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

  1. H0:θ=θ0 vs. H1:θ>θ0;
  2. H0:θ=θ0 vs. H1:θ<θ0;
  3. H0:θ=θ0 vs. H1:θθ0.

If we know a test statistic that, under H0, has an asymptotic normal distribution, that is

Z=ˆθθ0ˆσ(θ0)dN(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=ˆθMLEθ01/nI(θ0)=nI(θ0)(ˆθMLEθ0)dN(0,1).

Due to Corollary 4.3, another asymptotic pivot is

Z=nˆI(θ0)(ˆθMLEθ0)dN(0,1),ˆI(θ0)=1nni=1(logf(Xi;θ)θ|θ=θ0)2,

which is always straightforward to compute from the srs (X1,,Xn) from Xf(;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:

  1. H0:μ=μ0 vs. μ>μ0;
  2. H0:μ=μ0 vs. μ<μ0;
  3. H0:μ=μ0 vs. μμ0.

For that, employing the CLT (Theorem 2.5) we know that under H0:μ=μ0,

Z=ˉXμ0S/ndN(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:

  1. H0:θ=θ0 vs. H1:θ>θ0;
  2. H0:θ=θ0 vs. H1:θ<θ0;
  3. H0:θ=θ0 vs. H1:θθ0.

We use that, by Example 4.12, I(θ)=k/θ2. Then, we know that under H0:θ=θ0,

Z=ˆθMLEθ0θ0/nkdN(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 YBin(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=ˆpp0p0(1p0)/ndN(0,1).

Therefore, Z is a test statistic with observed value

Z=0.150.100.1×0.9/100=5/3.

The rejection region is

C={Z>z0.012.33}.

Since Z=5/31.67<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.