Hypothesis testing for discrete distributions has seldom been discussed in literature. The difficulty of constructing a test for hypotheses of parameters for discrete distribution is that it could be rarely possible to construct a test with an exactly specified significance level when the sample size is finite (see for examples, Welsh (1996, page 146) and Kotz and Johnson (1982)).
The most popular use of level α test for discrete random variables is the normal
approx-imation. However, it is an approximate level α test which in general is not a level α test in the case of finite sample, and its exact significance level is not known without further calculation. One other possibility for the discrete case is the Fisher’s fiducial approach. If we consider hypothesis H0 : p = p0 for a binomial random variable, we may reject H0 then if and only if Pp0(X ≤ x) < α/2 or Pp0(X ≥ x) < α/2 (see Garthwaite et al. (2002, p103)) where x is observed. We prefer to use the p-value to interpret a hypothesis testing problem in replace of the significance level α test since we are dealing with significance test. We re-state the HDS test in an appropriate way for dealing with discrete distribution.
Definition 3.5. Consider the hypothesis H0 : θ = θ0. The HDS test for discrete distribution defines the p-value with observation X = x0 as
phd= X
L(θ0,x)≤L(θ0,x0)
L(θ0, x).
Generally there are many versions for constructing Fisherian significance test. The choice of a test statistic may affects the conclusion of a test as it behaves in continuous case. The following example presents the results from our concern.
Example 3.4. Let X be a random variable with binomial distribution b(n, p). Consider the hypothesis H0 : p = p0 and assume that an observation x0 is available. There are two popularly used Fisherian significance tests defining the p values. First, the normal approximation defines p-value as
px0 = P (|Z| ≥ |x0− np0| pnp0(1 − p0))
where Z has the standard normal distribution N(0, 1). On the other hand, taking X as a test statistic, it has a binomial distribution b(n, p0) when H0 is true. Typically, the test statistic is applied to construct a one-sided Fisherian significance test (see Garthwaite et al.
(2002)) with p value as follows
px0 =
Suppose that the observation is x0 = 18 with sample size n=20. We consider hypothesis H0 : p = p0 = 0.7. In this situation, p-value px0 for the normal approximated Fisherian
significance test is 0.051 and it for the one-sided Fisherian significance test is 0.0355. There-fore, it often provides various conclusions when we use different Fisherian significance tests.
Otherwise, the HDS test defines the p-value as
phd = Xn
x=0
µn x
¶
px0(1 − p0)n−xI(
µn x
¶
px0(1 − p0)n−x≤ µn
x0
¶
px00(1 − p0)n−x0)
where I(a ≤ b) is the indicator function of the event if a ≤ b which generates the p-value as 0.0526.2
We present the results in Example 3.4 only for reflecting the fact that in the discrete distribution case there may also have several versions of test statistic for Fisherian significance test and various conclusions may be drawn from these tests. Only a consistent way in defining the significance test makes it sense in the interpretation of a p-value for statisticians.
However, as long as we decide to apply the HDS test to determine the bound of p-value so that we may classify a test as a significant one or a non-significant one, this classification should be further studied.
Whatever a Fisherian significance test can interpret, this test is not justified with any desired optimal property. However, it is generally not true that tests for parameters of discrete distributions are with the same p-value (see this point in Welsh (1996)). We slightly revise the definition to enlarge the class of tests for comparison that results the following theorem which may be analogously derived as we did for Theorem 3.2.
Theorem 3.6. Consider that the underlying distribution is discrete and the hypothesis is H0 : θ = θ0. For given observation X = x0, suppose that the HDS test has p-value, phd, and the number of elements x in the non-extreme set Ehdc is denoted by nhd. For any significance test with p-value smaller than or equal to phd and its element number in its corresponding non-extreme set is denoted by n0. Then, n0 ≥ nhd.
This optimality of smaller element number for the HDS test does provide a justification for its application to the discrete distributions.
Let’s consider the case that p0 = 0.5 and n = 50 to verify the result presented in Theorem 3.6, and we list the numbers of non-extreme sets for both HDS test and the one-sided Fisherian significance test that they have p-values of some interest ones less than or equal
to 0.05.
Table 1. Numbers of non-extreme points for HDS test and one-sided Fisherian significance test with approximated equal p-value.
p-value HDS test One-sided test
0.001 23 ∼ 25 37 ∼ 38
0.005 21 ∼ 23 35 ∼ 36
0.01 19 ∼ 21 34 ∼ 35
0.02 17 ∼ 19 33 ∼ 34
0.03 17 ∼ 19 33 ∼ 34
0.04 15 ∼ 17 32 ∼ 33
0.05 15 ∼ 17 32 ∼ 33
For interpretation, suppose that we are interesting in the tests with p-values around 0.02.
Then, the HDS test takes about number 17 to 19 of x’s in sample space in the non-extreme set for having p-value phds ≈ 0.02. However, one-sided Fisherian significance test takes about number 33 to 34 of x’s in sample space in the non-extreme set to have the same p-value which is about twice the number for HDS test.
Explicit formulations of p-value for Fisherian significance test and HDS test are generally not obtainable in the case of discrete distribution. For considering the HDS test, we show two results related to the computation of p-value when variable X follows a binomial distribution.
Theorem 3.7. Consider that random variable X has a binomial distribution b(n, p). Then the p-value of the HDS test for hypothesis H0 : p = p0 with observation X = x0 and it for hypothesis H0 : p = 1 − p0 with observation X = n − x0 are identical.
Proof. It is followed from the fact that L(x0, p = p0) = L(n − x0, p = 1 − p0). 2
The above theorem indicates that when the p-values of the HDS test for hypothesis H0 : p = p0 with p0 ≤ 0.5 are available, then those for cases p0 > 0.5 are automatically implied.
It is very often that the hypothesis about binomial p is the case that p0 = 0.5. We list some results of p-value for HDS test in the following theorem.
Theorem 3.8. Consider that random variable X has a binomial distribution b(n, p) and
indicating that L(x, p = 0.5) is monotone increasing on {0, 1, ..., k} and monotone decreasing on {k, k + 1, ..., 2k}. This indicates the results in (a) and (b).
indicating that L(x, p = 0.5) is monotone increasing on {0, 1, ..., k} and monotone decreasing on {k + 1, ..., 2k + 1} and the results in (c) and (d) are followed.2
For application, it is desired to have table of p-value for all cases of binomial distributions.
It is not difficult in computation of it for every situation of p0. For considering only p0 = 0.5, we here display p-values in Appendix Tables A.1-A.6 for cases with sample size n is less than or equal to 30.