# Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Material and Method

#### 2.1. Addressing the Computation of $CDF$ for $OS$(s)

Algorithm 1: Balancing the drawings from uniform $\mathrm{U}(0,1)$ distribution. |

Input data: n (2 ≤ n, integer) |

Steps: |

For i from 1 to n do v[i] ← Rand |

For j from 0 to n do |

For i from 1 to j do u[i] ← v[i]/2 |

For i from j+1 to n do u[i] ← v[i]/2+1/2 |

occ ← n!/j!/(n-j)! |

Output u[1],..., u[n], occ |

EndFor |

Output data: (n+1) samples (u) of sample size (n) and their occurrences (occ) |

Algorithm 2: Sampling an order statistic ($OS$). |

Input data: n (2 ≤ n, integer) |

Steps: |

For i from 1 to n do v[i] ← Rand |

For j from 0 to n do |

For i from 1 to j do u[i] ← v[i]/2 |

For i from j+1 to n do u[i] ← v[i]/2+1/2 |

OSj ← any Equations (5)–(12) with p${}_{1}\phantom{\rule{-0.166667em}{0ex}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}$ u[1],..., p${}_{n}\phantom{\rule{-0.166667em}{0ex}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}$ u[n] |

Output OSj, j |

EndFor |

Output data: (n+1) OS and their occurrences |

#### 2.2. Further Theoretical Considerations Required for the Study

## 3. Results and Discussion

#### 3.1. The Analytical Formula of $\mathrm{CDF}$ for $T\phantom{\rule{-0.166667em}{0ex}}S$

#### 3.2. Computations for the $CDF$ of $T\phantom{\rule{-0.166667em}{0ex}}S$ and Its Analytical Formula

Algorithm 3: Avoiding computational errors for $T\phantom{\rule{-0.166667em}{0ex}}S$. |

Input data: n (n ≥ 2, integer), x (1 ≤ x ≤ 1/n, real number, double precision) |

$y\leftarrow 1/x$; //${p}_{1/T\phantom{\rule{-0.166667em}{0ex}}S}\leftarrow $ Equation (19), ${\alpha}_{1/T\phantom{\rule{-0.166667em}{0ex}}S}\leftarrow $ Equation (21) |

if y <(n+1)/2 |

$p\leftarrow {\sum}_{k=0}^{\lfloor y\rfloor -1}{(-1)}^{k}\frac{{(y-1-k)}^{n}}{k!(n-1-k)!}\phantom{\rule{0.166667em}{0ex}};\alpha \leftarrow 1-p$ |

else if y >(n+1)/2 |

$\alpha \leftarrow {\sum}_{k=0}^{\lfloor n-y\rfloor}{(-1)}^{k}\frac{{(n-y-k)}^{n-1}}{k!(n-1-k)!}\phantom{\rule{0.166667em}{0ex}};p\leftarrow 1-\alpha $ |

else |

$\alpha \leftarrow 0.5\phantom{\rule{0.166667em}{0ex}};p\leftarrow 0.5$ |

Output data: $\alpha \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{\alpha}_{1/T\phantom{\rule{-0.166667em}{0ex}}S}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{p}_{T\phantom{\rule{-0.166667em}{0ex}}S}\phantom{\rule{-0.166667em}{0ex}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}{\mathrm{CDF}}_{T\phantom{\rule{-0.166667em}{0ex}}S}(x;n)$ and $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{p}_{1/T\phantom{\rule{-0.166667em}{0ex}}S}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\alpha}_{T\phantom{\rule{-0.166667em}{0ex}}S}\phantom{\rule{-0.166667em}{0ex}}\leftarrow \phantom{\rule{-0.166667em}{0ex}}1\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{p}_{T\phantom{\rule{-0.166667em}{0ex}}S}$ |

Algorithm 4: FreePascal implementation for calculating the $\mathrm{CDF}$ of $I\phantom{\rule{-0.166667em}{0ex}}H$. |

Input data: n (integer), x (real number, double precision); |

var k,i: integer; //$integer$ enough for n < 32,768 |

var z,y: mpf_t; //$double$ or $extended$ instead of $mpf\phantom{\rule{-0.166667em}{0ex}}\_t$ |

Begin //$\mathrm{CDF}$ for Irwin–Hall distribution |

mpf_set_default_prec(128); //or bigger, 256, 512,... |

mpf_init(y); mpf_init(z); //y := 0.0; |

for k := trunc(x) downto 0 do begin //main loop |

If(k mod 2 = 0) // z := 1.0 or z := −1.0; |

then mpf_set_si(z,1) //z := 1.0; |

else mpf_set_si(z,-1); //z := −1.0; |

for i := n − k downto 1 do z := z*(x − k)/i; |

for i := k downto 1 do z := z*(x− k)/i; |

y := y + z; |

end; |

pIH_gmp := mpf_get_d(y); mpf_clear(z); mpf_clear(y); |

End; |

Output data: p (real number, double precision) |

#### 3.3. Approximations of $CDF$ of $T\phantom{\rule{-0.166667em}{0ex}}S$ with Known Functions

#### 3.4. The Use of $CDF$ for $T\phantom{\rule{-0.166667em}{0ex}}S$ to Measure the Departure between an Observed Distribution and a Theoretical One

- Case study 1.

- Case study 2.

- Case study 3.

- Case study 4.

- Case study 5.

- Case study 6.

- Case study 7.

- Case study 8.

- Case study 9.

- Case study 10.

## 3.5. The Patterns in the Order Statistics

## 3.6. Another Rank Order Statics Method and Other Approaches

## 4. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Agreement estimating ${\mathrm{CDF}}_{T\phantom{\rule{-0.166667em}{0ex}}S}$ for n = 2...54 and $1000p$ = 1...999 with a step of 2.

**Figure 9.**Standard errors ($SE$) as function of sample size (n) for the approximation of $1/T\phantom{\rule{-0.166667em}{0ex}}S$ with $GL$ (Equation (29)).

Constant/Variable/Type Value | Meaning |
---|---|

stt ← record v:single; c:byte; end | (OSj, j) pair from Algorithm 2 stored in 5 bytes |

mem ← 12,800,000,000 | in bytes, 5*mem ← 64Gb, hardware limit |

buf ← 1,000,000 | the size of a static buffer of data (5*buf bytes) |

stst ← array[0..buf-1]of stt | static buffer of data |

dyst ← array of stst | dynamic array of buffers |

lvl ← 1000 | lvl + 1: number of points in the grid (see Figure 1) |

n | ${\mathit{p}}_{\mathit{i}}$ Calculated with Equation (19) | ${\mathit{p}}_{\mathit{i}}$ Calculated with Equation (21) | ${\mathit{p}}_{\mathit{i}}$ Calculated with Algorithm 4 |
---|---|---|---|

34 | 3.0601572482628 × 10${}^{-8}$ | 3.0601603616294 × 10${}^{-8}$ | 3.0601364353173 × 10${}^{-8}$ |

35 | 6.0059397209079 × 10${}^{-8}$ | 6.0057955311142 × 10${}^{-8}$ | 6.0057052975471 × 10${}^{-8}$ |

36 | 1.1567997676343 × 10${}^{-8}$ | 1.1572997605838 × 10${}^{-8}$ | 1.1567370749831 × 10${}^{-8}$ |

37 | 8.9214456109544 × 10${}^{-8}$ | 8.9215230398577 × 10${}^{-8}$ | 8.9213063043724 × 10${}^{-8}$ |

38 | 1.1684682533384 × 10${}^{-8}$ | 1.1681544866285 × 10${}^{-8}$ | 1.1677646550768 × 10${}^{-8}$ |

39 | 1.2101651325053 × 10${}^{-8}$ | 1.2181659126285 × 10${}^{-8}$ | 1.2100378665608 × 10${}^{-8}$ |

40 | 1.1041708665520 × 10${}^{-7}$ | 1.1043952711846 × 10${}^{-7}$ | 1.1036003349029 × 10${}^{-7}$ |

41 | 7.2871410520319 × 10${}^{-8}$ | 7.2755412302319 × 10${}^{-8}$ | 7.2487977100103 × 10${}^{-8}$ |

42 | 1.9483807018501 × 10${}^{-8}$ | 1.9626447735907 × 10${}^{-8}$ | 1.9273186509959 × 10${}^{-8}$ |

43 | 3.1128379331196× 10${}^{-8}$ | 1.7088238120170 × 10${}^{-8}$ | 1.3899520242290 × 10${}^{-8}$ |

44 | 8.7810761126831× 10${}^{-8}$ | 3.8671367222236× 10${}^{-8}$ | 1.0878689813951 × 10${}^{-8}$ |

45 | 1.1914784602127 × 10${}^{-7}$ | 3.1416715528555 × 10${}^{-7}$ | 5.8339481916925 × 10${}^{-8}$ |

46 | 2.0770754629042 × 10${}^{-6}$ | 1.2401177918843 × 10${}^{-6}$ | 4.4594953399233 × 10${}^{-8}$ |

47 | 5.0816356972050 × 10${}^{-7}$ | 4.1644326761832 × 10${}^{-7}$ | 1.8942487765410 × 10${}^{-8}$ |

48 | 1.5504732794049 × 10${}^{-6}$ | 5.5760558048026 × 10${}^{-6}$ | 5.7292512517324 × 10${}^{-8}$ |

49 | 1.1594466754136 × 10${}^{-5}$ | 6.4164330856396 × 10${}^{-6}$ | 1.7286761495408 × 10${}^{-7}$ |

50 | 1.0902858025759 × 10${}^{-5}$ | 8.0190771776360 × 10${}^{-6}$ | 8.5891058550425 × 10${}^{-8}$ |

51 | 6.4572577668164 × 10${}^{-6}$ | 1.6023753568028 × 10${}^{-4}$ | 1.9676739380922 × 10${}^{-8}$ |

52 | 1.0080944275181 × 10${}^{-4}$ | 9.1080176774820 × 10${}^{-5}$ | 1.0359121739272 × 10${}^{-7}$ |

53 | 9.3219609856284 × 10${}^{-4}$ | 2.7347575817507 × 10${}^{-4}$ | 1.5873847007230 × 10${}^{-8}$ |

54 | 4.8555844748161 × 10${}^{-4}$ | 1.6086902937472 × 10${}^{-3}$ | 9.2930071189138 × 10${}^{-9}$ |

55 | 6.2446720485774 × 10${}^{-4}$ | 1.6579954395873 × 10${}^{-3}$ | 1.2848119194342 × 10${}^{-7}$ |

**Table 3.**Descriptive for the agreement in the calculation of the CDF of TS (Equation (12) vs. Algorithm 4).

n | $\mathit{SE}$ | $\mathit{minep}$ | $\mathit{maxep}$ | n | $\mathit{SE}$ | $\mathit{minep}$ | $\mathit{maxep}$ | n | $\mathit{SE}$ | $\mathit{minep}$ | $\mathit{maxep}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 3.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −2.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 1.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 20 | 5.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 38 | 3.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −7.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

3 | 3.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −2.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 21 | 3.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 39 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −7.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

4 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −2.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 22 | 6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 40 | 1.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −7.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

5 | 4.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −2.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 23 | 5.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −5.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 41 | 8.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −7.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

6 | 2.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 24 | 4.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 42 | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −7.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

7 | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 25 | 4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 43 | 3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

8 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 26 | 1.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −6.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 44 | 3.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

9 | 3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 27 | 1.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 45 | 7.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

10 | 4.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 28 | 7.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 46 | 6.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

11 | 5.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 29 | 7.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 47 | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

12 | 7.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 30 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 48 | 7.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 8.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

13 | 5.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 31 | 4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 49 | 1.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

14 | 5.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 32 | 5.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 50 | 9.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

15 | 1.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −3.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 33 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 51 | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

16 | 6.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 34 | 5.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 52 | 1.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

17 | 8.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 35 | 7.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 53 | 4.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

18 | 5.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 36 | 3.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 54 | 3.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

19 | 5.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −4.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 37 | 9.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | −6.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 55 | 1.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | −6.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

Case | Parameter | $\mathit{A}\phantom{\rule{-0.166667em}{0ex}}\mathit{D}$ | $\mathit{K}\phantom{\rule{-0.166667em}{0ex}}\mathit{S}$ | $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{M}$ | $\mathit{K}\phantom{\rule{-0.166667em}{0ex}}\mathit{V}$ | $\mathit{W}\phantom{\rule{-0.166667em}{0ex}}\mathit{U}$ | $\mathit{H}1$ | $\mathit{g}1$ | $\mathit{T}\phantom{\rule{-0.166667em}{0ex}}\mathit{S}$ | $\mathit{FCS}$ |
---|---|---|---|---|---|---|---|---|---|---|

1 | $Statistic$ | 1.137 | 1.110 | 0.206 | 1.715 | 0.182 | 5.266 | 0.494 | 4.961 | 15.80 |

${\alpha}_{Statistic}$ | 0.288 | 0.132 | 0.259 | 0.028 | 0.049 | 0.343 | 0.112 | 0.270 | 0.045 | |

2 | $Statistic$ | 0.348 | 0.549 | 0.042 | 0.934 | 0.039 | 7.974 | 0.496 | 6.653 | 6.463 |

${\alpha}_{Statistic}$ | 0.894 | 0.884 | 0.927 | 0.814 | 0.844 | 0.264 | 0.109 | 0.107 | 0.596 | |

3 | $Statistic$ | 0.617 | 0.630 | 0.092 | 1.140 | 0.082 | 4.859 | 0.471 | 5.785 | 4.627 |

${\alpha}_{Statistic}$ | 0.619 | 0.742 | 0.635 | 0.486 | 0.401 | 0.609 | 0.451 | 0.627 | 0.797 | |

4 | $Statistic$ | 0.793 | 0.827 | 0.144 | 1.368 | 0.129 | 3.993 | 0.482 | 4.292 | 8.954 |

${\alpha}_{Statistic}$ | 0.482 | 0.420 | 0.414 | 0.190 | 0.154 | 0.524 | 0.255 | 0.395 | 0.346 | |

5 | $Statistic$ | 0.440 | 0.486 | 0.049 | 0.954 | 0.047 | 104.2 | 0.500 | 103.2 | 5.879 |

${\alpha}_{Statistic}$ | 0.810 | 0.963 | 0.884 | 0.850 | 0.742 | 0.359 | 0.034 | 0.533 | 0.661 | |

6 | $Statistic$ | 0.565 | 0.707 | 0.083 | 1.144 | 0.061 | 83.32 | 0.499 | 82.17 | 5.641 |

${\alpha}_{Statistic}$ | 0.683 | 0.675 | 0.673 | 0.578 | 0.580 | 0.455 | 0.247 | 0.305 | 0.687 | |

7 | $Statistic$ | 1.031 | 1.052 | 0.170 | 1.662 | 0.149 | 52.66 | 0.494 | 51.00 | 11.24 |

${\alpha}_{Statistic}$ | 0.320 | 0.202 | 0.333 | 0.067 | 0.106 | 0.471 | 0.729 | 0.249 | 0.188 | |

8 | $Statistic$ | 0.996 | 0.771 | 0.132 | 1.375 | 0.127 | 22.201 | 0.460 | 27.95 | 5.933 |

${\alpha}_{Statistic}$ | 0.322 | 0.556 | 0.451 | 0.248 | 0.162 | 0.853 | 0.980 | 0.978 | 0.655 | |

9 | $Statistic$ | 0.398 | 0.576 | 0.058 | 1.031 | 0.051 | 31.236 | 0.489 | 32.04 | 2.692 |

${\alpha}_{Statistic}$ | 0.853 | 0.869 | 0.828 | 0.728 | 0.694 | 0.577 | 0.746 | 0.507 | 0.952 | |

10 | $Statistic$ | 0.670 | 0.646 | 0.092 | 1.170 | 0.085 | 11.92 | 0.460 | 14.66 | 3.549 |

${\alpha}_{Statistic}$ | 0.583 | 0.753 | 0.627 | 0.488 | 0.373 | 0.747 | 0.874 | 0.879 | 0.895 |

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**MDPI and ACS Style**

Jäntschi, L.
Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions. *Mathematics* **2020**, *8*, 216.
https://doi.org/10.3390/math8020216

**AMA Style**

Jäntschi L.
Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions. *Mathematics*. 2020; 8(2):216.
https://doi.org/10.3390/math8020216

**Chicago/Turabian Style**

Jäntschi, Lorentz.
2020. "Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions" *Mathematics* 8, no. 2: 216.
https://doi.org/10.3390/math8020216