Statistics in geophysics inferential statistics III

Tests not requiring assumptions involving specific parametric distributions for the data or for the sampling distribution of the test statistics are called nonparametric.. Nonparametric

Statistics in Geophysics: Inferential Statistics III

Steffen Unkel

Department of Statistics Ludwig-Maximilians-University Munich, Germany

Tests not requiring assumptions involving specific parametric distributions for the data or for the sampling distribution of the test statistics are called nonparametric

Nonparametric methods areappropriateif

1 we know or suspect that the parametric assumption(s) required for a particular test are not met;

2 a test statistic that is suggested or dictated by the problem at hand is a complicated function of the data, and its sampling distribution is unknown and/or cannot be derived analytically.

Only a few nonparametric tests for location will be presented here

One-sample Wilcoxon signed-rank test

Let X1, , Xn be a random sample with continuous cdf FX(·)

Suppose that it is desired to test that the 0.5 quantile, xmed,

of the population sampled from is a specific value, say δ0 Consider the test problems:

(a) H 0 : x med = δ 0 vs H 1 : x med 6= δ 0

(b) H 0 : x med ≥ δ 0 vs H 1 : x med < δ 0

(c) H 0 : x med ≤ δ 0 vs H 1 : x med > δ 0

For i = 1 , n, let Di = Xi− δ0 and define

Zi =



1 if Di > 0

0 if Di < 0 .

Test statistic

W+=

n

X

i =1

RiZi, where Ri is the rank of |Di|

Rejection region:

(a) W+> w1−α/2+ or W+< wα/2+

(b) W+< wα+

(c) W+> w1−α+ ,

where wα+ denotes the α-quantile of the distribution of W+

One-sample Wilcoxon signed-rank test

For sufficiently large samples: Approximation by

N n(n+1)4 ,n(n+1)(2n+1)24

 Test statistic:

+− n(n+1)4 q

n(n+1)(2n+1) 24

a

∼ N (0, 1)

Rejection region:

(a) Z > z1−α/2 or Z < zα/2

(b) Z < z α

(c) Z > z 1−α ,

where zα is the α-quantile of the standard normal distribution

We assume that the sampling situation is such that we observepaired data (X1, Y1), , (Xn, Yn)

For i = 1, , n, the differences Di = Xi− Yi arise from a continuous distribution and each pair (Xi, Yi) is chosen randomly and independent

The null hypothesis is that themedian difference, δ, between pairs of observations is zero

Consider the test problems:

(a) H 0 : δ = 0 vs H 1 : δ 6= 0

(b) H 0 : δ ≥ 0 vs H 1 : δ < 0

(c) H 0 : δ ≤ 0 vs H 1 : δ > 0

Wilcoxon signed-rank test for paired data

Define

Zi =



1 if Di > 0

0 if Di < 0 Test statistic:

W+=

n

X

i =1

RiZi, where Ri is the rank of |Di|

Rejection region:

(a) W + > w1−α/2+ or W + < wα/2+

(b) W + < w +

α (c) W + > w1−α+ ,

where wα+ denotes the α-quantile of the distribution of W+

Given two samples of independentdata, the aim is to test for

a possible difference in location

The null hypothesis is that the two data samples have been drawn from the same distribution

Under H0 there are n + m observations making up a single distribution, where n (m) denote the number of observations

in sample 1 (sample 2)

The test statistic is a function of the ranks of the data values within the n + m observations that are pooledunder H0

Wilcoxon rank-sum test

Let X1, , Xn and Y1, , Ym be two random samples from populations with continuous cdfs FX(·) and FY(·),

respectively

Consider the test problems:

(a) H0: xmed = ymed vs H1: xmed 6= y med

(b) H0: xmed ≥ y med vs H1: xmed < ymed

(c) H0: xmed ≤ ymed vs H1: xmed > ymed .

Arrange the n + m observations of the pooled sample

X1, , Xn, Y1, , Ym in ascending order

Define

Vi =



1 if the i -th order statistic belongs to the X sample

0 if the i -th order statistic belongs to the Y sample

Test statistic:

Wn,m =

n+m

X

i =1

iVi =

n

X

i =1

R(Xi) , where R(Xi) is the rank of Xi in the pooled sample

Rejection region:

(a) Wn,m> w1−α/2(n, m) or Wn,m< wα/2(n, m)

(b) Wn,m< wα(n, m)

(c) Wn,m> w1−α(n, m),

where wα denotes the α-quantile of the distribution of Wn,m For sufficiently large samples: Approximation by

N (n(n + m + 1)/2, nm(n + m + 1)/12)