A generalized soft likelihood function in combining multi-source belief distribution functions

Abstract

Likelihood function has significant advantages in the fields of statistical inference. Based on this theory, Yager proposed a soft likelihood function to make it more widely used. However, Yager’s method can only deal with probabilities expressed by crisp values, and has strict restrictions on the form of data. Due to human subjectivity and lack of effective information, it is inevitable that data uncertainty will be involved. In order to deal with the uncertain data more flexibly and intuitively and solve the complex problems faced in real-world applications, a generalized soft likelihood function in combining multi-source belief distribution functions is proposed in this paper. Different from other existing methods, this paper uses a distribution function to represent uncertain information, which can retain more original information and improve the credibility of the results. The expectation and variance are used to rank the obtained evidences, and the evidence that contributes more to the results is ranked higher. Finally, the reliable likelihood results are obtained. The proposed method extends the method of Yager and can work well in more uncertain environment. Several numerical examples and comparative experimental simulation are used to illustrate the efficiency of the proposed soft likelihood function.

Appendix A: The derivation process of likelihood results based on distribution function

Suppose there are m pieces of belief value that have been arranged in descending order and obey the distribution function. They are F₁,F₂,F₃, ⋯ , F_m. The weights obtained based on OWA operator are w₁, w₂, w₃ ⋯, w_m, here ${w_{i}} \in [0,1],{\sum }_{i} {{w_{i}} = 1}$.

(1) Assuming that the function F is a gaussian distribution function with mean μ and standard deviation δ, here δ > 0,μ ∈ [0, 1]. We can get the likelihood result L is

$$ \begin{array}{@{}rcl@{}} \sum\limits_{k = 1}^{m} \frac{\frac{1}{(2\pi )^{\frac{k - 1}{2}}}\sqrt{\frac{1}{\prod\limits_{i = 1}^{k} \delta_{i}^{2}}} \exp \left[ - \frac{1}{2}(\sum\limits_{i = 1}^{k} \frac{\mu_{i}^{2}}{\delta_{i}^{2}} - \frac{(\sum\limits_{i = 1}^{k} \frac{\mu_{i}}{\delta_{i}^{2}})^{2}}{\sum\limits_{i = 1}^{k} \frac{1}{\delta_{i}^{2}}}) \right]}{\sqrt{2}{\pi}}\\ \exp \left[ - (x - \frac{\sum\limits_{i = 1}^{k} \frac{\mu_{i}}{\delta_{i}^{2}}^{2}} {\sum\limits_{i = 1}^{k} \frac{1}{\delta_{i}^{2}}})*\sum\limits_{i = 1}^{k} \frac{1}{\delta_{i}^{2}} \right] \text{*} (\frac{k}{m} - (\frac{k - 1}{m})^{\frac{1 - \alpha}{\alpha}}). \end{array} $$

The specific proof process is as follows.

$$ \begin{array}{l} {\text{L}} = {{\text{F}}_{1}}*{{\text{w}}_{1}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}*{{\text{w}}_{2}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}*{{\text{w}}_{3}} + {\cdots} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}{\text{*}} {\cdots} {^{\ast}}{{\text{F}}_{\text{m}}}*{{\text{w}}_{\text{m}}} \\ {\text{ = }}\frac{1}{{\sqrt {2\pi } {\delta_{1}}}}{\text{exp - }}\left[ { - \frac{{{{(x - {\mu_{1}})}^{2}}}}{{2{\delta_{1}}^{2}}}} \right]{\text{*}}{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{{{{\text{S}}_{12}}}}{{\sqrt {2\pi {\delta_{12}}^{2}} }}\exp \left[ { - \frac{{{{(x - {\mu_{12}})}^{2}}}}{{2{\delta_{12}}^{2}}}} \right]{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ + \frac{{{{\text{S}}_{123}}}}{{\sqrt {2\pi {\delta_{123}}^{2}} }}\exp \left[ { - \frac{{{{(x - {\mu_{123}})}^{2}}}}{{2{\delta_{123}}^{2}}}} \right]{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) + {\cdots} \\ + \frac{{{{\text{S}}_{1 {\ldots} m}}}}{{\sqrt {2\pi {\delta_{1 {\ldots} m}}^{2}} }}\exp \left[ { - \frac{{{{(x - {\mu_{1 {\ldots} m}})}^{2}}}}{{2{\delta_{1 {\ldots} m}}^{2}}}} \right]{\text{*1}}(- {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ = \frac{1}{{\sqrt {2\pi } {\delta_{1}}}}{\text{exp - }}\left[ { - \frac{{{{(x - {\mu_{1}})}^{2}}}}{{2{\delta_{1}}^{2}}}} \right]{\text{*}}({(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{1 - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ + \frac{{\exp \left[ { - \frac{1}{2}(\sum\limits_{i = 1}^{2} \frac{{{\mu_{i}^{2}}}}{{{\delta_{i}^{2}}}} - \sum\limits_{i = 1}^{2} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} )} \right]}}{{\sqrt {2\pi \prod\limits_{i = 1}^{2} {{\delta_{i}^{2}}} } }}\exp \left[ { - \frac{{{{(x - \left[ {\sum\limits_{i = 1}^{2} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} } \right]\frac{{{\delta_{1}^{2}}{\delta_{2}^{2}}}}{{{\delta_{1}^{2}} + {\delta_{2}^{2}}}})}^{2}}}}{{2\frac{{{\delta_{1}^{2}}{\delta_{2}^{2}}}}{{{\delta_{1}^{2}} + {\delta_{2}^{2}}}}}}} \right]{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ + \frac{{\sqrt {\frac{1}{{\prod\limits_{i = 1}^{3} {{\delta_{i}^{2}}} }}} \exp \left[ { - \frac{1}{2}(\sum\limits_{i = 1}^{3} \frac{{{\mu_{i}^{2}}}}{{{\delta_{i}^{2}}}} - \sum\limits_{i = 1}^{3} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} )} \right]}}{{2{\pi^{\frac{2}{3}}}}}\exp \left[ { - \frac{{{{(x - \left[ {\sum\limits_{i = 1}^{3} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} } \right]\frac{{{\delta_{1}^{2}}{\delta_{2}^{2}}{\delta_{3}^{2}}}}{{{\delta_{3}^{2}}({\delta_{1}^{2}}{\text{ + }}{\delta_{2}^{2}}) + {\delta_{1}^{2}}{\delta_{2}^{2}}}})}^{2}}}}{2}} \right]{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) + {\cdots} \\ + \frac{{\frac{1}{{{{(2\pi )}^{\frac{{m - 1}}{2}}}}}\sqrt {\frac{1}{{\prod\limits_{i = 1}^{m} {{\delta_{i}^{2}}} }}} \exp \left[ - \frac{1}{2}(\sum\limits_{i = 1}^{m} \frac{{{\mu_{i}^{2}}}}{{{\delta_{i}^{2}}}} - \frac{{{{(\sum\limits_{i = 1}^{m} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} )}^{2}}}}{{\sum\limits_{i = 1}^{m} {\frac{1}{{{\delta_{i}^{2}}}}} }}) \right]}}{{\sqrt {2\pi } }}\exp \left[ { - {{(x - \frac{{\sum\limits_{i = 1}^{m} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} }}{{\sum\limits_{i = 1}^{m} {\frac{1}{{{\delta_{i}^{2}}}}} }})}^{2}}*\sum\limits_{i = 1}^{m} {\frac{1}{{{\delta_{i}^{2}}}}} } \right]{\text{*1}}(- {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\text{ = }}\sum\limits_{k = 1}^{m} {\frac{{\frac{1}{{{{(2\pi )}^{\frac{{k - 1}}{2}}}}}\sqrt {\frac{1}{{\prod\limits_{i = 1}^{k} {{\delta_{i}^{2}}} }}} \exp \left[ { - \frac{1}{2}(\sum\limits_{i = 1}^{k} \frac{{{\mu_{i}^{2}}}}{{{\delta_{i}^{2}}}} - \frac{{{{(\sum\limits_{i = 1}^{k} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} )}^{2}}}}{{\sum\limits_{i = 1}^{k} {\frac{1}{{{\delta_{i}^{2}}}}} }})} \right]}}{{\sqrt {2\pi } }}\exp \left[ { - {{(x - \frac{{\sum\limits_{i = 1}^{k} {\frac{{{\mu_{i}}}}{{{\delta_{i}}^{2}}}} }}{{\sum\limits_{i = 1}^{k} {\frac{1}{{{\delta_{i}^{2}}}}} }})}^{2}}*\sum\limits_{i = 1}^{k} {\frac{1}{{{\delta_{i}^{2}}}}} } \right]{\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}){\text{\ } }} \end{array}$$

(2) Assuming that the function F is a exponential distribution function with expection $\frac {1}{\lambda }$ and deviation $\frac {1}{{{\lambda ^{2}}}}$, here x ∈ (0, 1], λ_i ∈ (0, 1]. We can get the likelihood result L is

$${\text{L}} = \sum\limits_{k = 1}^{m} (\prod\limits_{i = 1}^{k} {{\lambda_{i}}} ){e^{- (\sum\limits_{i = 1}^{k} {{\lambda_{i}}} )x}}{\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}})$$

The specific proof process is as follows.

$$\begin{array}{l} {\text{L}} = {{\text{F}}_{1}}*{{\text{w}}_{1}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}*{{\text{w}}_{2}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}*{{\text{w}}_{3}} + {\cdots} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}{\text{*}} \cdots {\text{*}}{{\text{F}}_{\text{m}}}*{{\text{w}}_{\text{m}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\lambda_{1}}{e^{- {\lambda_{1}}x}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + {\lambda_{12}}{e^{- ({\lambda_{1}} + {\lambda_{2}})x}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\lambda_{123}}{e^{- ({\lambda_{1}} + {\lambda_{2}} + {\lambda_{3}})x}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\lambda_{1 {\cdots} m}}{e^{- ({\lambda_{1}} + {\lambda_{2}} + {\cdots} + {\lambda_{m}})x}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\lambda_{1}}{e^{- {\lambda_{1}}x}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + ({\lambda_{1}} \cdot {\lambda_{2}}){e^{- ({\lambda_{1}} + {\lambda_{2}})x}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + ({\lambda_{1}} \cdot {\lambda_{2}} \cdot {\lambda_{3}}){e^{- ({\lambda_{1}} + {\lambda_{2}} + {\lambda_{3}})x}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + (\prod\limits_{i = 1}^{m} {{\lambda_{i}}} ){e^{- (\sum\limits_{i = 1}^{m} {{\lambda_{i}}} )x}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{m} (\prod\limits_{i = 1}^{k} {{\lambda_{i}}} ){e^{- (\sum\limits_{i = 1}^{k} {{\lambda_{i}}} )x}}{\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \end{array} $$

(3) Assuming that the function F is a poisson distribution function with expection λ and deviation λ, here $x \in N + ,{\kern 1pt} {\kern 1pt} {\lambda _{i}} \in N +$. We can get the likelihood result L is

$${\text{L}}{\kern 1pt} = \sum\limits_{k = 1}^{m} {\frac{{{{(\prod\limits_{i = 1}^{k} {{\lambda_{i}}} )}^{x}}}}{{{{(x!)}^{k}}}}{e^{- (\sum\limits_{i = 1}^{m} {{\lambda_{i}}} )x}}} {\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}})$$

The specific proof process is as follows.

$$\begin{array}{l} {\text{L}} = {{\text{F}}_{1}}*{{\text{w}}_{1}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}*{{\text{w}}_{2}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}*{{\text{w}}_{3}} + {\cdots} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}{\text{*}} \cdots {\text{*}}{{\text{F}}_{\text{m}}}*{{\text{w}}_{\text{m}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{{\lambda_{1}}^{x}}}{{x!}}{e^{- {\lambda_{1}}x}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{{{\lambda_{12}}^{x}}}{{{{(x!)}^{2}}}}{e^{- ({\lambda_{1}} + {\lambda_{2}})x}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{\lambda_{123}}^{x}}}{{{{(x!)}^{3}}}}{e^{- ({\lambda_{1}} + {\lambda_{2}} + {\lambda_{3}})x}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{{({\lambda_{1 {\cdots} m}})}^{x}}}}{{{{(x!)}^{m}}}}{e^{- ({\lambda_{1}} + {\lambda_{2}} + {\cdots} + {\lambda_{m}})x}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{{\lambda_{1}}^{x}}}{{x!}}{e^{- {\lambda_{1}}x}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{{{{({\lambda_{1}} \cdot {\lambda_{2}})}^{x}}}}{{{{(x!)}^{2}}}}{e^{- ({\lambda_{1}} + {\lambda_{2}})x}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{{({\lambda_{1}} \cdot {\lambda_{2}} \cdot {\lambda_{3}})}^{x}}}}{{{{(x!)}^{3}}}}{e^{- ({\lambda_{1}} + {\lambda_{2}} + {\lambda_{3}})x}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{{(\prod\limits_{i = 1}^{k} {{\lambda_{i}}} )}^{x}}}}{{{{(x!)}^{m}}}}{e^{- (\sum\limits_{i = 1}^{m} {{\lambda_{i}}} )x}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{m} {\frac{{{{(\prod\limits_{i = 1}^{k} {{\lambda_{i}}} )}^{x}}}}{{{{(x!)}^{k}}}}{e^{- (\sum\limits_{i = 1}^{m} {{\lambda_{i}}} )x}}} {\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \end{array} $$

(4) Assuming that the function F is a Chi-square distribution function with expection k and standard deviation 2k, here x ∈ [0, 1], k_i ∈ N +. We can get the likelihood result L is

$$ {\kern 1pt} {\text{L}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^{m} {\frac{1}{{{2^{\sum\limits_{i = 1}^{k} {\frac{{{k_{i}}}}{2}} }}\prod\limits_{i = 1}^{k} {{\varGamma} (\frac{{{k_{i}}}}{2})} }}{x^{\sum\limits_{i = 1}^{k} {\frac{{{k_{i}}}}{2}} - k}}{e^{- \frac{x}{2}}}} {\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) $$

The specific proof process is as follows.

$$\begin{array}{l} {\text{L}} = {{\text{F}}_{1}}*{{\text{w}}_{1}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}*{{\text{w}}_{2}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}*{{\text{w}}_{3}} + {\cdots} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}{\text{*}} \cdots {\text{*}}{{\text{F}}_{\text{m}}}*{{\text{w}}_{\text{m}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \!\!\!\!\ = \frac{1}{{{2^{{{\text{k}}_{1}}}}{{\varGamma}_{1}}(\frac{k}{2})}}{x^{{{\text{k}}_{1}} - 1}}{e^{- \frac{x}{2}}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{1}{{{2^{{{\text{k}}_{12}}}}{{\varGamma}_{12}}(\frac{k}{2})}}{x^{{{\text{k}}_{12}} - 2}}{e^{- \frac{x}{2}}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{1}{{{2^{{{\text{k}}_{123}}}}{{\varGamma}_{123}}(\frac{k}{2})}}{x^{{{\text{k}}_{123}} - 3}}{e^{- \frac{x}{2}}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{1}{{{2^{{{\text{k}}_{1 {\cdots} m}}}}{{\varGamma}_{1 \cdots m}}(\frac{k}{2})}}{x^{{{\text{k}}_{1 {\cdots} m}} - m}}{e^{- \frac{x}{2}}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{1}{{{2^{\frac{{{k_{1}}}}{2}}}{\varGamma} (\frac{{{k_{1}}}}{2})}}{x^{\frac{{{k_{1}}}}{2} - 1}}{e^{- \frac{x}{2}}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{1}{{{2^{\frac{{{k_{1}} + {k_{2}}}}{2}}}{\varGamma} (\frac{{{k_{1}}}}{2}) \cdot {\varGamma} (\frac{{{k_{2}}}}{2})}}{x^{\frac{{{k_{1}} + {k_{2}}}}{2} - 2}}{e^{- \frac{x}{2}}}\\{\text{*}} ({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{1}{{{2^{\frac{{{k_{1}} + {k_{2}} + {k_{3}}}}{2}}}{\varGamma} (\frac{{{k_{1}}}}{2}) \cdot {\varGamma} (\frac{{{k_{2}}}}{2}) \cdot {\varGamma} (\frac{{{k_{3}}}}{2})}}{x^{\frac{{{k_{1}} + {k_{2}} + {k_{3}}}}{2} - 3}}{e^{- \frac{x}{2}}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{1}{{{2^{\sum\limits_{i = 1}^{m} {\frac{{{k_{i}}}}{2}} }}\prod\limits_{i = 1}^{m} {{\varGamma} (\frac{{{k_{i}}}}{2})} }}{x^{\sum\limits_{i = 1}^{m} {\frac{{{k_{i}}}}{2}} - m}}{e^{- \frac{x}{2}}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^{m} {\frac{1}{{{2^{\sum\limits_{i = 1}^{k} {\frac{{{k_{i}}}}{2}} }}\prod\limits_{i = 1}^{k} {{\varGamma} (\frac{{{k_{i}}}}{2})} }}{x^{\sum\limits_{i = 1}^{k} {\frac{{{k_{i}}}}{2}} - k}}{e^{- \frac{x}{2}}}} {\text{*}}({\frac{k}{m}^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{{k - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \end{array} $$

(5) Assuming that the function F is a F distribution function with expection $\frac {{{n_{2}}}}{{{n_{2}} - 2}}$ and deviation $\frac {{2{n_{2}}^{2}({n_{1}} + {n_{2}} - 2)}}{{{n_{1}}{{({n_{2}} - 2)}^{2}}({n_{2}} - 4)}}$, here x ∈ [0, 1], ${n_{1i}} \in N + ,{\kern 1pt} {\kern 1pt} {n_{2i}} \in N +$. We can get the likelihood result L is

$$ \begin{array}{@{}rcl@{}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{L}} &=& \sum\limits_{k = 1}^{m} \left( \prod\limits_{i = 1}^{k} {\frac{{{\varGamma} (\frac{{{n_{1i}}}}{2} + \frac{{{n_{2i}}}}{2}) \cdot {{(\frac{{{n_{1i}}}}{{{n_{2i}}}})}^{\frac{{{n_{1i}}}}{2}}}}}{{{\varGamma} (\frac{{{n_{1i}}}}{2}){\varGamma} (\frac{{{n_{2i}}}}{2})}}} {x^{\frac{{\sum\limits_{i = 1}^{k} {{n_{1i}}} }}{2} - k}}\right.\\ &&\left.{\vphantom{\prod\limits_{i = 1}^{k}}} \cdot \prod\limits_{i = 1}^{k} {{{(1 + \frac{{{n_{1i}}}}{{{n_{2i}}}}x)}^{- \frac{{{n_{1i}} + {n_{2i}}}}{2}}}} {\text{*}}({{\frac{k}{m}}^{\frac{{1 - \alpha }}{\alpha }}} - {{(\frac{{k - 1}}{m})}^{\frac{{1 - \alpha }}{\alpha }}}) \right) \end{array} $$

The specific proof process is as follows.

$$\begin{array}{l} {\text{L}} = {{\text{F}}_{1}}*{{\text{w}}_{1}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}*{{\text{w}}_{2}} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}*{{\text{w}}_{3}} + {\cdots} + {{\text{F}}_{1}}{\text{*}}{{\text{F}}_{2}}{\text{*}}{{\text{F}}_{3}}{\text{*}} \cdots {\text{*}}{{\text{F}}_{\text{m}}}*{{\text{w}}_{\text{m}}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{{N_{1}}}}{{{B_{1}}(\frac{{{n_{1i}}}}{2},\frac{{{n_{2i}}}}{2})}}{x^{\frac{{{n_{11}}}}{2} - 1}}{C_{1}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} + \frac{{{N_{12}}}}{{{B_{12}}(\frac{{{n_{1i}}}}{2},\frac{{{n_{2i}}}}{2})}}{x^{\frac{{{n_{11}} + {n_{12}}}}{2} - 2}}{C_{12}}{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{N_{123}}}}{{{B_{123}}(\frac{{{n_{1i}}}}{2},\frac{{{n_{2i}}}}{2})}}{x^{\frac{{{n_{11}} + {n_{12}} + {n_{13}}}}{2} - 3}}{C_{123}}{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{N_{1 {\cdots} m}}}}{{{B_{1 \cdots m}}(\frac{{{n_{1i}}}}{2},\frac{{{n_{2i}}}}{2})}}{x^{\frac{{{n_{11}} + {n_{12}} + {n_{13}}}}{2} - m}}{C_{1 {\cdots} m}}{\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{{\varGamma} (\frac{{{n_{11}}}}{2} + \frac{{{n_{21}}}}{2})}}{{{\varGamma} (\frac{{{n_{11}}}}{2}){\varGamma} (\frac{{{n_{21}}}}{2})}} \cdot {(\frac{{{n_{11}}}}{{{n_{21}}}})^{\frac{{{n_{11}}}}{2}}}{x^{\frac{{{n_{11}}}}{2} - 1}} \cdot {(1 + \frac{{{n_{11}}}}{{{n_{21}}}}x)^{- \frac{{{n_{11}} + {n_{21}}}}{2}}}*{(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{\varGamma} (\frac{{{n_{11}}}}{2} + \frac{{{n_{21}}}}{2})}}{{{\varGamma} (\frac{{{n_{11}}}}{2}){\varGamma} (\frac{{{n_{21}}}}{2})}} \cdot \frac{{{\varGamma} (\frac{{{n_{12}}}}{2} + \frac{{{n_{22}}}}{2})}}{{{\varGamma} (\frac{{{n_{12}}}}{2}){\varGamma} (\frac{{{n_{22}}}}{2})}} \cdot {(\frac{{{n_{11}}}}{{{n_{21}}}})^{\frac{{{n_{11}}}}{2}}}\cdot{(\frac{{{n_{12}}}}{{{n_{22}}}})^{\frac{{{n_{12}}}}{2}}}{x^{\frac{{{n_{11}} + {n_{12}}}}{2} - 2}} \cdot \left[ {{{(1 + \frac{{{n_{11}}}}{{{n_{21}}}}x)}^{- \frac{{{n_{11}} + {n_{21}}}}{2}}} \cdot {{(1 + \frac{{{n_{12}}}}{{{n_{22}}}}x)}^{- \frac{{{n_{12}} + {n_{22}}}}{2}}}} \right]\\{\text{*}}({(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{1}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{\varGamma} (\frac{{{n_{11}}}}{2} + \frac{{{n_{21}}}}{2})}}{{{\varGamma} (\frac{{{n_{11}}}}{2}){\varGamma} (\frac{{{n_{21}}}}{2})}} \cdot \frac{{{\varGamma} (\frac{{{n_{12}}}}{2} + \frac{{{n_{22}}}}{2})}}{{{\varGamma} (\frac{{{n_{12}}}}{2}){\varGamma} (\frac{{{n_{22}}}}{2})}} \cdot \frac{{{\varGamma} (\frac{{{n_{13}}}}{2} + \frac{{{n_{23}}}}{2})}}{{{\varGamma} (\frac{{{n_{13}}}}{2}){\varGamma} (\frac{{{n_{23}}}}{2})}} \cdot {(\frac{{{n_{11}}}}{{{n_{21}}}})^{\frac{{{n_{11}}}}{2}}}\cdot{(\frac{{{n_{12}}}}{{{n_{22}}}})^{\frac{{{n_{12}}}}{2}}}\cdot{(\frac{{{n_{13}}}}{{{n_{23}}}})^{\frac{{{n_{13}}}}{2}}}{x^{\frac{{{n_{11}} + {n_{12}} + {n_{13}}}}{2} - 3}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} \left[ {{{(1 + \frac{{{n_{11}}}}{{{n_{21}}}}x)}^{- \frac{{{n_{11}} + {n_{21}}}}{2}}} \cdot {{(1 + \frac{{{n_{12}}}}{{{n_{22}}}}x)}^{- \frac{{{n_{12}} + {n_{22}}}}{2}}} \cdot {{(1 + \frac{{{n_{13}}}}{{{n_{23}}}}x)}^{- \frac{{{n_{13}} + {n_{23}}}}{2}}}} \right]{\text{*}}({(\frac{3}{m})^{\frac{{1 - \alpha }}{\alpha }}} - {(\frac{2}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \prod\limits_{i = 1}^{m} {\frac{{{\varGamma} (\frac{{{n_{1i}}}}{2} + \frac{{{n_{2i}}}}{2}) \cdot {{(\frac{{{n_{1i}}}}{{{n_{2i}}}})}^{\frac{{{n_{1i}}}}{2}}}}}{{{\varGamma} (\frac{{{n_{1i}}}}{2}){\varGamma} (\frac{{{n_{2i}}}}{2})}}} {x^{\frac{{\sum\limits_{i = 1}^{m} {{n_{1i}}} }}{2} - m}} \cdot \prod\limits_{i = 1}^{m} {{{(1 + \frac{{{n_{1i}}}}{{{n_{2i}}}}x)}^{- \frac{{{n_{1i}} + {n_{2i}}}}{2}}}} {\text{*}}(1 - {(\frac{{m - 1}}{m})^{\frac{{1 - \alpha }}{\alpha }}}) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{m} {(\prod\limits_{i = 1}^{k} {\frac{{{\varGamma} (\frac{{{n_{1i}}}}{2} + \frac{{{n_{2i}}}}{2}) \cdot {{(\frac{{{n_{1i}}}}{{{n_{2i}}}})}^{\frac{{{n_{1i}}}}{2}}}}}{{{\varGamma} (\frac{{{n_{1i}}}}{2}){\varGamma} (\frac{{{n_{2i}}}}{2})}}} {x^{\frac{{\sum\limits_{i = 1}^{k} {{n_{1i}}} }}{2} - k}} \cdot \prod\limits_{i = 1}^{k} {{{(1 + \frac{{{n_{1i}}}}{{{n_{2i}}}}x)}^{- \frac{{{n_{1i}} + {n_{2i}}}}{2}}}} {\text{*}}({{\frac{k}{m}}^{\frac{{1 - \alpha }}{\alpha }}} - {{(\frac{{k - 1}}{m})}^{\frac{{1 - \alpha }}{\alpha }}})} ) \end{array}$$