6.4 Asymptotic tests
As evidenced in Section 5.4, another primary source of inferential tools beyond normal populations are asymptotic results. Assume that we want to test the hypotheses
- H0:θ=θ0 vs. H1:θ>θ0;
- H0:θ=θ0 vs. H1:θ<θ0;
- H0:θ=θ0 vs. H1:θ≠θ0.
If we know a test statistic that, under H0, has an asymptotic normal distribution, that is
Z=ˆθ−θ0ˆσ(θ0)d⟶N(0,1),
then the asymptotic critical regions are given by
Ca={Z>zα},Cb={Z<−zα},Cc={|Z|>zα/2}.
Note that despite not knowing the exact distribution of Z under H0, Z is a test statistic because its distribution is known asymptotically.
An especially relevant instance of (6.4) is given by likelihood theory (Theorem 4.1 under H0):
Z=ˆθMLE−θ01/√nI(θ0)=√nI(θ0)(ˆθMLE−θ0)d⟶N(0,1).
Due to Corollary 4.3, another asymptotic pivot is
Z=√nˆI(θ0)(ˆθMLE−θ0)d⟶N(0,1),ˆI(θ0)=1nn∑i=1(∂logf(Xi;θ)∂θ|θ=θ0)2,
which is always straightforward to compute from the srs (X1,…,Xn) from X∼f(⋅;θ) (analogous if X is discrete).
Other test statistics can be obtained from non-normal asymptotic distributions (see, e.g., the forthcoming Theorem 6.2).
Let us see some examples of asymptotic tests.
Example 6.12 Let (X1,…,Xn) be a srs of a rv X with mean μ and variance σ2, both unknown. We want to test:
- H0:μ=μ0 vs. μ>μ0;
- H0:μ=μ0 vs. μ<μ0;
- H0:μ=μ0 vs. μ≠μ0.
For that, employing the CLT (Theorem 2.5) we know that under H0:μ=μ0,
Z=ˉX−μ0S′/√nd⟶N(0,1).
Therefore, Z is a test statistic and H0 is rejected if the observed value of Z belongs to the corresponding critical region (Ca, Cb, or Cc).
Example 6.13 Let (X1,…,Xn) be a srs of a rv Γ(k,1/θ) with k known and θ unknown. We want to test:
- H0:θ=θ0 vs. H1:θ>θ0;
- H0:θ=θ0 vs. H1:θ<θ0;
- H0:θ=θ0 vs. H1:θ≠θ0.
We use that, by Example 4.12, I(θ)=k/θ2. Then, we know that under H0:θ=θ0,
Z=ˆθMLE−θ0θ0/√nkd⟶N(0,1).
Therefore, Z is a test statistic and H0 is rejected if the observed value of Z belongs to the corresponding critical region (Ca, Cb, or Cc).
Example 6.14 A certain machine has to be repaired if more than 10% of the items that it produces per day are defective. A srs of n=100 items of the daily production contains 15 that are defective and the foreman decides that the machine has to be repaired. Is the sample supporting his decision at a significance level α=0.01?
Let Y be the number of defective items that were found. Then Y∼Bin(n,p). We want to test
H0:p=0.10vs.H1:p>0.10.
Because of the CLT (Theorem 2.5), Y has a normal asymptotic distribution, so under H0:p=p0 it follows that
Z=ˆp−p0√p0(1−p0)/nd⟶N(0,1).
Therefore, Z is a test statistic with observed value
Z=0.15−0.10√0.1×0.9/100=5/3.
The rejection region is
C={Z>z0.01≈2.33}.
Since Z=5/3≈1.67<2.33, the sample does not provide enough evidence supporting the foreman decision, that is, that the actual percentage of defective items the machine is producing is above 10%. The machine should not be repaired and the larger proportion of defective items in the batch can be attributed to chance.