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.
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