two- parameter Sujatha distribution.

Transkript

1 A New-Two Parameter Sujatha Distribution with Properties and Applications Received: Received in revised form: Accepted: Available Online: Correspondence: ABSTRACT includes one parameter Akash distribution and Sujatha distribution as particular cases. The shapes of probability density function for varying values of parameters have been studied. Its statis- - two- parameter Sujatha distribution. Keywords: Sujatha distribution; Akash distribution; moments; statistical properties; estimation of parameters; applications ÖZET Anahtar Kelimeler: uygulamalar Shanker (6a) proposed a one parameter lifetime distribution named tive distribution function (cdf) f( ; ) ( + + ) e ; >, > + + ( + + ) F ( ; ) + e ;, > > + + (.) (.) 96

2 where (, ) and a gamma (, ) distributions for modeling of lifetime data. It has proposed by Lindley (958). (6a) are μ + +, ( ) ( + + ) 6 4 μ + +, μ ( + + ) ( + + ) μ + +, ( + + ) ( + + ) 4 5 μ ( ) ( + + ) μ ( ) 4 ( + + ) μ 4 4 Shanker (6a) has discussed its important statistical properties including shapes of density function for Shanker (6a) has discussed to model lifetime data from biomedical science and engineering. Shanker (6b) has introduced Po- estimation of parameter and applications to model count data. Shanker and Hagos (5) have discussed zero counts. 4 Shanker and Hagos (6) have also studied size-biased Poisson- Sujatha distribution and its 5 6 f (, ) ( + ) e ; >, > + (.) ( ) and a gamma (, ) distributions. Shanker - 97

3 by Shanker (5). 6 Shanker (5) has also discussed the applications of Akash distribution for modeling Lindley distribution. 6 is pdf 7 f( ;, α) ( α + + ) e ; >, >, α α + + application of TPSD. - stress-strength reliability have been discussed. The estimation of the parameters has been discussed using and TPSD. (.4) A new two-parameter Sujatha distribution (NTPSD) having parameters and α pdf. f4( ; α, ) ( + α+ ) e ; >, >, α + α + (.) where is a scale parameter and α is a shape parameter. It reduces to Sujatha distribution (.) and Akash distribution (.) for α and α ) distributions as follow ( ; α, ) ( ; ) + ( ;,) + ( ) ( ;,) f p g p g p p g 4 where α p, p, g ( ; ) e ; >, > + α + + α + (.) g( ;, ) e ; >, >, g( ;, ) e ; >, >. Γ Γ ( + + ) α F (, α, ) + e ;,, α > > > + α + (.) 98

4 Behaviors of the pdf and the cdf of NTPSD for various combination of parameters and α are shown in FIGURE : α 99

5 FIGURE : α

6 The r t { + ( + ) α + ( + )( + )} r! r r r μ r ; r,,,... r α ( + + ) (.) + α + 6 ( + α + ) μ μ + α + μ μ ( + α + ) 6( + 4α + ) ( + + ) α 4( + 5α + ) μ α + 4 α + 4α α + ( + + ) α The relationship between moments about the mean and moments about the origin gives the moments about mean of NTPSD as ( α + α + α + + α + α + + α + + ) ( + α + ) μ μ α + α + 44α + 4α α + α + 44α + + α + α + α + α + + α ( + + ) ( ) are thus obtained as σ α α α CV. μ + α + 6 β β ( α + α + α + + α + α + + α + + ) 4 ( ) μ β / / μ α α α α + α + 44α + 4α α + α + 44α + 8 μ α α α 4 μ + 6α + 768α α ( ) β γ of NTPSD

7 4 σ α α α γ μ + α + + α + ( 6) constants of Sujatha distribution and Akash at α and α respectively. β, β and γ, for various combination of parameters and α have been For a given value of α increases. Similarly for value.5 increases as the value of α increases.but for values 5α increases. Since β > positively skewed lifetime data. α TABLE : andα TABLE : ( β ) α α α TABLE : ( β ) o α

8 α TABLE 4: ( γ ) o α Since β >, curve. Thus NTPSD is suitable for lifetime data which are leptokurtic. As long as and α 5 σ > μ and for 5and α 5 ( σ < μ ). β, β and γ and α Let X be a continuous random variable with pdf f ( ) and cdf h h Δ ( < +Δ > ) Δ F p X X f lim m E X X > F() t dt F and [ ] h( ) and m( ) of NTPSD (.) are thus obtained as h and ( + α+ ) ( ) α α F.The hazard rate function (also m of X are respectively de- ( t+ + ) + α + t α t m e dt + ( α ) ( α ) e + α + ( 4) ( 6) + α α + + α α +

9 FIGURE : β, β and γ α h + α + ( ) f ( ) and m + α + 6 μ. ( + α + ) h and m of NTPSD reduce to the corresponding h and m of Sujatha distribution at α. Behaviors of h( ) and m( ) of NTPSD (.) for various combination of parame- 4

10 FIGURE 4: α 5

11 FIGURE 5: α It is obvious that h is monotonically increasing function of, andα where as m is monotonically decreasing function of, andα. 6

12 A random variable X is said to be smaller than a random variable Y in the (i) stochastic order ( X st Y ) if F Fy for all (ii) hazard rate order ( X hr Y ) if h hy for all (iii) mean residual life order ( X mrl Y ) if m my for all f (iv) likelihood ratio order ( X lry ) if decreases in fy The following results due to Shaked and Shanthikumar (994) are well known for establishing stochastic ordering of distributions 8 X lr Y X hr Y X mlr Y st y following theorem: Theorem: Suppose X ~NTPSD (, α ) and ~NTPSD (, ) Y α. If and α α ) then X lr Y and bence X hr Y, X m rl Y and X st Y. Proof: We have ( α ) ( α ) fx ;, + α + + α+ ( ) e ; > fy ;, α + α ( α ) ( α ) ( + + ) fx ;, + α + + α+ ln ln + ln fy ;, α α + + ( ) > and α α ( (or This gives ( ;, ) ( ) ( ) d f α α α α α d f X ln ( ). Y ;, α α α ( + + )( + + ) ( > andα α ) or ( α > α and ) ( ;, α) ( ;, ) d fx ln. d f α < Y This means that X lr Y and bence X hr Y, X mrl Y and X st Y. - PSD over Sujatha distribution and Akash distribution. Stress-strength reliability of a component shows the life of the component. When the stress of the com- X component will function satisfactorily till X > Y R PY ( < X) is a measure of the component - 7

13 Suppose X and Y be independent strength and stress random variables having NTPSD (.) with parameters,, α R of NTPSD (.) can be obtained as ( α ) and R PY< X PY< X X f d ( ; α) ( ; α ) f F d,, ( ) + ( ) ( αα α 6α 4 4α 8α α ) ( ) ( αα α α + α + + ) + ( + α + ) 5 ( + α + )( + α + )( + ) α α αα α α α αα + α + α + + α + + α + α stress-strength reliability of NTPSD at α α. δ ( X) μ f d and δ where E( X) μ and M Median( X). the following relationships μ X M f d, respectively. The measures δ and X δ can be calculated using δ X μf μ f d (5..) and δ μ M X f d (5..) X ( ; α, ) ( + α + ) μ μ αμ μ μ αμ μ α f d μ e μ (5..) ( + α + ) + α α α + M M M M M M M M 6 e f4( ; α, ) d μ 8 (5..4) δ X δ and X

14 δ ( X ) ( + α + ) μ + αμ+ + μ+ α + 6 e μ (5..5) δ ( + α + ) M M + αm + M + M + αm + + M + α + e μ 6 ( X ) q B ( p) f d f d f d μ f d pμ pμ pμ q q (5..6) The Bonferroni and Lorenz curves and Bonferroni and Gini indices given by Bonferroni (9) have app- demography and medical science (Bonferroni CE. Elementi di Statistica Generale. Firenze: Seeber; 9). (5..) q L p f d f d f d f d μ μ μ q q and μ (5..) p B p F d pμ (5..) and p L p F d μ (5..4) μ E( X) and q F ( p). B B p dp (5..5) and G L p dp (5..6) respectively. q ( ; α, ) { ( α ) ( α ) ( α) 6} ( + α + 6) q + q + q + q + q+ + q+ + e f d 4 q (5..7) 9

15 B p and p { } + α α α + 6 q q q q q q e 6 L p + α + { } + α α α + 6 q q q q q q e + α + 6 q q (5..8) (5..9) are obtained as B { } + α α α + 6 q q q q q q e + α + 6 { } + α α + q q q q q q 6 e G + + α + 6 α α α α α α ( + + ) ( + + ) ( + + ) α α ( + + ) q q (5..) (5..) mum likelihood have been discussed. α + 4 ( α ) + + (6..) + α + + α + 4 α + 5 m + + α + + α + + α + ( α + ) 4 5 m + α ( + + )

16 ( α + ) α + m (6..) Equations (6..) and (6..) give the following cubic equation in as α m + 4 m α α + (6..) - of can be obtained and substituting the value of α of α can be obtained as ( ) ( ) 6 + α ( ) (6..4) Suppose,,,... n be random sample from NTPSD (.). The log likelihood function is thus obtained as ( + α) ln L n n n + α + ( i i ) n ln L n ln ln( + α + ) + ln + α + n i non-linear equations ( ˆ α, ˆ) of ( α, ) are then the solutions of the following (6..) ln L n + α + α + + α + n i i i i (6..) ( ˆ α, ˆ) of ( α, ) can be computed directly by solving the log likelihood equation ˆ and ˆ α are obta- ( α, ) of the parameters ( α, ).. before service in a bank and the second dataset is the relief time (in minutes) of patients receiving anal- -

17 The values of lnl- tistic) for the above data sets have been computed for the considered distributions. The formulae for computing AIC and K-S Statistics are AIC lnl + k, BIC lnl + k lnn and K-S Sup F n F k the number of n Fn ( ) empirical distribution function. The best distribution corresponds to lower values of lnl( ˆ α, ˆ ) along with their standard ( ˆ α, ˆ ), lnl Table 5. Since the lnl considered distributions. TABLE 5 : ( ˆ α, ˆ ), lnl Data set Distribution MLE s S.E lnl AIC K-S P-value ˆ ˆ α ˆ ˆ α ˆ ˆ ˆ ˆ ˆ α ˆ ˆ α ˆ ˆ ˆ

18 A new two-parameter Sujatha distribution (NTPSD) has been introduced which includes Akash distribution and Sujatha distribution as particular cases. Raw moments and central moments of NTPSD ters. Behaviors of hazard rate function and mean residual life function of NTPSD have been studied of parameters have been discussed with two well-known methods namely method of moment and ributions and TPSD. The values of -InL- distribution for modeling real lifetime data. Acknowledgements Authors are grateful to the Editor-In-Chief of the journal and the three anonymous reviewers for their constructive comments which improved the quality and the presentation of the paper. Source of Finance - negatively affect the evaluation process of this study. Authorship Contributions Rama Shanker; Design: Rama Shanker; Rama Shanker; Literature Review: fay; Critical Review: Rama Shanker; References and Fundings: Rama Shanker