(5), Bayesian inference derives a posterior distribution (P( x | Y)) of variables (x), offsetting constants, based on a

likelihood function (P(Y | x)) and a prior distribution of the variables ([pi](x)).

Notations f(*): Probability density function E(*): Mean function Var(*): Variance function [gamma](y, x): Lower incomplete gamma function [GAMMA](y, x): Upper incomplete gamma function F(*): Cumulative distribution function S(*): Survival function h(*): Hazard rate function L(*):

Likelihood function l(*): Log-likelihood function [psi](*): Digamma function [psi]'(*): Trigamma function I(*): Expected Fisher information matrix [mu]: Positive parameter [phi]: Positive parameter [alpha]: Positive parameter [beta]: Real parameter [lambda]: Positive parameter Q(*): Quantile function [THETA]: Vector of parameters H(*: Observed Fisher information matrix [[mu].sub.k]: kth moment G(*: TTT-plot [D.sub.n]: Kolmogorov-Smirnov statistic [y.sup.*]: Predictive value.

The

likelihood function of the whole failure time data set could be obtained by multiplying the three individual contributions as

If we let [L.sub.i] = log f([y.sub.i]; [[theta].sub.i], [THETA]) as a contribution of [y.sub.i] to the

likelihood function L = [[summation].sub.i][L.sub.i], then

Obtaining maximum

likelihood function under population ([OMEGA]).

Assuming that [theta]' was a parameter value of [theta], the logarithmic

likelihood function concerning [theta]' was

(i) Individual speed follows a normal (Gaussian) distribution with mean [mu] and variance [[sigma].sup.2], since a normal distribution reduces

likelihood function complexity [23].

One main challenge in these inferential tasks is attributed to the burden of computational load of minimizing/maximizing corresponding object functions, including

likelihood functions or the squared differences [8-11].

The local

likelihood functions are formulated using four of the 37 original datasets, one from each fault case.

The estimation for the parameters of DWD via maximum likelihood estimation technique presented the independent observations are x1,x2,....xn then the

likelihood function of the DWD shown:

Since the multidimensional

likelihood function is a nonlinear function of time delays and has many local maxima, the exact ML estimate needs multidimensional grid search.

Another approach [10] was to employ linear transform (LT) on the phase noise term and derive an approximate expression of the

likelihood function. By using the

likelihood function, a modified LLR metric was obtained.