned by this function is generally good when it is close to 0, but is not very reliable when it exceeds, say, 0.4. If \(m \le(N + 1)d\), the method returns 1. Else, if \(Nd \le10\), it returns the approximation given by Glaz [71] . If \(Nd > 10\), it computes ( DistScan2 ) or ( DistScan1 ) and returns the result if it does not exceed 0.4, otherwise it computes the approximation from [71] , returns it if it is less than 1.0, and returns 1.0 otherwise. The relative error can reach 10% when \(Nd \le10\) or when the returned value is less than 0.4. For \(m > Nd\) and \(Nd > 10\), a returned value that exceeds \(0.4\) should be regarded as unreliable. For \(m = 3\), the returned values are totally unreliable. (There may be an error in the original formulae in [71] ). Restrictions: \(N \ge2\) and \(d \le1/2\).

Parameters

n	sample size ( \(\ge2\))
d	length of the test interval ( \(\in(0,1)\))
m	scan statistic

Returns

the complementary distribution function of the statistic evaluated at m

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The documentation for this class was generated from the following file:

• FBar.java