Chi-Squared Tests for Specific Distributions
9.1 Tests for Poisson, binomial, and “binomial” approximation of Feller’s distribution
Let 
be independent and identically distributed random variables. Consider the problem of testing the composite hypothesis 
according to which the distribution of 
is a member of a parametric family with the Poisson distribution
(9.1)
Let 
be the observed frequencies meaning the number of realized values of 
that fall into a specific class or interval 
, where the fixed integer grouping classes 
are such that 
. As before, let 
be a column vector of standardized grouped frequencies with its components as
If the nuisance parameter 
is estimated effectively from grouped data by 
, then the standard Pearson’s sum 
will follow in the limit the chi-square distribution with 
degrees of freedom. If, on the other hand, the parameter 
is estimated from raw (ungrouped) data, for example, by the maximum likelihood estimate (MLE), then the standard Pearson test must be modified. Let 
be a 
(s being the dimensionality of the parameter space) matrix with its elements as
For the null hypothesis in (9.1), the 
matrix 
possesses its elements as
Let 
be the MLE of 
based on the raw data. Then, the well-known Nikulin-Rao–Robson (NRR) modified chi-squared test statistic, distributed in the limit as 
, can be expressed as
(9.2)
where 
and 
are the estimators of the Fisher information matrix (scalar) and of the column vector 
, respectively.
Next, let us consider the binomial null hypothesis. Let the probability mass function be specified as
(9.3)
where 
and 
. In this case, the Fisher information matrix (scalar) is 
and the elements of the matrix 
are
Let 
be the MLE estimator of 
. Then, the modified chi-squared test statistic, distributed in the limit as 
, can be expressed as
(9.4)
where 
and 
.
Now, let the probability distribution of the null hypothesis follow Feller’s 1948, pp. 105–115) discrete distribution with cumulative distribution function
(9.5)
where 
is the complement gamma function.
If the parameter 
and 
, then the distribution function in (9.5) can be approximated as
Using the results of Bol’shev (1963) a more accurate approximation of (9.5) can be obtained as
(9.6)
where 
and 
. Note that (9.6) looks like the binomial distribution function with the exception that the parameter 
can be any real positive number. Consider the probability mass function of (9.6) given by
(9.7)
where the parameter 
. In this case, there are three possibilities to construct a modified chi-squared test for testing a composite null hypothesis about the distribution in (9.6). First one is to use MLEs of 
and 
and the NRR statistic 
. Since the MLEs of 
and 
cannot be derived easily, the modified test 
based on MMEs (see Eq. (4.9)) or Singh’s 
(see Eq. (3.25)) can be used.
For the model in (9.7) and 
, the DN statistic 
will follow in the limit 
and 
. If 
, then, as before (see Eq. (4.12)), we will have
In this case, 
, 
, and
To specify the above tests, we of course will need explicit expressions of all the matrices involved.
The elements of the Fisher information matrix 
for the model in (9.7) are
where 
is the largest integer contained in 
and 
is the psi-function. It is known that series expansions for the psi-function converge very slowly. But, for integer values of x, a recurrence 
can be used, from which it follows that
This result permits us to calculate all expressions containing 
with a very high accuracy.
The elements of the matrix 
are
The elements of the matrix 
are
The elements of the matrix 
are
The components 
and 
of the vector 
for Singh’s test 
for the model in (9.7) are
and
It has to be noted that the test in (3.25) is computationally much more complicated than the statistic 
for large samples.
For the model in (9.7), 
and 
. Denoting the first two sample moments 
and 
and then equating them to population moments, the MMEs of 
and 
are obtained as
(9.8)
From (9.8), we see that negative values of 
and 
are possible, but the proportion of such estimates will be almost negligible for samples of size 
. It seems that 
test can be used for analyzing Rutherford’s data, but the question about 
-consistency of the MMEs in (9.8) is still open.
To examine the rate of convergence of estimators 
and 
for sample sizes 
, we simulated 3,000 estimates of 
and 
assuming that 
and 
, values that correspond to Rutherford’s data. The power curve fit of 
, the average value of estimates 
for 3000 runs, in Figure 9.1 shows that 
and 
. The power curve fit of 
in Figure 9.2 gives 
and 
. To check for the distribution of the statistic 
under the null “Feller’s” distribution (
and 
), we simulated 
values of 
. The histogram of these values is well described by the 
distribution (see Figure 9.3). The average value 
also does not contradict the assumption that the statistic 
follows in the limit the chi-squared distribution with one degree of freedom. Another important property of any test statistic is its independence from the unknown parameters. To check for this feature of the test 
, we simulated 
values of 
assuming that 
(two times less than for the null hypothesis 
) and 
(two times more than for the null hypothesis 
). The results (Figure 9.4) show that the simulated values do not contradict the independence, because the histogram is again well described by 
distribution.
Figure 9.1 Simulated average value of 
(circles) and the power function fit (solid line) as function of the sample size n.
Figure 9.2 Simulated average value of 
(circles) and the power function fit (solid line) as function of the sample size n.
Figure 9.3 The histogram of the 1000 simulated values of 
for the null hypothesis (
and 
) and the 
distribution (solid line).
Figure 9.4 The histogram of the 1000 simulated values of 
for 
and the 
distribution (solid line).
The above results evidently allow us to use the HRM statistic 
for Rutherford’s data analysis.
9.2 Elements of matrices K, B, C, and V for the three-parameter Weibull distribution
For r equiprobable cells of the model in (4.25), the borders of equiprobable intervals are defined as:
Then, the elements of the matrix 
are as follows:
where 
and 
is the psi-function. For the required calculation of 
, we used the series expansion
where 
is the Euler’s constant.
Similarly, the elements of the matrices 
and 
are as follows:
where the population moments are
and 
is the incomplete gamma function. For the required calculation of 
, we used the following series expansion (Prudnikov et al., 1981, p. 705):
Finally, the elements of the matrix 
are as follows:
9.3 Elements of matrices J and B for the Generalized Power Weibull distribution
Elements 
of the Fisher information matrix 
are as follows:
Next, the elements of matrix 
are as follows:
9.4 Elements of matrices J and B for the two-parameter exponential distribution
Consider the two-parameter exponential distribution with cumulative distribution function
(9.9)
where the unknown parameter 
. It is easily verified that the matrix 
for the model in (9.9) is
(9.10)
Based on the set of n i.i.d. random variables 
, the MLE 
of the parameter 
equals 
, where
(9.11)
Consider r disjoint equiprobable intervals
For these intervals, the elements of the matrix 
(see Eq. (3.4)) are
Using the matrix in (9.10) and the above elements of the matrix 
with 
replaced by the MLE 
in (9.11), the NRR test 
(see Eq. (3.8)) can be used. While using Microsoft Excel, the calculations based on double precision is recommended.
9.5 Elements of matrices B, C, K, and V to test the logistic distribution
Let 
, and 
, be borders of r equiprobable random grouping intervals. Then, the probabilities of falling into each interval are 
.
The elements of the 
matrix 
, for 
, are as follows:
Next, the elements of the 
matrix 
are as follows:
where 
is Euler’s dilogarithm function that can be computed by the series expansion
and by the expansion
for 
(Prudnikov et al., 1986, p. 763).
Finally, we have the matrices 
and 
as
9.6 Testing for normality
System requirements for implementing the software of Sections 9.6, 9.7, 9.8, 9.9, 9.10 are Windows XP, Windows 7, MS Office 2003, 2007, 2010.
1. Open file Testing Normality.xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and the desired number of equiprobable intervals 
. The recommended number of intervals for the NRR test 
in (3.8), under close alternatives (such as the logistic), is 
. The recommended number of intervals for the test 
in (3.24) is 
(see Section 4.4.1). Note that the power of 
can be more than that of the NRR test;
5. Numerical values of 
and 
are in cells F2 and G2, respectively. Cells F3 and G3 contain the corresponding percentage points at level 0.05. The P-values of 
and 
are in cells F4 and G4, respectively.
9.7 Testing for exponentiality
9.7.1 Test of Greenwood and Nikulin (see Section 3.6.1)
1. Open file Testing Exp GrNik.xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and desired number of equiprobable intervals. The recommended number of equiprobable intervals is 
;
5. The numerical value of 
(see Eq. (3.44)) is in cell F2. The percentage point at level 0.05 and the P-value are in cells F3 and F4, respectively.
9.7.2 Nikulin-Rao-Robson test (see Eq. (3.8) and Section 9.4)
1. Open file Testing NRR 2-param EXP.xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and desired number of equiprobable intervals. The recommended number of equiprobable intervals is 
;
5. The numerical value of 
is in cell F2. The percentage point at level 0.05 and the P-value are in cells F3 and F4, respectively.
9.8 Testing for the logistic
1. Open file Testing Logistic.xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and desired number of equiprobable intervals 
. The recommended number of equiprobable intervals, for close alternatives (such as normal), is 
;
5. Numerical values of 
in (4.9) and 
in (4.13) are in cells E2 and F2, respectively. Cells E3 and F3 contain the corresponding percentage points at level 0.05. The P-values of 
and 
are in cells E4 and F4, respectively.
9.9 Testing for the three-parameter Weibull
1. Open file Testing Weibull3.xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and desired number of equiprobable intervals 
. The recommended number of equiprobable intervals for the Exponentiated Weibull and Power Generalized Weibull alternatives is 
;
5. Numerical values of 
in (4.9) and 
in (4.14) (see also Section 9.2) are in cells F2 and G2, respectively. Note that the power of 
is usually higher than that of 
. Cells F3 and G3 contain the corresponding percentage points at level 0.05. The P-values of 
and 
are in cells F4 and G4, respectively.
9.10 Testing for the Power Generalized Weibull
1. Open file Test for PGW (Left-tailed).xls;
2. Enter your sample data in column “I” starting from cell 1;
3. Click the button “Run,” introduce the sample size and desired number of equiprobable intervals 
. The recommended number of equiprobable intervals for the Exponentiated Weibull, Generalized Weibull, and Three-Parameter Weibull alternatives is 
;
4. Click OK. Note that the power of 
in (3.50) is usually higher than that of 
in (3.48);
5. Numerical values of 
and 
(see Eqs. (3.48), (3.50) and Section 9.3) are in cells F2 and G2, respectively. Cells F3 and G3 contain the corresponding percentage points at level 0.05. The P-values of 
and 
are in cells F4 and G4, respectively.
9.11 Testing for two-dimensional circular normality
1. Open file Testing Circular Normality.xls;
2. Enter your sample data in columns “I” and “J” starting from cell 1;
3. Click the button “Compute,” introduce the sample size and the desired number of equiprobable intervals. The recommended number of intervals for the two-dimensional logistic alternative is 
, while the recommended number of intervals for the two-dimensional normal alternative is 3;
5. Numerical values of 
and 
(see Section 3.5.3) are in cells F2 and G2, respectively. Cells F3 and G3 contain the corresponding percentage points at level 0.05. The P-values of 
and 
are in cells F4 and G4, respectively.
References
1. Bol’shev LN. Asymptotical Pearson’s transformations. Theory of Probability and its Applications. 1963;8:129–155.
2. Feller W. On probability problems in the theory of counters. In: Courant Anniversary Volume. New York: Interscience Publishers; 1948;105–115.
3. Prudnikov AP, Brychkov YA, Marichev OI. Integrals and Series. Nauka, Moscow: Elementary Functions; 1981.
4. Prudnikov AP, Brychkov YA, Marichev OI. Integrals and Series. Nauka, Moscow: Additional Chapters; 1986.