Definition 1.1 Let N(0, 1) denote the one dimensional normal distribution. This distribution has density n(x) = e−x 2/2/ √ 2π. LetNd(0, 1) denote the d-dimensional Gaussian distribution, induced by picking each coordinate independently from the standard normal distribution N(0, 1). Let Exp(λ) denote the exponential distribution, with parameter λ. The density function of the exponential distribu...

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped f...

Genetic sequence data typically exhibit variability in substitution rates across sites. In practice, there is often too little variation to fit a different rate for each site in the alignment, but the distribution of rates across sites may not be well modeled using simple parametric families. Mixtures of different distributions can capture more complex patterns of rate variation, but are often ...

In a regular distribution family, Cramér-Rao inequality holds, and the maximum likelihood estimator (MLE) converges to a normal distribution whose variance is the inverse of the Fisher information because the Fisher information converges and is well-defined in this family. However, in a non-regular location shift family which is generated by a distribution of R whose support is not R (e.g., a W...

The exponentiated gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called Transmuted exponentiated gamma (TEG) distribution is introduced. A new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the TEG distribution is prov...

It is well-known that maximum likelihood (ML) estimators of the two parameters in a Gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The Gamma distribution is a special case of a generalized Gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized Gamma ...