The generalized gamma distribution (GGD)

was constructed by Stacy 108 in an attempt to combine the power of two distributions: The Gamma

distribution and the Weibull distribution.

It is a newer distribution (1962) than the normal distribution (1774).

The generalized gamma distribution remains a renowned distribution as this one

stands exceptionally adaptable. This distribution is too expedient because it

comprises as special cases of quite a few distributions: the exponential

distribution, the log-normal distribution, the Weibull distribution, the Levy

distribution.

In cooperation the

exponential in addition to the more general Weibull distributions have

wide-spread usages in the exploration of engineering data. It is impartially

common habitation, however, to come across facts which are discordant through

these distributions and further accustomed probability models. Such data

kindles inference relating to the letdown systems besides added corporeal

marvels which may be involved. It also emboldens exploration to expand the

group of distributions which are useful to the enthusiastic statistician.

The parameter estimation

of the generalized distribution was discussed by Stacy

and Mihran 107, Gomes et al. in 2008 47 and the Scheme of Parameter

Estimation was discoursed by Gui Gao et al. 51. The log–logistic distribution

(LLD) and log normal distribution (LND) are commonly used to examine the

lifetime data. The LLD is derived from logistic distribution (LD) by applying

the logarithmic to LD in a similar way as LND is obtained from normal

distribution.

The LLD has

an alike shape related to LND but precisely more adaptable. It is renowned that

the log-logistics model is capable to model societal progressions with

monotonically declining, as well as non-monotonic, inverse type failure rates.

Several illustrations of flood frequency analysis used LLD was conferred by Ahmad

et al. 2, social diffusion model process using LLD explored by Diekmann

Andreas 29, 30 and economic reliability using LLD was discussed by Kantam R R

L 60.

Generalization

of LD and LLD was contested by Ojo 92. Recently the Balakrishnan skew

logistic distribution and its properties are discussed by Asgharzadeh et al. 5.

Skew multivariate models related to hidden truncation 10 and Censor order

statistics from log-logistic distribution with applications to inference 9 was

conferred by Balakrishnan. The parameter estimation using maximum entropy was

conversed by Singh 106. Fahim Ashkarsdaf and Smail Mahdi 36 discussed the

comparison to two fitting methods for LLD.

Oxytocin is a pituitary neuropeptide kind produced and discharge

by hypothalamus, at that time it is expected at the rear of pituitary to the

far side of hypothalamic–pituitary–adrenal axis at latter unconfined on the way

to blood. It can cause uterine contractions in labor and stimulate milk

secretion, 133. At present oxytocin

consequence on abirritaion has careworn far-reaching consideration. In addition,

its role of delivery promotion, oxytocin can also be espoused in the pain

factor in the backbone and the periaqueductal gray of the brain (Yu, Long-Chaun

132). Habenular nucleus (Hb) is extensive in the dorsal diencephalon of

vertebrates, and as an important nucleus which participates in analgesia in

central nervous system, it can be divided into medial habenular nucleus (MHb)

and lateral habenular nucleus (LHb) {Fu et al.2010 42, Huang Min et al. 201157,

Fu, 2006 41}, in which LHb, as the junction of limbic system like amygdale,

hippocampus, preoptic area, etc. and midbrain, can regulate sleep and physiological

process of cardiovascular activity (Zhao and Wang, 132).

The natural roles of oxytocin in labor, milk letdown and social

affection were well-conserved 21, 46, 136. In modern ages, various

single-dose studies have been conducted on the effects of oxytocin on social

cognition in healthy humans 64 and 78.

In our model 116, we study the generalized

gamma distribution in fuzzy environment and is used to analysis the effect of dissimilar doses of

oxytocin for the changes of hind

paw with drawl latency of normal mature mice. The mean and

variance values of the fuzzy generalized gamma distribution for different doses

of oxytocin are calculated.

2.1.

Fuzzy

Generalized Gamma Distribution Model

2.2.1.

Generalized

Gamma Distribution

The random variable X

follows GGD with real positive parameters

is symbolized by

. The probability density function (p.d.f.) of GGD is given

by

The

expected value and variance value of X are given by

2.2.2.

Fuzzy

Generalized Gamma Distribution

In many life time applications, randomness

is not the only characteristic of improbability. In numerous ?elds of

application, remaining the fuzziness of environment and the laxity of

observers, it is from time to time farfetched to get hold of precise

observations of lifetime. The obtained lifetime data may be “polluted” and

imprecise most of the time. Besides, controlled by human and further resources

in research, exclusively for new equipment’s, exceptionally long-life

equipment’s, and non-mass manufacture merchandises, for which there is no

comparative reliability information available, more often than not, the

lifetime is based upon subjective evaluation or rough estimate. That leads to

the fuzziness of lifetime data.

Now consider the GGD with fuzzy parameter

that is switched with the parameters

. The probability of a random variable X follows

Fuzzy Generalized Gamma distribution (FGGD) is symbolized by

. The fuzzy p.d.f. of a random variable

is defined byIn our model 120, we

presented log-logistic distribution with fuzzy conditions and is used to

analysis the effect of different dosages of oxytocin in the adult mice. In

addition to that the mean values of the fuzzy log-logistic distribution for

different doses of oxytocin are calculated. Fuzzy mean values are

increasing in the lower alpha cuts and decreasing for upper alpha cuts for two

of the dissimilar doses. But fuzzy mean values are decreasing for lower alpha

cut and increasing for upper alpha cut for the other dose.2.4.1.

Log

Logistic Distribution

Let X be a random variable (r.v.)

follows the log logistic distribution with scale parameter

and shape parameter

is denoted by

has the following probability density function (p.d.f.) 2.4.1.

Fuzzy

Log Logistic Distribution

So consider a

random variable

follows fuzzy log logistic distribution (FLLD) with the fuzzy

numbers

as parameters is indicated by

. The p.d.f of FLLD is given byIn section 2.3 the FGGD was successfully established

and using the FGGD the changes of HWL of normal mature mice for dissimilar

doses oxytocin was analysed. The mean values were calculated for the doses of

1.25 nmol, 2.5 nmol and 5.0 nmol. Fuzzy mean values are increasing in the lower

alpha cuts and decreasing in the upper alpha cuts for the three doses. The

variance values are increasing in initial alpha cut values and decreasing after

the initial alpha cut values for the doses 1.25nmol and 5.0nmol. But the

variance values are increasing in lower alpha cut values and decreasing in

upper alpha cut values.

In section 2.4 FLLD was successfully established and

using the FLLD the changes of HWL of normal adult rats for dissimilar doses oxytocin

was evaluated. The mean values were calculated for the doses of 1.25 nmol, 2.5

nmol and 5.0 nmol. Fuzzy mean values are increasing in the lower alpha cuts and

decreasing for upper alpha cuts for the doses 1.25 nmol and 5 nmol. But fuzzy

mean values are decreasing for lower alpha cut and increasing for upper alpha

cut for the dose 2.5 nmol. The variance values are increasing for the lower

alpha cuts and decreasing for the upper alpha cuts for all three doses.