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ISYE 2028 A and B
Spring 2009
Lecture 17
Dr. Kobi Abayomi
April 25, 2009
1 Introduction: Non-Parametric Tests
We call X ∼ Fθ(X) a parametric distribution for X. We have already investigated inference on θ:
- Hypothesis Tests: Ho : θ = θo vs. Ha : θ 6 = θo
- Confidence Intervals: θ ∈ θˆ ± Zα/ 3 s.e.(θˆ)
For ‘Large’ sample we used the Central Limit Theorm (CLT) for the sampling distribution of the estimator x. In other settings (t dist, F-dist) we needed the explicit assumption X ∼ N (μ, σ^2 ).
If the ‘parameter’ we look for inference on is a quantile, we can use functions of the order statis- tics. We call such methods ‘Non-Parametric’, in the sense that the distribution X need only be continuous.
2 Confidence interval on Quantiles
Take X 1 , ..., Xn F (X), parameter unvailable or unknown, and the associated order statistics X(1), ..., X(n). Let’s look at the probability two order statistics cover a quantile ξp
P(X(i) < ξp < X(j))
Since P(X < ξp) = F (ξp) = p, by definition of the CDF...
P(X(i) < ξp < X(j)) =
j∑− 1
k−i
n! k!(n − k)! pk(1 − p)n−k
This is the probability that a quantile ξp is between two particular order statistics; observed data will yield a confidence interval for the quantile.
For Example:
Let X 1 , ..., X 4 ∼ F (x).
Generate the order statistics: X(1), ..., X(4).
P(X(1) < ξ. 5 < X(4)) =
k=
k!(4 − k)!
(1/2)^4 =. 875
This means that the X(1) and X(4) yield a 87.5% Confidence Interval for the median.
In general, though, non-parametric methods are less efficient (i.e. will have greater variance for equal sample size) than parametric methods. When parametric methods are available, use them.
3 The Sign Test
Remember the χ^2 test for Goodness of Fit? We had a random variable X that could take values in A 1 , ..., Ak
Ho : P(X ∈ Ai) = pi
vs.
Ha : P(X ∈ Ai) 6 = pi f or at least 1 i
Let X ∼ F (x), and consider the test Ho : F (ξ) = p 0 ; that is we wish to test the location of the p 0 th quantile. The chi-squared test with k = 2 tests against all alternatives, say we want to test
Ho : F (ξ) = p 0
vs.
Ha : F (ξ) > p 0
We need a different test - we want to test the fit at just a quantile (not several jointly) and the alternative hypothesis includes the direction of departure.
Let Y ≡ the number of ‘successes’ in n independent trials, or draws of X. A ‘success’ is a value of Y that is less than ξpo.
∑^10
y=
C y^10 (1/2)y(1/2)^10 −y^ ≈. 3438
We fail to reject the null hypothesis.
4 Wilcoxon Signed Rank Test
The signed rank test uses the signs and magnitude of the deviations Xi − ξ.
Assume: X ∼ F (x).
Take: X 1 , ..., Xn.
Rank |X 1 |, ..., |Xn|, this yields
Ri ≡ rank of the magnitude of Xi
For Example: for X 1 = 5, X 2 = − 6 , X 3 = −1, R 1 = 3, R 2 = 3, R 3 = 1. Thus R 1 , ...Rn is an arrangement of the first n positive integers.
Let
Zi =
− 1 if Xi < 0 1 if Xi < 0
Since P(Xi = 0) = 0 we need not concern ourselves with Xi = 0, either Zi = 1 or Zi = −1, it does not matter.
We call
W =
∑^ n
i=
Zi · Ri
the wilcoxon statistic and it turns out under the null hypothesis H 0 : F (ξ) =. 5
T =
W
n(n + 1)(2n + 1)/ 6
∼ N (0, 1)
Example
Test H 0 : ξ. 5 = 75 against Ha : ξ. 5 > 75 at α = .01.Say n = 18 and we calculate observed w = 135 and
n(n + 1)(2n + 1)/6 = 45.92.
Then .01 = P(W/ 45. 92 > 2 .376) = P(W > 106 .8). We reject H 0.
5 Exercises
