Abstract

Amer et al. [1] considered the distributions of the sum and the difference of two independent and identically distributed random variables with the common Quasi Lindley distribution. They derived, very nicely, the above mentioned distributions and provided certain important mathematical and statistical properties as well as simulations and applications of the new distributions. Wang and Ma [2] considered the sum of the gamma random variables under the assumption of independence of the summands and presented very interesting results. In this short note, we like to show that the assumption of independence can be replaced with a much weaker assumption of sub-independence in both papers. Then we present certain characterizations of the distributions derived by Amer et al. [1], called 2SQLindley and 2DQLindley distributions.

1. Introduction

As we have done in a couple of our previous papers, to make this short note self contained, we will copy some parts of our work ³ here.

We may in some occasions have asked ourselves if there is a concept between "uncorrelatedness" and "independence" of two random variables. It seems that the concept of "sub-independence" is the one: it is much stronger than uncorrelatedness and much weaker than independence. The notion of sub-independence seems important in the sense that under usual assumptions, Khintchine’s Law of Large Numbers and Lindeberg-Levy’s Central Limit Theorem as well as other important theorems in probability and statistics hold for a sequence of s.i. (sub-independent) random variables. While sub-independence can be substituted for independence in many cases, it is difficult, in general, to find conditions under which the former implies the latter. Even in the case of two discrete identically distributed rv’s (random variables) X and Y, the joint distribution can assume many forms consistent with sub-independence.

Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent (and often identically distributed) random variables rather than the joint distribution of the summands. Therefore, the full force of independence of the summands will not be required. In other words, it is the convolution of the marginal distributions which is needed, rather than the joint distribution of the summands which, in the case of independence, is the product of the marginal distributions. The concept of sub-independence, which is weaker than that of independence, is shown to be sufficient to yield the conclusions of these theorems and results. This is precisely the reason for the statement: "why assume independence when you can get by with sub-independence".

The concept of sub-independence can help to provide solution for some modeling problems where the variable of interest is the sum of a few components. Examples include household income, the total profit of major firms in an industry, and a regression model where and are uncorrelated, however, they may not be independent. For example, in Bazargan et al. ⁴, the return value of significant wave height is modeled by the sum of a cyclic function of random delay and a residual term They found that the two components are at least uncorrelated but not independent and used sub-independence to compute the distribution of the return value.

Let and be two (random variables) with joint and marginal (cumulative distribution functions) and respectively. Then and are said to be independent if and only if

(1.1)

or equivalently, if and only if

(1.2)

where and respectively, are the corresponding joint and marginal (characteristic functions). Note that (1.1) and (1.2) are also equivalent to

(1.3)

The concept of sub-independence, as far as we have gathered, was formally introduced by Durairajan (1979) and developed by Hamedani in the past 40 years, stated as follows: The and with and are s.i. (sub-independent) if the of is given by

(1.4)

or equivalently if and only if

(1.5)

The drawback of the concept of sub-independence in comparison with that of independence has been that the former does not have an equivalent definition in the sense of (1.3) which some believe, to be the natural definition of independence. We found such a definition which is stated below. We shall give the definition for the continuous case (Definition 1.1).

We observe that the half-plane can be expressed as a countable disjoint union of rectangles:

where and are intervals. Now, let be a continuous random vector and for let

and

Definition 1.1. The continuous and are s.i. if for every

(1.6)

To see that (1.6) is equivalent to (1.4), observe that ( of (1.6))

(1.7)

where Now, if and are s.i. then

where are probability measures on defined by

and is the product measure.

We also observe that ( of (1.6))

Now, (1.7) and (1.8) will be equal if which is true since the points in are obtained by shifting each point in over to the right by units and then up by units.

If and are s.i., then unlike independence, and are not necessarily s.i. for any real This demonstrates how weak is the concept of sub-independence in comparison with that of independence. Please observe the following simple example.

Example 1.1. Let and have the joint given by

where is an appropriate constant. (The characteristic function is the Fourier transform of (probability density function), so the corresponding joint is given by

where

Then and are s.i., standard normal , and hence is normal with mean 0 and variance 2, but and are not s.i. and consequently does not have a normal distribution.

The concept of sub-independence defined above can be extended to as follows.

Definition 1.2. The are s.i. if for each subset of

2. Remarks

i) If the and are with common Quasi Lindley distribution with parameters the characteristic function of is

The of is

and since and are we have

ii) If the and are with common Quasi Lindley distribution with parameters and if and are the characteristic function of is

The of under the assumption of of and is

iii) In view of i) and ii), the assumption of "independence" in Amer et al. paper can be replaced with that of "sub-independence".

iv) Equation (2.3) of Wang and Ma holds for gamma

v) In Theorem 3.2 of Wang and Ma, the distribution of is a gamma distribution with parameters under the assumption that are and are equal to

vi) In Theorem 5.1 of Wang and Ma, the distribution of is a Chi-square distribution with parameter under the assumption that are

vii) For a detailed treatment of the concept of sub-independence, we refer the interested reader to Hamedani ³.

3. Characterizations of the 2S-Lindley and 2D-Lindley Distributions

Amer et al. ¹ introduced the distributions of the sum and differences of two (now, ) Quasi Lindley random variables with parameters (called 2SQLindley and 2DQLindley) with their respective given by

(3.1)

and

(3.2)

Following our ³ work, to understand the behavior of the data obtained through a given process, we need to be able to describe this behavior via its approximate probability law. This, however, requires to establish conditions which govern the required probability law. In other words we need to have certain conditions under which we may be able to recover the probability law of the data. So, characterization of a distribution is important in applied sciences, where an investigator is vitally interested to find out if their model follows the selected distribution. Therefore, the investigator relies on conditions under which their model would follow a specified distribution. A probability distribution can be characterized in different directions one of which is based on the truncated moments. This type of characterization initiated by Galambos and Kotz ⁵ and followed by other authors such as Kotz and Shanbhag ⁶, Glänzel et al. ⁷, Glänzel ⁸, Glänzel and Hamedani ⁹ and Kim and Jeon ¹⁰, to name a few. For example, Kim and Jeon ¹⁰ proposed a credibility theory based on the truncation of the loss data to estimate conditional mean loss for a given risk function. It should also be mentioned that characterization results are mathematically challenging and elegant. In this section, we present characterizations 2S-Lindley and 2D-Lindley distributions based on the conditional expectation (truncated moments) of certain function of the random variable.

We will employ Theorem 1 of Glänzel ⁸ given in the Appendix A. As shown in Glänzel ¹¹, this characterization is stable in the sense of weak convergence.

Proposition 3.1. Let be a continuous random variable and let and for Then has (3.1) if and only if the function defined in Theorem 1 is of the form

Proof. If has (3.1), then

and

and hence

We also have

Conversely, if is of the above form, then

and

Now, according to Theorem 1, has density (3.1).

Corollary 3.1. Suppose is a continuous random variable. Let be as in Proposition 3.1. Then has density (3.1) if and only if there exist functions and defined in Theorem 1 for which the following first order differential equation holds

Corollary 3.2. The differential equation in Corollary 3.1 has the following general solution

where is a constant.

Proof. If has pdf (3.1), then clearly the differential equation holds. Now, if the differential equation holds, then

or

or

from which we arrive at

A set of functions satisfying the above differential equation is given in Proposition 3.1 with Clearly, there are other triplets satisfying the conditions of Theorem 1.

Remark 3.1. Similar results can be stated for the 2DQLindley distribution as well.

References

Appendix A

Theorem 1. Let be a given probability space and let be an interval for some Let be a continuous random variable with the distribution function and let and be two real functions defined on such that

is defined with some real function Assume that and is twice continuously differentiable and strictly monotone function on the set Finally, assume that the equation has no real solution in the interior of Then is uniquely determined by the functions and particularly

where the function is a solution of the differential equation and is the normalization constant, such that

Note: The goal is to have the function as simple as possible.

We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, ¹¹), in particular, let us assume that there is a sequence of random variables with distribution functions such that the functions and satisfy the conditions of Theorem 1 and let for some continuously differentiable real functions and Let, finally, be a random variable with distribution Under the condition that and are uniformly integrable and the family is relatively compact, the sequence converges to in distribution if and only if converges to where

This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions and respectively. It guarantees, for instance, the ‘convergence’ of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if

A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions and, specially, should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.

In some cases, one can take which reduces the condition of Theorem 1 to We, however, believe that employing three functions and will enhance the domain of applicability of Theorem 1.