This post is a continuation of the preceding post on (a,b,0) class. This post introduces the class of discrete discrete distributions called the (a,b,1) class.

The discussion in this post has a great deal of technical details. A concise summary of the (a,b,0) class and (a,b,1) class is found here.

**The (a,b,1) Class**

A counting distribution is a discrete probability distribution that takes on the non-negative integers. Let be a random variable that is a counting distribution. For each integer , let if is the counting distribution being considered. Recall that a counting distribution is a member of the (a,b,0) class of distributions if the following recursive relation holds for some constants and .

**(1)**……….

For a member of the (a,b,0) class, the initial probability is fixed (the sum of all the must sum to 1). Thus a member of the (a,b,0) class has two parameters, namely and in the recursive relation (1). A counting distribution is a member of the (a,b,1) class of distributions if the following recursive relation holds for some constants and .

**(2)**……….

The recursion in the (a,b,1) class begins at . That means that the initial probability must be an assumed value. The probability is then the value such that the sum is . Thus a member of the (a,b,1) class has three parameters: , and .

There are two subclasses in the (a,b,1) class of distributions. They are determined by whether or The (a,b,1) distributions in the first category are called the zero-truncated distributions. The (a,b,1) distributions in the second category are called zero-modified distributions.

This is how we will proceed. Using a given distribution in the (a,b,0) class as a starting point, we show how to derive the zero-truncated distribution. From a given zero-truncated distribution, we show how to derive the zero-modified distribution.

The name of the (a,b,1) distribution has the same (a,b,0) name with either zero-truncated or zero-modified as the prefix. For example, if the starting point is the negative binomial distribution in the (a,b,0) class, then the derived distributions in the (a.b.1) class are the zero-truncated negative binomial distribution and the zero-modified negative binomial distribution.

There are only three distributions in the (a,b,0) class – Poisson, binomial and negative binomial. Then the (a,b,1) class contains the zero-truncated and zero-modified versions of these three distributions. However, the (a,b,1) class contained distributions that are not modifications of the (a,b,0) distributions. We discuss three such additional distributions – extended truncated negative binomial (ETNB) distribution, logarithmic distribution and Sibuya distribution. These three distributions that are not derived from an (a,b,0) distribution are discussed in a separate section below.

We present three examples demonstrating how using a base (a,b,0) negative binomial distribution to derive a zero-truncated negative binomial distribution (Example 1) and a zero-modified negative binomial distribution (Example 2). We also give an example for ETNB distribution (Example 3).

**Notations**

To facilitate discussion, let’s fix some notations. To clearly denote the distributions, notations without superscripts and subscripts refer to the (a,b,0) distributions. Notations with the superscript T (or subscript T) refer to the zero-truncated distributions in the (a,b,1) class. Likewise notations with the superscript M (or subscript M) refer to the zero-modified distributions in the (a,b,1) class.

For example, the following are the probability function (pf) and the probability generating function (pgf) of a distribution from the (a,b,0) class.

**(3)**……….

The following shows the notations for the pf and pgf for a zero-truncated distribution from the (a,b,1) class.

**(4)**……….

The following shows the notations for the pf and pgf for a zero-modified distribution from the (a,b,1) class.

**(5)**……….

Whenever it is convenient to do so, is a random variable from (a,b,0) while and are to denote random variables for the zero-truncated distribution and zero-modified distribution, respectively.

**Zero-Truncated Distributions**

The focus in this section is on the zero-truncated distributions that originate from the (a,b,0) class. The three distributions indicated above (ETNB, logarithmic and Shibuya) are discussed in a separate section below.

Suppose we start with a distribution from the (a,b,0) class, with the notations , and as indicated above. We show how to derive the corresponding zero-truncated distribution in the (a,b,1) class. For the zero-truncated distribution, there are two ways to compute probabilities. One is the recursion relation:

**(6)**……….

The recursion relation (6) is identical to the one in (2). The recursion begins at . The zero-truncated probabilities can also be derived from the (a,b,0) probabilities as follows:

**(7)**……….

The probabilities in (7) can be regarded as conditional probabilities – the probability that given that . From a procedural standpoint, the probabilities are the (a,b,0) probabilities multiplied by to make the probabilities sum to 1. With the probabilities established, the probability generating function (pgf) and mean and moments of the zero-truncated distribution can also be expressed in terms of the corresponding quantities of the (a,b,0) distribution.

**(8)**……….

**(9)**……….

**(10)**……..

**(11)**……..

The goal of the above items is to inform on the zero-truncated distribution based on information from the (a,b,0) distribution. They can also be derived based on definitions using the probability function (7). The following shows the factorial means of the zero-truncated distribution.

**(12)**……..

……..

**(13)**……..

The first factorial mean is identical to the mean of . The in is the value of zero probability for the corresponding member in the (a,b,0) class. The higher factorial moments are derived recursively as in the (a,b,0) case. The raw moments can be derived using the factorial moments. The variance, as derived from the factorial moments, is:

**(14)**……..

*Example 1*

It is helpful to go through an example. First, we set up an (a,b,0) distribution – an negative binomial distribution with parameters and . The (a,b,0) parameters are and . The following gives the pf and the recursive relation for this negative binomial distribution, as well as the mean, variance and pgf.

……….

……….

……….

……….

……….

The following table shows the first 5 probabilities for the zero-truncated negative binomial distribution.

*Table – Zero-Truncated Negative Binomial*(a,b,0) | Zero-Truncated | |
---|---|---|

0 | ||

1 | ||

2 | ||

3 | ||

4 | ||

5 |

The probabilities are generated by either the (a,b,0) pf or the recursive relation. The probabilities are generated by the recursive relation (7) or by the recursive relation (6). The following lists the mean, variance and pgf of the zero-truncated negative binomial example.

……….

……….

……….

**Zero-Modified Distributions**

The goal of this section is to derive a zero-modified distribution from a zero-truncated distribution, either derived from an (a,b,0) distribution as discussed in the preceding section, or a truncated distribution not originated from (a,b,0) class.

We now take a zero-truncated distribution as a given and derive the probabilities and other distributional quantities. As in the case of zero-truncated distribution, one way to generate probabilities is through the recursion process:

**(15)**……..

The probability is an assumed value. The probability is the value that ensures that all the probabilities sum to 1. As indicated the recursion begins at . Another way to calculate probabilities is through the zero-truncated distribution:

**(16)**……..

Of course, if the zero-truncated distribution is based on a distribution from the (a,b,0) class, we can express the zero-modified probabilities as, after plugging (7) into (16):

**(17)**……..

Further distributional quantities can now be derived:

**(18)**……..

**(19)**……..

**(20)**……..

The result (18) is the pgf of the zero-modified distribution based on the pgf of the given zero-truncated distribution. In words, (19) says that the mean of the modified distribution is times the mean of the given zero-truncated distribution. In words, (20) says that the variance of the zero-modified distribution is times the variance of the given zero-truncated distribution plus times the square of the mean of the truncated distribution.

If the given zero-truncated distribution is actually obtained from a member of the (a,b,0) class, then the above three results can be expressed in terms of (a,b,0) information, after plugging the corresponding information for into (18), (19) and (20).

**(21)**……..

**(22)**……..

**(23)**……..

*Example 2*

Consider the zero-truncated negative binomial distribution considered in Example 1. We now generate information on the corresponding zero-modified negative binomial distribution with the assumed value of . The following table gives several probabilities.

*Table – Zero-Modified Negative Binomial*(a,b,0) | Zero-Truncated | Zero-Modified | |
---|---|---|---|

0 | 0.2 | ||

1 | |||

2 | |||

3 | |||

4 | |||

5 |

The zero-modified probabilities are calculated according to (16). With the assumed value , . We simply multiply each zero-truncated probability by 0.8. Using (18), (19) and (20), we obtain the mean, variance and pgf of the zero-modified negative binomial example.

……….

……….

……….

**Additional Zero-Truncated Distributions**

As indicated earlier, the (a,b,1) class contains distributions other than the ones derived from the three (a,b,0) distributions. These distributions also have the zero-truncated versions as well as the zero-modified versions. We discuss the truncated versions. They are: the extended truncated negative binomial (ETNB) distribution, the logarithmic distribution and the Sibuya distribution. The extended truncated negative binomial (ETNB) distribution is resulted from relaxing the r parameter of the negative binomial distribution. The logarithmic distribution and Sibuya distribution are derived from the ETNB distribution. The modified versions of these three distributions can then be obtained by going through the process outlined in the preceding section.

**ETNB**

Recall that the (a,b,0) negative binomial distribution has two parameters and . The following gives the parameters and used in the (a,b,0) recursion and the first two probabilities.

……….

……….

The extended negative binomial distribution is resulted from extending the parameter so that is applicable in addition to the usual . With the extension of , the ETNB probabilities are generated according to the truncated probabilities of (7). In effect, we are pretending that we are starting from a base (a,b,0) negative binomial distribution even the parameter could be such that . Thus the two parameters of the zero-truncated ETNB distribution are given by the following:

**(24)**……..

What do we do with the ETNB parameters indicated in (24)? Using these and , we can generate the “negative binomial” probabilities according to the recursive relation (1) with . However, with being negative, these values of are not probabilities (in fact they are negative). However, the value of is also negative when is negative. Using (7), the zero-truncated probabilities are positive. Thus the “negative binomial” distribution using a negative is not really a distribution. It is just a device to define ETNB distribution.

Using the idea in the preceding paragraph, we can also come up with direct formula for the ETNB probabilities . The following gives the first three probabilities.

……….

……….

……….

Based on the pattern of the above three probabilities, the ENTB probability , , is:

**(25)**……..

All other distributional quantities such as pgf and means and higher moments can be derived based on the ETNB pf For example, the mean, variance and pgf are:

**(26)**……..

**(27)**……..

**(28)**……..

**Logarithmic Distribution**

This is a truncated distribution that is derived from ETNB by letting . The following shows the information that is needed for the recursive generation of probabilities.

**(29)**……..

**(30)**……..

The parameter is obtained by letting in the for ETNB. The logarithmic is from taking the limit of the ETNB as (using the L’Hopital’s rule). The rest of the pf for can be generated from the recursive relation (6). Unlike a zero-truncated distribution that is derived from an (a,b,0) distribution, the distributional quantities of the logarithmic distribution cannot be derived from an (a,b,0) distribution. Thus in order to gain more information about the logarithmic, its pf must be used. The mean and variance for the logarithmic distribution are:

**(31)**……..

**(32)**……..

**Sibuya Distribution**

This is a truncated distribution that is derived from ETNB by letting and making . The following shows the information that is needed for the recursive generation of probabilities.

**(33)**……..

**(34)**……..

All of the three items are obtained by letting in the corresponding items in ETNB. To see that , rewrite the ETNB as follows:

……….

As , the ratio goes to 1. As , goes to 0 because is negative. Thus the above goes to . With the and in (34) and the in (35), the rest of the Shibuya pf can be generated by the recursive relation in (6). Note that the mean does not exist for the Shibuya distribution. The following is the pgf of the Sibuya distribution.

**(35)**……..

Once these three zero-truncated distributions are obtained, we can derive the zero-modified versions of these distributions in the process described earlier.

*Example 3*

We demonstrate how ETNB is calculated. Let and . Then the parameters for the “artificial” negative binomial distribution are:

……….

The for the artificial negative binomial distribution is , making . We generate the fake negative binomial probabilities recursively using the and . Then we multiply by the to get the zero-truncated probabilities according to (7).

*Table – Zero-Truncated ETNB*Artificial | Zero-Truncated | |
---|---|---|

0 | ||

1 | ||

2 | ||

3 | ||

4 | ||

5 |

The column labeled artificial is obviously not probabilities. It is generated recursively using and . Then multiply the column labeled artificial by to obtain the ETNB probabilities, which can also be computed directly using (25).

Using (26) and (27), the ETNB mean and variance are and . With an assumed value of , we generate the first 5 zero-modified ETNB probabilities in the following table.

*Table – Zero-Modified ETNB*Artificial | Zero-Truncated | Zero-Modified | |
---|---|---|---|

0 | 0.1 | ||

1 | |||

2 | |||

3 | |||

4 | |||

5 |

With the assumed value of , the zero-modified probabilities are obtained by multiplying the zero-truncated probabilities by . Using (19) and (20), the zero-modified ETNB mean and variance are: and .

**Practice Problems**

The discussion in this post has a great deal of technical details. A concise summary of the (a,b,0) class and (a,b,1) class is found here.

Practice problems on (a,b,0) class

Practice problems on (a,b,1) class

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