Femke Stokkink Statistical Data Analysis
2666619
2022-03-21
Summary
1 Chapter 6: Nonparametric Methods
1.1 The One Sample Problem
Nonparametric tests:
• Nonparametric tests make no (parametric) assumptions about the underlying distribution F of the data.
So, for example no normality assumption.
• These tests are applicable for broad classes of distributions, and have actual level α. The distribution of
the test statistic under H0 is the same for each distribution F belonging to H0 .
• Nonparametric tests are robust with respect to the level: they have the intended level α for a large class
of distributions.
• Nonparametric tests are more efficient (have higher power) than parametric tests when the (normality)
assumptions are not fulfilled.
One sample problem: assume we have a sample X1 ,...,Xn from an unknown distribution F , and we want to
test the location of F . Which test would you use?
T-test:
• Assumption: X1 ,...,Xn ∼ N(µ,σ 2 )
• Null hypothesis H0 : µ = µ0
√
• Test statistic T = n X̄−µ
Sx
0
• Distribution: under H0 we have T ∼ tn−1 .
• This is a parametric test (assumes normality) for a composite H0 , consisting of all normal distributions
with expectation µ0
Sign Test:
• Assumption: underlying distribution F has a unique median m, such that P(Xi < m) = P(Xi > m) = 12 .
• Null hypothesis H0 : m = m0 . This is a composite null hypothesis.
• Test statistic T = #(Xi > m0 ) = Σni=1 1Xi >m0
• Distribution: under H0 we have T ∼ bin(n, 12 ). This is a nonparametric test, since T has this distribution
for all F in H0 .
• In case k of the Xi ’s are equal to m0 , delete these k values and perform the test conditionally on k values
equal to m0 , and T ∼ bin(n - k, 21 ) under H0 .
(Wilcoxon) Signed Rank Test:
• Assumption: Underlying distribution F is continuous and symmetric around m.
• Null hypothesis H0 : m = m0 . This is a composite null hypothesis.
• Test statistic V is based on the ranks Ri of the absolute differences |Xi - m0 |. V = Σni=1 Ri sgn(Xi − m0 )
• Distribution: relatively large values of V indicate that m is larger than m0 . Under H0 , V is distributed as
Σni=1 Qi Ri with
– Qi random variable, P(Qi = -1) = P(Qi = 1) = 12 .
– R1 ,...,Rn a random permutation of {1,...,n}.
• Since this distribution is the same for all distributions under H0 , this is a nonparametric test.
• Wilcoxon test is usually more powerfull
1
, 1.2 Asymptotic Efficiency
A more efficient test needs fewer observations to obtain the same power as the less efficient test. Like in the
case of (robust) estimators we will consider the asymptotic case for the number of observations n tending to
infinity, and we will see that the asymptotic variance (now of the test statistic) plays a role in determining the
asymptotic efficiency.
Let X1 ,...,Xn be observations from a distribution F . According to H0 , F belongs to a class F0 (for example, all
distributions with median m), whereas according to H1 , F belongs to a class F1 . The power of a test is the
function π(F ) = PF (H0 is rejected).
For a test to be good π(F ) should be small when F ∈ F0 and large when F ∈ F1 .
It is difficult to compare the power of two tests for every possible F , therefore we focus on shift alternatives:
alternatives that can be obtained by shifting a distribution that belongs to F0 over a certain distance θ. Such
alternatives therefore have a location that is shifted over this distance θ, whereas the scale stays the same. We
limit to right-sided testing problems, so θ > 0.
If, for example, one would consider hypotheses about the median, F0 could be a distribution with median m0 ,
and then F0 would have median m0 + θ. The power for the class of shift alternatives F0 can be written as
πn (θ) = Pθ (H0 is rejected).
For a suitable right-sided test the value πn (0) of the power function under H0 is small, whereas πn (θ) is ‘large’
for θ > 0, so under the alternative hypothesis.
Suppose that H0 is rejected for a large value of the test statistic Tn and assume Tn is asymptotically normally
distributed. This means that for large n, Tn is approximately distributed as N (µ(θ), σ 2 (θ)/n), when θ is the
true value of the parameter. here µ(θ) is the asymptotic mean and σ 2 (θ) the asymptotic variance of Tn
√ 0
The power of the test can then be rewritten to: πn (θ) ≈ 1 − Φ(ξ1−α − n µσ(0)
(0)
θ)
• ξ1−α is the 1-α quantile of the N(0,1) distribution.
µ0 (0)
• σ(0) is the slope of the test. The larger the slope, the better the test.
• A sequence of tests Tn is consistent when for a fixed α the power tends to 1 for each alternative when n
goes to ∞. In this case the sequence of tests is consistent for the shifted alternatives if µ(θ) > µ(0).
0
The asymptotic relative efficiency of the test Tn with respect to T̃m = are(Tn ,T̃m ) = ( µµ̃0 (0)/σ(0) 2
(0)/σ̃(0) ) =
m
n
• Here T̃m is the test statistic of a second test
• If are(Tn ,T̃m ) > 1, then Tn is more efficient than T̃m .
If for a certain sample the a.r.e of two tests is 2/π, then in order to obtain similar power in both tests, the
sample size ratio should be nntest1
test2
= π2 = 0.64
1.3 Two Sample Problems
Two samples can either be paired or independent:
• Paired samples: (X1 , Y1 ), (X2 , Y2 ),..., (Xn , Yn )
• Independent samples: X1 , X2 , ..., Xm and Y1 , Y2 , ..., Yn
For paired samples: if one is interested whether one sample is stochastically larger, consider differences Zi =
Yi - Xi . This then becomes a one sample problem.
Median test:
• Assumption: X1 , .., Xm ∼ F, Y1 , .., Yn ∼ G, F and G continuous and both are i.i.d.
2
2666619
2022-03-21
Summary
1 Chapter 6: Nonparametric Methods
1.1 The One Sample Problem
Nonparametric tests:
• Nonparametric tests make no (parametric) assumptions about the underlying distribution F of the data.
So, for example no normality assumption.
• These tests are applicable for broad classes of distributions, and have actual level α. The distribution of
the test statistic under H0 is the same for each distribution F belonging to H0 .
• Nonparametric tests are robust with respect to the level: they have the intended level α for a large class
of distributions.
• Nonparametric tests are more efficient (have higher power) than parametric tests when the (normality)
assumptions are not fulfilled.
One sample problem: assume we have a sample X1 ,...,Xn from an unknown distribution F , and we want to
test the location of F . Which test would you use?
T-test:
• Assumption: X1 ,...,Xn ∼ N(µ,σ 2 )
• Null hypothesis H0 : µ = µ0
√
• Test statistic T = n X̄−µ
Sx
0
• Distribution: under H0 we have T ∼ tn−1 .
• This is a parametric test (assumes normality) for a composite H0 , consisting of all normal distributions
with expectation µ0
Sign Test:
• Assumption: underlying distribution F has a unique median m, such that P(Xi < m) = P(Xi > m) = 12 .
• Null hypothesis H0 : m = m0 . This is a composite null hypothesis.
• Test statistic T = #(Xi > m0 ) = Σni=1 1Xi >m0
• Distribution: under H0 we have T ∼ bin(n, 12 ). This is a nonparametric test, since T has this distribution
for all F in H0 .
• In case k of the Xi ’s are equal to m0 , delete these k values and perform the test conditionally on k values
equal to m0 , and T ∼ bin(n - k, 21 ) under H0 .
(Wilcoxon) Signed Rank Test:
• Assumption: Underlying distribution F is continuous and symmetric around m.
• Null hypothesis H0 : m = m0 . This is a composite null hypothesis.
• Test statistic V is based on the ranks Ri of the absolute differences |Xi - m0 |. V = Σni=1 Ri sgn(Xi − m0 )
• Distribution: relatively large values of V indicate that m is larger than m0 . Under H0 , V is distributed as
Σni=1 Qi Ri with
– Qi random variable, P(Qi = -1) = P(Qi = 1) = 12 .
– R1 ,...,Rn a random permutation of {1,...,n}.
• Since this distribution is the same for all distributions under H0 , this is a nonparametric test.
• Wilcoxon test is usually more powerfull
1
, 1.2 Asymptotic Efficiency
A more efficient test needs fewer observations to obtain the same power as the less efficient test. Like in the
case of (robust) estimators we will consider the asymptotic case for the number of observations n tending to
infinity, and we will see that the asymptotic variance (now of the test statistic) plays a role in determining the
asymptotic efficiency.
Let X1 ,...,Xn be observations from a distribution F . According to H0 , F belongs to a class F0 (for example, all
distributions with median m), whereas according to H1 , F belongs to a class F1 . The power of a test is the
function π(F ) = PF (H0 is rejected).
For a test to be good π(F ) should be small when F ∈ F0 and large when F ∈ F1 .
It is difficult to compare the power of two tests for every possible F , therefore we focus on shift alternatives:
alternatives that can be obtained by shifting a distribution that belongs to F0 over a certain distance θ. Such
alternatives therefore have a location that is shifted over this distance θ, whereas the scale stays the same. We
limit to right-sided testing problems, so θ > 0.
If, for example, one would consider hypotheses about the median, F0 could be a distribution with median m0 ,
and then F0 would have median m0 + θ. The power for the class of shift alternatives F0 can be written as
πn (θ) = Pθ (H0 is rejected).
For a suitable right-sided test the value πn (0) of the power function under H0 is small, whereas πn (θ) is ‘large’
for θ > 0, so under the alternative hypothesis.
Suppose that H0 is rejected for a large value of the test statistic Tn and assume Tn is asymptotically normally
distributed. This means that for large n, Tn is approximately distributed as N (µ(θ), σ 2 (θ)/n), when θ is the
true value of the parameter. here µ(θ) is the asymptotic mean and σ 2 (θ) the asymptotic variance of Tn
√ 0
The power of the test can then be rewritten to: πn (θ) ≈ 1 − Φ(ξ1−α − n µσ(0)
(0)
θ)
• ξ1−α is the 1-α quantile of the N(0,1) distribution.
µ0 (0)
• σ(0) is the slope of the test. The larger the slope, the better the test.
• A sequence of tests Tn is consistent when for a fixed α the power tends to 1 for each alternative when n
goes to ∞. In this case the sequence of tests is consistent for the shifted alternatives if µ(θ) > µ(0).
0
The asymptotic relative efficiency of the test Tn with respect to T̃m = are(Tn ,T̃m ) = ( µµ̃0 (0)/σ(0) 2
(0)/σ̃(0) ) =
m
n
• Here T̃m is the test statistic of a second test
• If are(Tn ,T̃m ) > 1, then Tn is more efficient than T̃m .
If for a certain sample the a.r.e of two tests is 2/π, then in order to obtain similar power in both tests, the
sample size ratio should be nntest1
test2
= π2 = 0.64
1.3 Two Sample Problems
Two samples can either be paired or independent:
• Paired samples: (X1 , Y1 ), (X2 , Y2 ),..., (Xn , Yn )
• Independent samples: X1 , X2 , ..., Xm and Y1 , Y2 , ..., Yn
For paired samples: if one is interested whether one sample is stochastically larger, consider differences Zi =
Yi - Xi . This then becomes a one sample problem.
Median test:
• Assumption: X1 , .., Xm ∼ F, Y1 , .., Yn ∼ G, F and G continuous and both are i.i.d.
2
