SIA-Lognormal Power Distribution

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1 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat SI-Lognormal Power Distribution Safoora Samuel Student of MPhil Statistics Kinnaird College for Women Lahore, Pakistan bstract During the past few years, some researchers have worked on distributions that are invariant under the reciprocal transformation. Such distributions are now referred to as being Self-Inverse at Unity. Only very recently, a generalized version of this class of distributions has been introduced --- distributions Self-Inverse at where, an arbitrary positive number, represents the median of the distribution. The self-inversion property permits development of estimators of distribution parameters that are more efficient than their wellknown counterparts. The lognormal distribution with scale parameter zero belongs to the class of SIU distributions. In this paper, we obtain the SI-Lognormal-Power distribution derive some of its fundamental properties such as the first four moments, the quantile function the hazard function. The shape of the density provides optimism that this newly derived probability model will turn out to be a suitable cidate for modeling a variety of real-life data-sets the fact that it belongs to the class of SI distributions will enable efficient estimation of the shape parameter of this distribution. Keywords-Self-Inverse distributions; Lognormal distribution; SI-Lognormal-Power distribution; moments; hazard function. I. INTRODUCTION number of authors have focused on distributions that are invariant under the reciprocal transformation. (See [], [] [].) The nomenclature Self-Inverse at Unity (SIU) has been adopted for this class of distributions in [4]. generalized version of SIU distributions has been given in [5] which have been called Self-Inverse at (SI) where is an arbitrary positive number. The remarkable property of SI distributions is that, due to self-inversion, it is possible to modify the formulae of well-known estimators in order to obtain estimators of distribution parameters that are more efficient than the wellknown ones. II. DISTRIBUTIONS SELF-INVERSE T UNITY The self-inversion property can be defined as that property by which reciprocal of a non negative continuous rom variable possesses eactly the same distribution as the one possessed by the original rom variable. One of the fundamental properties of this class of distributions is that the ( p) th quantile is the reciprocal of the pth quantile the median is equal to unity. Some simple eamples are the half Cauchy distribution, the F distribution having lognormal distribution where. It is well-known Saleha Naghmi Habibullah Professor of Statistics Kinnaird College for Women Lahore, Pakistan salehahabibullah@gmail.com that each of these distributions finds applications in a number of areas. III. DISTRIBUTIONS SELF-INVERSE T The distribution of a non-negative continuous rom variable X will be regarded as being self-inverse at if the distribution of X/ is identical to the distribution of /X where is an arbitrary positive real number. The median of this distribution will be equal to. The property that the median of every SIU distribution is unity is, in fact, a limitation of this class of distributions. being an arbitrary positive number, it is obvious that the class of SI distributions is much wider than the class of SIU distributions. IV. LOGNORML DISTRIBUTION The lognormal distribution is one of the most wellknown distributions all over the world finds applications in a wide variety of disciplines including economics, finance, biology, medicine, engineering human behaviors as well. The probability density function of lognormal distribution is given by ln y g y ( y ;, ) e, y y () where can be regarded as the scale parameter the shape parameter. V. LOGNORML POWER DISTRIBUTION It has been shown in [] that application of the power transformation to an SIU distribution results in another SIU distribution. In this section, we apply the power transformation to the lognormal distribution with scale parameter equal to zero. r pplying the transformation Z Y to the lognormal distribution given in eq. (), we obtain the following probability density function: ln z r w( z). e, z, r () zr We call it the Lognormal Power distribution. It is easy to verify that this distribution is self-inverse at unity.

Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat VI. SI-LOGNORML POWER DISTRIBUTION pplying the transformation to the Lognormal Power distribution given in eq.

2 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat VI. SI-LOGNORML POWER DISTRIBUTION pplying the transformation to the Lognormal Power distribution given in eq. (), we obtain the probability density function where X Z ln r f ( ) e, () r, r,. for which we adopt the nomenclature SI-Lognormal Power distribution. The graph of the density is given in Figure. where t erf ( ) e dt 5 7 t e dt... 5.! 7.! VII. FUNDMENTL PROPERTIES In this section, we present some of the fundamental properties of the SI-Lognormal Power distribution. We begin with the well-known measures of central tendency.. rithmetic Mean The mean of the distribution is given by ln r r E( X ) e d e 5 r B. Geometric Mean The logarithm of the geometric mean G X of a distribution with rom variable X is the epected value of ln(x). s such, we have ln r E ln X ln. e d ln() r Fig. : Graph of the density function of the SI- Lognormal-Power distribution Clearly, the shape of the density function is unimodal positively skewed. The cumulative distribution function is given by ln erf 4 r The error function erf ( ) is defined as follows: Therefore, we have GX (6) It is interesting to note that the geometric mean of the distribution is equal to the median. C. Harmonic Mean The harmonic mean (H X) of a distribution of the rom variable X is the reciprocal of the epected value of /X. Therefore, we have H X ln r r e d e ( 7) r It is interesting to note that the arithmetic harmonic means are related by the equation: M = HM

3 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat VIII. QUNTILE FUNCTION The quantile function is one way of describing a probability distribution, it is an alternative to the probability density function (pdf), the cumulative distribution function (CDF) the characteristic function. By definition, the q th quantile is obtained by solving for X q the equation s such, we obtain X q f ( ) d q X ep erfinv(q ) r (8) q The first quartile the third quartile of the distribution come out to be.67449r Q e (9) Q e.67449r () It is noteworthy that the first third quartiles are related by the equation: Q = Q IX. MESURES OF DISPERSION The variance the stard deviation are regarded as some of the most important measures of dispersion. The variance of SI-Lognormal Power distribution is derived below: ln r r E( X ). e d e r Therefore r r X X X e e Var E E s such, the stard deviation of the SI-Lognormal- Power distribution is given by r r r r S.D. e e e e () The coefficient of variation is given by r r S.D. e e C.V.= Mean r e s such, we have r r r C.V. e e () It is interesting to note that the coefficient of variation of the SI-Lognormal Power distribution is independent of. X. MODE By definition, the mode is obtained by equating the first derivative of the density function to zero. Here, we have f r r e ln r e ln r ln r Hence, the mode of the distribution is given by ˆ r () X e XI. HIGHER MOMENTS ND MOMENT-RTIOS In this section, we obtain the third fourth moments of the SI-Lognormal Power distribution. The third fourth moments about the origin come out to be 9r E ( X ) e 4 4 8r E ( X ) e The third fourth moments about the mean are given by s such, we have 9r 5r r e e e (4)

4 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat 4 4 8r 5r r r e 4e 6e e (5) The two moment ratios are given by / /. For the SI- 4 Lognormal Power distribution, these come out to be e e e e e 9r 5r r r r 8r 5r r r e 4e 6e e r r e e (6) (7) The moment-ratios of the SI-Lognormal Power distribution are independent of. XII..QUNTILES-BSED MESURES OF CENTRE, SPRED, SKEWNESS ND KURTOSIS In this section, we obtain measures of central tendency, dispersion, skewness kurtosis based on the quantiles of the SI-Lognormal Power distribution. The Mid-Quartile Range is given by Q Q e e The Inter-Quartile Range is.67449r.67449r Q Q e e.67449r.67449r The Bowley s Coefficient of Skewness is given by Q Q Q e e Sk.67449r.67449r.67449r r (8) QQ e e The Percentile Coefficient of Kurtosis is Q Q e e.67449r.67449r.855r.855r D9 D e e (9) The formulae of the Bowley s Coefficient of Skewness the Percentile Coefficient of Kurtosis of the SI-Lognormal Power distribution do not involve. XIII..SURVIVL ND HZRD FUNCTIONS Here, we have The survival function is defined as S ( ) F ( ) ln S( ) erf r or in other words ln S( ) erfc () r where the complementary error function erfc( ) is defined as Here, we have h t erfc ( ) e dt The hazard function is defined as h f f( ) F S( ) ln r e ln r erfc r () The graph of the hazard function is given in Figure. The graph shows upside down bathtub shaped hazard rate. This is also called unimodal hazard rate.

5 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat ln r e ln erf r r r Fig. : Graph of the hazard function of the SI- Lognormal-Power distribution By definition, the Cumulative Hazard Function is given by ( ) log S ( ) s such, the CHF of the SI-Lognormal Power distribution is ln ( ) log() log() log erf r or ln ( ) log() log erfc r The Reverse Hazard Rate is defined as r f F XIV. CONCLUDING REMRKS In this paper, we have developed the SI-Lognormal Power distribution which can be regarded as a generalization of the lognormal distribution having scale parameter equal to zero. Some of the fundamental properties of the distribution such as moments moment-ratios, quartiles deciles, survival function hazard function have been obtained. The shape of the density being unimodal positively skewed, it can be epected that this newly derived probability distribution will turn out to be a pertinent model for real-life data-sets ehibiting an upside down bathtub-shaped hazard rate. More importantly, the selfinversion property provides the capability of developing an SI-estimator of that will be more efficient than the estimator obtained by the ordinary method of moments. This work is under way. REFERENCES [] Seshadri, V. (965). On Rom Variables which have the Same Distributions as their Reciprocals. Can. Math. Bull., 8(6), [] Saunders, S.C. (974). Family of Rom Variables Closed Under Reciprocation. J. mer. Statist ssoc., 69(46),5-59. [] Habibullah, S.N., Memon,.Z. hmad, M. ().On a Class of Distributions Closed Under Inversion, Lambert cademic Publishing (LP), ISBN [4] Habibullah, S.N. Saunders, S.C. (), Role for Self- Inversion, Proceedings of International Conference on dvanced Modeling Simulation (ICMS, Nov 8-, ) published by Department of Mechanical Engineering, College of Electrical Mechanical Engineering, National University of Science Technology (NUST), Islamabad, Pakistan, Copyright, ISBN [5] Habibullah, S.N. Fatima, S.S. (5), On a Newly Developed Estimator for More ccurate Modeling with an pplication to Civil Engineering, Proceedings of the th International Conference on pplications of Statistics Probability in Civil Engineering (ICSP) organized by CERR (Vancouver, BC, Canada, July - 5, 5). Sponsoring gency: Higher Education Commission, Pakistan. For the SI-Lognormal Power distribution, we have

6 Proceedings of nd International Multi-Disciplinary Conference 9- December 6, Gujrat

Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.

Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential. Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample