A new five-parameter Birnbaum–Saunders distribution for modeling bicoid gene expression data
Introduction
The Birnbaum–Saunders (BS) distribution is a probability distribution used extensively in reliability applications to model failure times. This distribution, which is also known as the fatigue life distribution, originated from a problem of material fatigue proposed by Birnbaum and Saunders [1]. This model has received great attention in recent decades due to its many attractive statistical and probabilistic properties. Although this distribution was originally proposed in the field of material fatigue and reliability analysis, it has been widely applied to various fields beyond the original context such as earth, environmental and medical sciences; see for example [2], [3], [4], [5], [6].
The classical BS distribution contains two parameters; shape (α) and scale (β). The random variable T that follows the two-parameter BS distribution is denoted by T ~ BS(α, β) and its cumulative distribution function (cdf) has the following formwhere Φ( · ) is the cdf of standard normal distribution. There is an interesting relationship between this model and the standard normal distribution. If T ~ BS(α, β), thenFor a detailed review on two-parameter BS distribution see [7]. Moreover, the random variable T follows the three-parameter BS distribution [8], denoted by T ~ BS(α, μ, β), if its cdf is given bywhere the parameter μ represents the location of distribution. Similar to the above situation with two parameters, the relationship between this model and the standard normal distribution can be represented as follows:
Another extension of BS distribution, is based on Johnson’s transformation and includes four parameters (FBS), denoted by T ~ FBS(δ, λ, ξ, γ) [8], which the following cdfwhereThe parameters δ, λ, ξ and γ are the shape, scale, location and non centrality parameters, respectively. The probability density function (pdf) of FBS distribution is as follows:where ϕ( · ) is the pdf of standard normal distribution and
If T ~ FBS(δ, λ, ξ, γ), then
The quantile function (qf) and hazard rate of FBS distribution was studied in [8]. If T ~ FBS(δ, λ, ξ, γ), then the qf and hazard rate of T are given bywhere is the pth quantile of standard normal distribution.
The FBS distribution includes the two-parameter and three-parameter versions of BS distribution as particular cases. Since it can be easily seen that if then and if then . Therefore, the pdf, qf and hazard rate of the two-parameter and three-parameter versions of BS distribution can be obtained from FBS distribution. For a detailed information on FBS distribution see [8].
Recently, the BS distribution has been used as a mathematical model to describe the distribution of bicoid gene expression profile. For instance, fifty-four commonly used distributions were evaluated in [9] and authors concluded that BS distribution with three parameters fits the bicoid profile more accurately than the other distributions. It should be noted that fitting an accurate statistical distribution to bicoid profile is of great importance because of significantly stochastic, highly volatile with a heavy tail and non-normal structure of these data.
The aim of this research is to introduce an extended version of BS distribution with five parameters. Our motivation for extending the BS distribution is recent findings in [10] where authors proposed the FBS distribution to describe the expression profile of bicoid gene. The high performance of FBS distribution in fitting an adequate statistical distribution to bicoid gene expression profile encouraged us to apply an extended version of it including five parameters. This new distribution is the extension of FBS distribution based on Marshall and Olkin’s method, which includes an extra parameter making it more flexible, thus widening the applications of this distribution.
This paper is organized as follows. In Section 2, the pdf, cdf, qf, hazard rate and Maximum Likelihood Estimation (MLE) of parameters of five-parameter BS distribution are proposed. Section 3 provides an extensive simulation study and in Section 4, the five-parameter BS distribution (5BS) is also evaluated on the real data including all the cleavage cycles in which bicoid gene is expressed in the embryo. The paper concludes with a concise summary in Section 5.
Section snippets
A new five-parameter BS distribution
In this section, we propose a new five-parameter BS distribution, which is the extension of FBS distribution using Marshall and Olkin’s method [11]. This new distribution with five parameters denoted by 5BS. The resulting distribution is more flexible than the original distribution owing to an additional parameter. The pdf, cdf, qf and hazard rate of 5BS distribution are carried out in this section. Finally, we propose the log-likelihood function and equations to obtain ML estimates of
Simulation study
A simulation study is first utilized to assess the applicability of the proposed 5BS distribution to real bicoid expression profiles. The Synthesis Diffusion Degradation (SDD) model is a common model for analyzing the bicoid profile and follows an exponential curve [14], [15], [16], [17]:where A is the amplitude, x is distance from the anterior, and λ is the length parameter. Here, an exponential curve with and is used. Furthermore, normally distributed random noise with
Real data application
In this section, we apply the 5BS distribution to describe the data of cleavage cycles 10, 11 and 12 in which bicoid gene is expressed in the embryo. A detailed description of different cleavage cycles can be found in [9]. Table 2 presents the p-values of KS and AD tests together with MLE of parameters obtained in fitting 5BS distribution to these data. The standard errors of ML estimators are reported in parenthesis. Similar to the simulation study, the results obtained here confirm that 5BS
Conclusion
A new five-parameter BS distribution, which is the extension of four BS distribution, was introduced and its parameters was estimated using MLE approach. Since there are no explicit solutions to the MLE equations, due to their non-linearity and complicated form, the MLE of five parameters were achieved numerically. Moreover, the pdf, cdf, qf and hazard rate of the proposed five-parameter BS distribution were theoretically discussed. Simulation results as well the outcome of fitting 5BS
Declaration of Competing Interest
None.
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