In the field of biostatistics it is often required to develop inferential tools dealing with the presence of nuisance parameters. The most adopted solution is to resort to pseudo-likelihood functions, having properties similar to the ones of a genuine likelihood. A possible choice is to use the integrated likelihood where the nuisance parameters are eliminated by means of integration with respect to a weight function. The selection of the weight function turns out to be crucial since it could have a strong impact on the properties of the resulting integrated likelihood. After having introduced the concept of pseudo-likelihood, the definition and the properties of the integrated likelihood, the focus will be on reviewing the main alternatives to choose the weight function according to different inference paradigms.

Let Pi (N) be the number of primes less than or equal to N, for any real number N, the New Prime Number Theorem can be expressed by the formulas as follows: Pi (N) = R (N) + K × ( Li (N) - R (N) ), 1 ≥ K ≥ -1 P (K) = 1.99471140200716338969973029967…×EXP (-12.5×K×K) Where the R (N) is the Riemann Prime Counting Function, the Li (N) is the logarithmic integral function; the P (K) is the Normal Distribution N (μ=0, σ=0.2).

Let Pi(N) be the number of primes less than or equal to N, Pi (2≤Pi≤Pm) be taken over the primes less than or equal to √N, then exists the formula as follows: Pi(N) = INT { N×∏ (1-1/Pi) } + m-1 = Li (N) - 0.5×Li (N^0.5) ± 0.5×Li (N^0.5) Li (N^0.5) ≥ Li (N) - Pi (N) ≥ 0 : ( The Riemann Hypothesis is proved ) Pi (N) = R (N) + K × (Li (N) - R (N) ), 1 ≥ K ≥ -1 . P (K) = 1.99471140200716338969973029967…×EXP (-12.5×K×K) Where the INT { } expresses the taking integer operation of formula spread out type in { }, the Li (N) is the logarithmic integral function, the R (N) is the Riemann Prime Counting Function, the P (K) is the Normal Distribution N (μ=0,σ=0.2).

The Exponential Transmuted Exponential distribution (ETE) appeared in [1] and in this paper we present a new generalization of the ETE distribution based on a Box-Cox transformation of the form ( ) 2 X 1 Z λ−µλ+ −σλµ = Where 2 0 , , R, 0,andZ is ETE distributed. We also show the new distribution is a good fit to some real-life data, indicating practical significance.