Generalized Bernoulli Process and Fractional Binomial Distribution

We first review the generalized Bernoulli process $\{X_{i},i\in\mathbb{N}\}$ developed in Lee (2021b). We will call it the generalized Bernoulli process I (GBP-I) to differentiate it from what we will develop in this paper. The GBP-I, $\{X_{i},i\in\mathbb{N}\}$, was defined with parameters, $p,H,c$ that satisfy the following assumption.

Under Assumption 1, the GBP-I is well defined with the following probabilities.

and for any disjoint sets $A,B\subset\mathbb{N},$

and

The following operators were defined in Lee (2021b) to express the above probabilities more conveniently.

Using the operators, the probabilities in the GBP-I can be expressed as

for disjoint sets $A,B\subset\mathbb{N}.$ The GBP-I is a stationary process with covariance function

When $H\in(.5,1),$ the GBP-I possesses LRD since $\sum_{i=1}^{\infty}Cov(X_{1},X_{i})=\infty.$ The sum of the first $n$ variables in the GBP-I was defined as the fractional binomial random variable whose mean is $np$, and if $H\in(.5,1),$ its variance is asymptotically proportional to $n^{2H}.$ We will call it fractional binomial distribution I, and denote it by $B_{n}(p,H,c)$ or simply $B_{n}$. When $c=0,$ $B_{n}(p,H,0)$ becomes the regular binomial random variable whose parameters are $n,p.$

We define the generalized Bernoulli process II (GBP-II), $\{X_{i}^{(n)},i=1,2,\cdots,n\}$, $n\in\mathbb{N}$, that has three parameters $H,c,\lambda,$ and the fractional binomial random variable II, $B_{n}^{\circ}(H,c,\lambda)=\sum_{i=1}^{n}X_{i}^{(n)}.$ To ease our notation, $X_{i}$ and $X_{i}^{(n)}$ are used interchangeably, and $B_{n}^{\circ}$ will replace $B_{n}^{\circ}(H,c,\lambda)$ when there is no confusion. For the GBP-II, it is assumed that the parameters satisfy the following condition.

We will show that the GBP-II is well-defined under Assumption 2 with the following probabilities. For $n\in\mathbb{N},$ $\{X_{i}^{(n)},i=1,2,\cdots,n\}$ is defined with $P(X_{i}=1)=p_{n},P(X_{i}=0)=1-p_{n}$ for $i=1,2,\cdots,n,$ and for any $1\leq i_{0}<i_{1}<i_{2}<\cdots<i_{k}\leq n,$

where $p_{n}=\lambda n^{2H-2}.$

If $|A|=1$, then we define $L_{H}^{\circ}(A):=1.$ If $|A|=0$ (when $A=\emptyset$), then $L_{H}^{\circ}(A):=c/p_{n}.$ For example, if $A=\{1,2,4,7\},L_{H}^{\circ}(A)=|(2-1)(4-2)(7-4)|^{2H-2}.$

The joint probabilities in GBP-II is defined with (3) and (1-2) for any disjoint sets $A,B\subset\{1,2,\cdots,n\}$, i.e., by the inclusion-exclusion principle. This can be succinctly written as

To show that the GBP-II is well-defined, we have to verify that $(4)\geq 0$ for any disjoint sets $A,B\subset\{1,2,\cdots,n\},$ for any $n\in\mathbb{N}.$

From the result of Proposition 2.1, the GBP-II is well defined stationary binary sequence whose correlation function obeys a power law asymptotically.

We call the stationary process $\{X_{i}^{(n)},i=1,2,\cdots,n\}$ defined in Theorem 2.2 the generalized Bernoulli process II (GBP-II). Also, the sum of the first $n$ variables in the GBP-II, $B_{n}^{\circ}$, is called the fractional binomial random variable II.

We will show the connection between GBP-II and the fractional Poisson process by using the moment generating function (mgf). First, we modify the GBP-II, and define $\{X_{i}^{*},i=1,2,\cdots\}$ such that

and in general, for $A,B\subset\mathbb{N},A\cap B=\emptyset,$

Note that $D_{H}^{\circ}(A\cup\{0\},B)=D_{H}^{\circ}(A^{(1)}\cup\{1\},B^{(1)})$ where $A^{(j)}=\{i+j:i\in A\}$ and $B^{(j)}=\{i+j:i\in B\}.$ Therefore, by Proposition 2.1, $(5)>0$, and $\{X_{i}^{*},i=1,2,\cdots\}$ is well defined, which we will call GBP-II^∗.

Roughly speaking, $\{X_{i}^{*},i=1,2,\cdots\}$ can be considered as what is observed after the first 1 in the GBP-II for large $n$. In fact, the probability distribution (5) is the limiting distribution of the conditional probability of the GBP-II that is observed after the first observation of ”1”. If we define a random variable $T_{0}$ as the time when the first ”1” appears in the GBP-II, then for any $t,i\in\mathbb{N},$

and for $A,B\subset\mathbb{N},A\cap B=\emptyset,$

which is the same as the distribution of $\{X_{i}^{*},i\in\mathbb{N}\}$ defined in (5).

Define $B_{n}^{\circ,{*}}(H,c)$ as the sum of the first $n$ variables in the GBP-II^∗,

and call it the fractional binomial random variable II${}^{*}.$ In both of the fractional binomial distributions-II, II^∗, the moments are asymptotically proportional to the powers of $n^{2H-1}.$

It turns out that a scaled fractional binomial II^∗ has the same limiting distribution as a scaled fractional Poisson distribution. More specifically, the scaled fractional binomial II^∗ and scaled fractional Poisson, $B_{n}^{\circ,{*}}(H,c)/n^{2H-1}$ and $N_{2H-1,c\Gamma(2H-1)}(n)/n^{2H-1},$ converge in distribution to the second family of Mittag-Leffler random variable of order $2H-1,$ as $n\to\infty.$

We will compare the asymptotic properties of GBPs through asymptotic moments and tail behavior of return time.

First, we define GBP-I^∗ in a similar way that we defined the GBP-II^∗. GBP-I^∗, $\{X_{i}^{*},i=1,2,\cdots\}$, is what is observed after the first “1” in the GBP-I. Then it has

for $A,B\in\mathbb{N},A\cap B=\emptyset,$ where $T_{0}$ is the first time when “1” appeared in the GBP-I. We also define $B_{n}^{*}(p,H,c)$ as the sum of the first $n$ variables in the GBP-I^∗, and call it the fractional binomial random variable I^∗.

Let’s consider the number of 0’s between two successive 1’s plus one as a return time, also called interarrival time, and denote the return time from the $(i-1)^{th}$ 1 to the $i^{th}$ 1 in the GBP-I as $T_{i}.$ In the GBP-I, return times, $\{T_{i},i=2,3,\cdots,\},$ are i.i.d. with

We will drop the subscript and use $T$ as a random variable for return time in the GBP-I.

In the GBP-II $\{X_{i}^{(n)}\}$, we define a return time in the same way and denote $T_{i}^{(n)}$ as the return time to the $i^{th}$ 1. Note that $T_{i}^{(n)},i=2,3,\cdots,$ are independent but not identically distributed, since the length of the sequence $\{X_{i}^{(n)},i=1,\cdots,n\}$ is fixed as $n$. However, they are asymptotically i.i.d. when $n\to\infty,$ with its limiting distribution

which is in fact the distribution of return time $T^{\circ}$ in the GBP-II^∗, if we define return time in the GBP-II^∗, $\{X_{i}^{*},i\in\mathbb{N}\},$ in the same way. In the GBP-II^∗, denote $T_{i}^{\circ}$ as the return time to the $i^{th}$ 1, it is easily derived that return times $T_{1}^{\circ},T_{2}^{\circ},\cdots$ are i.i.d. with distribution (6).

It is found that the return time of the GBP-I has a finite mean, whereas the return time of the GBP-II^∗ has an infinite mean. Also, the return time follows heavy-tail distribution in both the GBP-I and the GBP-II^∗ with

for a large $t$, some constant $c$, and $\alpha\in(1,3)$ for the GBP-I, and $\alpha\in(0,1)$ for the GBP-II.

The monthly unemployment rate (seasonally adjusted, percent) from January 1948 to December 2022 was obtained from the U.S. Bureau of Labor Statistics (https://fred.stlouisfed.org/series/UNRATE) and shown in the left graph of Figure 5. Using the decomposition method, the trend of the time series of the unemployment rate was extracted, and the remaining component of the detrended unemployment rate was used for data analysis. With the detrended time series, a sequence of indicator variables was made to specify if the detrended unemployment rate is above a cutoff (indicator 1) or not (indicator 0) at each time. We tried three cutoffs, .08%, .21%, .28%, and three indicator sequences were obtained. In Figure 5, the graph on the right shows the detrended unemployment rates and horizontal lines at the three cutoffs.

For each indicator sequence, we checked its autocorrelation in the left graphs of Figure 6. It is observed that in the indicator sequence with cutoff .08, the correlation decreases relatively fast, and as the cutoff increases, the decay rate of the correlation slows down.

The GBP-I, GBP-II, Bernoulli process, and Markov chains were applied to each indicator sequence. In each of the four stochastic models, the return times, times between successive 1’s, are independent and identically distributed as

where $k\in\mathbb{N}$, $p,q\in(0,1)$, and $I_{\{k=0\}},I_{\{k>0\}}$ are indicator variables.

In each indicator sequence, we checked whether the return times were correlated. Autocorrelograms on the right in Figure 6 show that at a 5% significance level, there is no significant evidence that return times are correlated. Therefore, we could proceed to fit each of the distributions in (7) to the return times of each indicator sequence.

The results on the MLE of the parameters and the AIC of each model are shown in Tables 1-3. From Table 1, when the cutoff is .08, the Markov chain shows the best fit, having the smallest AIC, followed by the GBP-I. Table 2 shows that with the cutoff of .21, GBP-I has the smallest AIC, although the difference from the second smallest AIC from the Markov chain is very small. In Table 3, when the cutoff is .28, the GBP-I best fits the data, followed by the GBP-II and the Markov chain, in that order. The graphs in Figure 7 show the histogram of the return time overlaid with the fitted probability models.

From the results, it is observed that the relative performance of the GBP-I to the Markov chain improves as the cutoff increases. It seems to be related to the fact that as the cutoff increases, the estimated parameter of $H$ in the GBP-I also increases. Especially, with the largest cutoff, the estimate of $H$ was $.58$, which indicates the presence of long-range dependence (LRD) in the indicator sequence. Since GBPs can incorporate LRD in a binary sequence unlike a Markov chain, it is not surprising that GBP shows a better fit than a Markov chain in the presence of LRD.

We use a dataset in horticulture in Ridout et al. (1998). The dataset contains the number of roots produced by 270 micropropagated shoots of the columnar apple cultivar Trajans which were cultured under different experimental conditions, and the distribution of the number of roots is over-dispersed and has excess zeros. In Ridout et al. (1998), it was shown that zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) models fitted the data better than Poisson and negative binomial models with covariates on the experimental conditions. Here we only use data on the number of roots without covariates, and fit fractional binomial models, ZIP, and ZINB. The results on the MLE of parameters and AIC of each model are provided in Table 4. The FB-I shows the lowest AIC, followed by the FB-II, zero-inflated models, FB-II^∗, and FB-I^∗, in that order. Figure 8 shows the fitted distribution of each model with the data distribution. It was observed that FB-I and FB-II fitted the data well over the entire range of distribution, whereas the zero-inflated models overestimated the variable around the middle of the range and underestimated it in the latter half of the range. FB-I^∗, II^∗ show the worst fit to the data distribution, and also since the estimated $p$ is almost zero in FB-I^∗, the fitted distribution of FB-I^∗ and FB-II^∗ becomes almost identical.