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1.16.1
This class provides methods to compute (or approximate) the distribution functions of special types of goodness-of-fit test statistics. More...
Static Public Member Functions | |
| static double | kolmogorovSmirnovPlusJumpOne (int N, double a, double x) |
| Similar to umontreal.ssj.probdist.KolmogorovSmirnovPlusDist but for the case where the distribution function \(F\) has a jump of size. | |
| static double | scan (int N, double d, int m) |
| Returns \(F (m)\), the distribution function of the scan statistic with parameters \(N\) and \(d\), evaluated at \(m\). | |
This class provides methods to compute (or approximate) the distribution functions of special types of goodness-of-fit test statistics.
Definition at line 33 of file FDist.java.
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static |
Similar to umontreal.ssj.probdist.KolmogorovSmirnovPlusDist but for the case where the distribution function \(F\) has a jump of size.
\(a\) at a given point \(x_0\), is zero at the left of \(x_0\), and is continuous at the right of \(x_0\). The Kolmogorov-Smirnov statistic is defined in that case as
\[ D_N^+(a) = \sup_{a\le u\le1} \left(\hat{F}_N (F^{-1}(u)) - u\right) = \max_{\Rule{0.0pt}{7.0pt}{0.0pt} \lfloor1+aN \le j \le N} \left(j/N - F (V_{(j)})\right). \tag{KSPlusJumpOne} \]
where \(V_{(1)},…,V_{(N)}\) are the observations sorted by increasing order. The method returns an approximation of \(P[D_N^+(a) \le x]\) computed via
\begin{align} P[D_N^+(a) \le x] & = 1 - x \sum_{i=0}^{\lfloor N (1-a-x)\rfloor} \binom{N}{i} \left(\frac{i}{N} + x \right)^{i-1} \left(1 - \frac{i}{N} - x \right)^{N-i}. \tag{DistKSJ1} \\ & = x \sum_{j=0}^{\lfloor N (a+x) \rfloor} \binom{N}{j} \left(\frac{j}{N} - x \right)^j \left(1 - \frac{j}{N} + x \right)^{N-j-1}. \tag{DistKSJ2} \end{align}
The current implementation uses formula ( DistKSJ2 ) when \(N (x+a) < 6.5\) and \(x+a < 0.5\), and uses ( DistKSJ1 ) when \(Nx \ge6.5\) or \(x+a
\ge0.5\). Restriction: \(0 < a < 1\).
| N | sample size |
| a | size of the jump |
| x | positive or negative Kolmogorov-Smirnov statistic |
Definition at line 68 of file FDist.java.
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static |
Returns \(F (m)\), the distribution function of the scan statistic with parameters \(N\) and \(d\), evaluated at \(m\).
For a description of this statistic and its distribution, see FBar.scan(int,double,int), which computes its complementary distribution \(\bar{F} (m) = 1 - F (m-1)\).
| N | sample size ( \(\ge2\)) |
| d | length of the test interval ( \(\in(0,1)\)) |
| m | scan statistic |
Definition at line 156 of file FDist.java.
1.16.1