A stochastic approach using Garima distribution

In this article we propose a power Garima distribution which includes Garima distribution as particular case. Its statistical properties including behavior of its probability density function for varying values of parameters, moments, hazard rate function and mean residual life function has been discussed. The estimation of the parameters of the distribution has been discussed using maximum likelihood estimation.

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Introduction

In this research paper a continuous distribution named “Garima distribution” has been suggested for modelling data from real life time database. The important properties including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations have been discussed. The condition under which Garima distribution is over dispersed, uniform dispersed, and under-dispersed are presented along with other one parameter continuous distributions. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments.

The application of the proposed distribution has been explained using a numerical example. Power Garima Distribution (PGD) The probability density function of power Garima distribution (PGD) is given by     x θx x θx x; f ( ) ( ) ) 1.7 ( 0 ,0 ,0 ; e 1 1 2 ,    − − + + + =        and the cumulative distribution function of power Garima distribution is given by Maths 2022; 7(6): 157-163 https://www.mathsjournal.com R Subathira Department of Statistics, Annamalai University, Annamalai Nagar, Tamil Nadu, India N Vijayshankar Assistant Professor, Department of Statistics, Annamalai University, Annamalai Nagar, Tamil Nadu, India R. Subathira Department of Statistics, Annamalai University, Annamalai Nagar, Tamil Nadu, India https://www.mathsjournal.com Fig 1: Graphs of pdf of PGD for varying values Fig 2: Graphs of the cdf of PGD for varying values of the Reliability Measures Survival function The survival function is also known as reliability function. it can be computed as complement of the cumulative distribution function. The survival function or the reliability function of power Garima distribution is given by ) , ; ( 1 ) (   x lF x S − = x S ,)1 2 ( ,)1 ( ,)1 ( 1 1 ) (   )1 2 ( )1 ( )1 (  θx θx θx   − =                        + + + + +    +  + +  + +  θ    https://www.mathsjournal.com Hazard function The hazard function is also known as hazard rate, instantaneous failure rate or force of mortality and is given by x lf x h ) , ; ( ) (   = x S ) ( + 1        ) 1( − + + θx e θx x h(x) =   , )1 2 ( , )1 ( , )1 ( )1 2 ( )1 ( )1 ( θx θx θx                                   + + + + + − +  + +  + +       Reliability function of Power Garima Distribution Hazard function of power Garima Distribution Structural Properties Here, I discuss some statistical properties of power Garima distribution. https://www.mathsjournal.com Moments Let X represents the random variable of power Garima distribution with parameters θ and α, then the rth order moment E(X r) of power Garima distribution is obtained as   = = ) , ; ( ) ( ' dx x lf r x r X E r    0 1        ) 1( − + + + + = dx θx e θx r x      0 )1 2 ( )1 ( )1 ( +  + +  + +  θ    + 1     = dx θx e θx r x     ) 1( )1 2 ( )1 ( )1 (    0   − + + + +  + +  + +  θ    + 1     2 = dx θx e r x dx θx e r x dx θx e r x         + − + + − +   +      0 )1 2 ( )1 ( )1 ( ) 3.7 ( 0 0     −   +  + +  + +  θ      1 ) ( Put t x t x =  = dt   dt dx dt dx x 1 1 1 Also − = − =  = −    x   t After simplification, equation (7.3) becomes )1 2 ( )1 ( )1 (  r r r       ) ( ' + +  + + +  + + +  = = r X E r  r ) 4.7 ( )1 2 ( )1 ( )1 (              +  + +  + +  Substitute r = 1 and 2 in equation (7.4), we get mean and the second moment of power Garima distribution. ) 2 2 ( ) 2 ( ) 2 (         +  + +  + +  = = ) ( '1 X E )1 2 ( )1 ( )1 ( 1              +  + +  + +  ) 3 2 ( ) 3 ( ) 3 (         +  + +  + +  = = ) 2 ( ' 2 X E )1 2 ( )1 ( )1 ( 2              +  + +  + +  https://www.mathsjournal.com 2   ) 3 2 ( ) 3 ( ) 3 (   ) 2 2 ( ) 2 ( ) 2 (              +  + +  + +  = ) 2 ( Variance )1 2 ( )1 ( )1 ( 2   )1 2 ( )1 ( )1 ( 1      +  + +  + +  − +  + +  + +  θ θ                          +  + +  + +    Moment Generating Function Moment generating function is the expected function of the random variable. we begin with the well-known definition of the moment generating function given by ( ) dx (x; lf tx e tx e E t X M ) , 0 ) (     = =    2 1 dx x lf (tx) tx   ) , ; ( ... !2 0 + + + =       l j j  dx x f x j t   = = ) , ; ( ! j 0 0 j t    = = 0 ' ! j j j   )1 2 ( )1 ( )1 ( j j j j t        + +  + + +  + + +  =   = 0 ! j j j )1 2 ( )1 ( )1 (  +  + +  + +                            )1 2 ( )1 ( )1 (   1 ) (   j t t X M  j =      =       + +  + + +  + + +  +  + +  + +  ! )1 2 ( )1 ( )1 ( j j j j j            0 Characteristic function Characteristic function is defined as the function of any real-valued random variable completely defines the probability distribution of a random variable.

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Graphs of the cdf of PGD for varying values of the

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Graphs of pdf of PGD for varying values

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Graphs of pdf of PGD for varying values

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Original paper figureSource PDF · page 2 ↗

The characteristic function of power Garima distribution is given by ) ( ) ( it X M t x =  )1 2 ( )1 ( )1 ( 1 ) (   j it it X M        j j j j )1 2 ( )1 ( )1 ( j = =     + +  + + +  + + +  +  + +  + +      0           j ! Maximum Likelihood Estimation and Fisher’s Information matrix In this section, we will discuss about the maximum likelihood estimation of power Garima distribution for estimating its parameters and also Fisher’s Information matrix have been discussed. Let x1, x2,…,xn be a random sample of size n from the power Garima distribution, then the likelihood function can be written as n ) , ; ( ) (    x lf x L = = i 1 https://www.mathsjournal.com  + 1         1 − + +     n i θx e i θx ix x L ) ( =  1 )1 2 ( )1 ( )1 ( i θ = +  + +  + +                    n + 1   n    x L ) (     ( )  i θx e i θx ix 1 )1 2 ( )1 ( )1 (   i = 1 =     − + + +  + +  + +                 + 1 n         n  x L ) (     ( )  i θx e i θx ix n i = 1 =     1 )1 2 ( )1 ( )1 (   − + + +  + +  + +           The log likelihood function is given by n   log )1 2 ( )1 ( )1 ( log log 1 log   = − + +  = i ix i θx n (7.5) 1 1 ) 1 log( 1 i = i ix n n L              + + +  + +  + +  − + =         n The maximum likelihood estimates of θ and α can be obtained by differentiating equation (7.5) with respect to θ and α and must satisfy the following normal equations      1   ix       )1 ( 1 log =    )1 2 ( )1 ( )1 ( i = −  i ix n +  − + =        n       = + + 0 1 1 1 θ n n L    + + +  + +  + +        i θx                  log  ix i θx n    1 = + +  + +  + +  − − =                   i ix ix n 0 1 log )1 2 ( )1 ( )1 ( 3 log = = − = + + +  i ix n n L     log 1 1 i i θx         n   Where ψ (.) is the digamma function. To obtain the confidence interval we use the asymptotic normality

Results

We have that if ( ) ( ) follows as

Result

the state can we , , of MLE the denotes ˆ ,ˆ ˆ       = = ( )) ( ,0 ) ˆ ( 1 2    − → − I N n by given is matrix n informatio s Fisher' is ) ( Where  I 2 2     log log L E L E 2                1 ) (     − = 2 2 n I   log log L E L E 2                               Where https://www.mathsjournal.com 2   2  2 2    1   x n n L E + − 2 2 2 2      n )1 ( 1 ) ( log      =   1 )1 2 ( )1 ( )1 (          +  +       + − =   1 + + i θx ( ) ( )  i i                         +  + +  + +    2 2 2  n n L E 2 2 3   )1 2 ( )1 ( )1 ( ' ) ( 3 log                       +  + +  + +  − =      2 2 n n   2 log ) (log 1  x θx x θx θx  − + + + i i 2 ( )      −   = − 1 i 1 i i i i i x x θx log 1 + +   ( ) ( ) ( ) i i     2 2   )1 ( log log   n L E L E )1 2 ( )1 ( )1 (               =      +  − =   +  + +  + +                     n n   1 log log 1  x x θx x x θx  + − + + + i i 2    −   = = 1 1 i i i i i i i x x θx log 1 + +  ( ) ( ) ( )   i i   we unknown, being Since function. diagamma of derivative order first the is (.)' where   . and for interval confidence asymptotic obtain to used be can this and )ˆ ( by ) ( estimate 1 1     − − I I

Conclusion

In this paper, we have discussed a new generalization of power Garima distribution using stochastic approach. The different mathematical and statistical properties of the newly introduced distribution have been discussed. The maximum likelihood estimator is also used for estimating the parameters of the proposed distribution and also the fisher’s information matrix have been discussed. Finally, the application of power Garima distribution fits better than the Garima, exponential and lindley distributions.

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