Alpha-Skew-Logistic Distribution

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: 9-765X Volume 0, Issue Ver VI (Jul-Aug 0), PP 6-6 Alpha-Sew-Logistic Distribution Partha Jyoti Haaria a and Subrata Charaborty b a Department of Mathematics, NIT, Meghalaya, India b Department of Statistics, Dibrugarh University, Dibrugarh , Assam, India Abstract: Alpha sew Logistic distribution is introduced following the same methodology as those of Alpha sew normal distribution (Elal-Olivero, 00) and Alpha sew Laplace distributions (Harandi and Alamatsa, 0) Cumulative distribution function (cdf), moment generating function (mgf), moments, sewness and urtosis of the new distribution is studied Some related distributions are also investigated Parameter estimation by method of moment and maximum lielihood are discussed Closeness of the proposed distribution with alpha-sew-normal distribution is studied The suitability of the proposed distribution is tested by conducting data fitting experiment and comparing the values of log lielihood, AIC, BIC Lielihood ratio test is used for discriminating between Alpha sew Logistic and logistic distributions Mathematics Subject Classification: 60E05; 6E5; 6H0 Keywords: Alpha-Sew normal distribution; Multimodal distribution; Alpha-sew normal; Alpha-sew Laplace; Kullbac Leibler (KL) distance I Introduction Aalini (985) first introduced the path breaing sew-normal distribution, which is nothing but a natural extension of normal distribution derived by adding an additional asymmetry parameter and the probability density function (pdf) of the same is given by f Z ( ; ) ( ) ( ); R, R () where and are the pdf and cdf of standard normal distribution respectively Here, is the asymmetry parameter Following Aalini s (985) seminal paper, a lot of research wor has so far been carried out to present different sew normal distributions derived from the underlying symmetric one to model asymmetric behavior of empirical data suitable under different situations (For a complete survey on univariate sew normal distributions see Charaborty and Haaria (0)) Huang and Chen (007), proposed the general formula for the construction of sew-symmetric distributions starting from a symmetric (about 0) pdf h() by introducing the concept of sew function G (), a Lebesgue measurable function such that, 0 G ( ) and G ( ) G( ) ; Z R, almost everywhere A random variable Z is said to be sew symmetric if the pdf is of the following form f Z ( ) h( ) G( ) ; R () Choosing an appropriate sew function G() in the equation () one can construct different sew distributions with both unimodal and multimodal behavior Charaborty et al, [0, 0a, and 0b] introduced new sew logistic, sew Laplace and sew normal distributions which are multimodal with peas occurring at intervals by considering a periodic sew function involving trigonometric sine function and studied their different properties including parameter estimation and data fitting Alpha sew normal distribution was introduced by Elal-Olivero (00) as a new class of sew normal distribution that includes both unimodal as well as bimodal normal distributions Using exactly the similar approach Harandi and Alamatsa (0) investigated a class of alpha sew Laplace distributions and very recently Gui (0) introduced alpha sew normal slash distribution and studied its properties The logistic distribution which very often is seen as a competitor of normal and Laplace distribution is an important probability distribution which attracted wide attention from both theoretical and application point of view (see Johnson et al, 00 and Balarishnan, 99) Again, lie other sewed distributions, the sewed version of the logistic distribution also has wide range of applications in real life (for detail see Wahed and Ali (00); Nadarajah and Kot, (006 and 007); Nadarajah (009); Charaborty et al, (0)) In the present study we have borrowed the idea from Elal-Olivero (00) and Harandi and Alamatsa (0) to propose and investigate an alpha-sew-logistic distribution which generalies the logistic distribution and flexible enough to adequately model both unimodal as well as bimodal data in the presence of positive or negative sewness 6 Page

2 II Alpha Sew Logistic Distribution In this section we introduce the proposed alpha sew logistic distribution and investigate its shape and mode numerically Definition If an rv Z has the density function {( ) }exp( ) f Z ( ; ) ; R, R () {6 ( )}( exp( )) where, then we say Z is an alpha-sew-logistic rv with parameter and denote it by Z ~ ASLG( ) Particular cases: i for 0, we get the standard Logistic distribution is given by exp( ) f Z ( ) ; R () ( exp( )) and is denoted by Z ~ L(0,) ii If the we get a bimodal logistic (BLG) distribution given by f Z exp( ) ( ) ; R (5) ( exp( )) and is denoted by Z BLG iii If Z ~ ASLG( ) then Z ~ ASLG(- ) d Plots of the pdf The ASLG pdf for different choices of the parameter are plotted in figure Figure Plots of pdf of ASLG( ) From the figures it can be seen that the distribution has at most two modes (see also section ) Also the smaller of the two peas when present appears in the right (left) of the higher one depending on the value of < (>) 0 Mode of ASLG Here we numerically verify the fact that ASLG has at most two modes The first differentiation of the pdf f Z ( ; ) of ASLG ( ) with respect to is given by 7 Page

3 Df exp( )[ {exp( )( ( ( )) ) ( ( ) ) } exp( ) ] ( ; ) (6) Z (6 )(exp( ) ) In order to chec that ASLG( ) has at most two modes, the contour of the equation Df Z ( ; ) 0 is drawn in figure It can be seen that there is at most three eros of Df Z ( ; ) which establishes that ASLG( ) has at most two modes Note that for 08 < < 08 ASLG( ) remains unimodal (see the second plot in figure ) Figure Contour plots of the equation Df Z ( ; ) 0 III Distributional Properties Cumulative Distribution Function (cdf) Theorem The cdf of ASLG( ) is given by ( ) F Z ( ; ) ( )log{ exp( )} Li[ - exp()] (7) (6 ) exp( ) Proof: Where Li FZ ( ; ) (6 ) (6 ) {( ) }exp( ) d ( exp( )) exp( ) ( exp( )) d exp( ) ( exp( )) d log( exp( (6 ) exp( ) exp( ) )) log( exp( )) Li[ exp( ) ( ) ( )log{ exp( )} (6 ) exp( ) - exp()] ( is the poly-logarithm function (see Prudniov et al (986)) n n ) Li exp( ) ( exp( )) [ - exp()] The cdf is plotted in figure for studying variation in its shape with respect to the parameter Corollary In particular by taing limit or the cdf of BLG distribution in equation (5) is easily obtained from (7) as {exp( ) ( exp( )log(( exp( ))} 6(( exp( )) Li [ exp()] ( exp( )) d 8 Page

4 Figure Plots of cdf of ASLG( ) Moment Generating Function (mgf) and Moments Moment Generating Function (mgf) Theorem The mgf of ASLG( ) is given by 6 M X ( t) { M X ( t)} M Z ( t) t { t cot( t)} t{ cos( t)} sin( t) ; t, t (6 ) t (8) where X ~ L(0,) and M X ( t) B( t, t) ( t) ( t) is the mgf of L (0,) Proof: M Z ( t) (6 ) (6 ) {( ) }exp( ) exp( t) d ( exp( )) exp( t )exp( ) d ( exp( )) exp( t )exp( ) d ( exp( )) exp( t )exp( ) d ( exp( )) M X ( t) M ( ) { t cot( t)} X t (6 ) t The result follows on simplification M X { t ( t)} t{ cos( t)} sin( t) Corollary In particular, by taing limit as or of the M Z (t) in equation (8), we get the mgf of BLG distribution as { M X ( t)} ( t) t{ cos( t)} sin( t) Remar Alternative expression for the mgf of ASLG( ) can be obtained in terms of gamma function and its derivatives as / / [( t) ( t) { ( t) ( t) ( t) ( t)} (6 ) // / // { ( t) ( t) ( t) ( t)}] (9) Theorem The th order moment of ASLG( ) distribution is given by 9 Page

5 E( Z ( ) ( ) ( ) ( ) ( ) ( ); if even 6 ) (6 ) ( ) ( ) ( ); if odd where s s (s) j is the Riemann eta functions and ( s) t exp( t) dt j Proof: Recall that, for X ~ L(0,) ie for the standard logistic random variable X Now, E( X E( Z ) ( (/ ) 0; )) ( ) ( ); if is even f ( ; ) d (6 ) if is odd 0 ( (Balarishnan, 99) exp( ) ) ( exp( )) exp( ) exp( ) exp( ) d d d (6 ) ( exp( )) ( exp( )) ( exp( )) d (0) Case I when is even exp( ) exp( ) E( Z ) d d (6 ) ( exp( )) ( exp( )) 6 ( ) ( ) ( ) ( ) ( ) ( ) (6 ) Case II when is odd exp( ) E( Z ) d (6 ) ( exp( )) 6 ( ) ( ) ( ) (6 ) The result (0) follows immediately Moments From (0) using the following particular values of the Riemann eta functions, () / 6, () / 90 and (6) / 95 (Prudniov et al 986), we get 9 / E( Z ), 6 Which is proportional to variance of standard logistic distribution and is positive (or negative) if () 0 E( Z / 0 7 ) 5(6 ) / E( Z ), is positive (or negative) if depending on whether () 0 5(6 ) E( Z / (98 55 ) ) 5(6 ) (60 7 ) Var( Z) 5(6 ) (80 96 ) Since, 5(6 ) L (0,) (80 96 ) 5(6 ) therefore, variance of ASLG( ) is greater than or equal to that of 0 Page

6 Remar From the compact formulas of the moments obtained above we can see that / 886, which means that the third moment is increasing (decreasing) function of the mean according as the 0 00 parameter () 0 Further, 07 and hence the pdf of in equation () can be written in ( ) terms of as f Z {( 0090 ) }exp( ) ( ; ) ; R, R (6 0889)( exp( )) Bounds for mean and variance: The following bounds for mean, variance can be derived using numerically optimiing E( Z ) and Var ( Z ) with respect to i 9 E( Z ) 9 and ii 9 Var( Z ) 8 Plots of mean and variance: We have plotted the mean and the variance respectively in figure and 5 to study their behavior Theses plots also verify the bounds presented above Figure Plots of mean Figure 5 Plots of variance Remar Moments of BLG Distribution can be derived easily by taing limit as or in the moments of ASLG( ) in section as E( Z ) 0, Var( Z ) 7 / 5 8 Sewness and Kurtosis The sewness and Kurtosis of ASLG ( ) is given by: Sewness, / ( ) 6 Kurtosis, ( ) 5 { 5C 90C ( 0 6C) } / {0C ( 0 C) } {C(C 80) 800} 5 C{ C(65C 7) 0} 890C 7{ (C 0) 0C} (7C 0) where, C (6 ) / Note that, for () 0 all odd order moments are positive (negative) while the even order moments are always positive Bounds for sewness and urtosis: The following bounds for sewness and urtosis can be derived using numerically optimiing and with respect to i 6 6 and ii 07 7 Plots of sewness and urtosis: We have plotted the sewness and urtosis respectively in figure 6 and 7 to study their behavior Theses plots also verify the bounds presented above Remar Sewness and Kurtosis of BLG Distribution can be derived easily by taing limit as or in the results of ASLG( ) in section as, Page

7 Figure 6 Plots of sewness Figure 7 Plots of and urtosis IV Truncated alpha-sew-logistic-distribution as a life time distribution Here we present alpha-sew-logistic-distribution truncated below 0 as a potential life time distribution The pdf of the truncated-alpha-logistic-distribution TASLG( ) truncated below 0 is given by provided, {( t) }exp( t) f T ( t; ) ; t 0 () ( exp( t)) ( 6 6) The survival function S T ( t; ) and the haard rate functions h T ( t; ) of TASLG( ) can be easily expressed in terms of the pdf f Z ( t; ) and cdf F Z ( t, ) of ASLG( ) as FZ ( t, ) f ST ( t; ) and ht ( t; ) F (0, ) S Z T T ( t; ) fz ( t; ) ( t; ) F ( t, ) For admissible values of the parameter, we have plotted the h T ( t; ) in figure 8 to study its behavior graphically Z Figure 8 Plots of haard rate function TASLG( ) From the figure 8 it can be seen that for 0 the haard rate is increasing always while for 0 7 it assumes bathtub shape For values of (0,07) the haard rate taes different shapes Thus depending on the range of the values of the parameter, the haard rate function of TASLG ( ) assumes different useful shapes lie the Weibull model and thus has the potential to be a flexible life time model Remar 5 Clearly, for 0, TASLG ( ) reduces to standard half logistic distribution (Balarishnan, 99) As such the present TASLG( ) may be seen as a generaliation of the standard half logistic distribution (Balarishnan, 99) V Closeness of alpha sew normal and alpha sew logistic distributions Similarities of normal and logistic have been examined in the statistical literature (Johnson et al (00), pp 9,) Since the alpha sew logistic distribution proposed here has the same structure as that of the alpha sew normal distribution of Elal-Olivero (00), we present results of a brief numerical investigation carried out to chec their closeness Accordingly, the difference of the pdfs of ASN( a,0, ) and ASLG(,0, ), have been studied to find combinations of parameters such that the two pdfs almost coincide This is achieved by minimiing the sum of the squared difference of the pdfs over a finite set of equally spaced points in the range of these two distributions with respect to one parameter while others are ept fixed The KL distance Page

8 (Burnham and Anderson, 00) of these pairs are then checed to re-emphasie their closeness We found many close pairs, two such close pairs are I ASN( 5, 0,) and ASLG (656, 0,) : Distance between the pdfs = and KL distance = 0075 II ASN( 855, 0,6) and ASLG ( 5,0, ) : Distance between the pdfs = and KL distance = 0057 We have presented the graphs of the pdfs, cdfs and difference of cdfs of the first pair of distributions respectively in Figure 9, 0 and to depict their closeness Figure 9 Plots of pdf Figure 0 Plots of cdf Figure Plots of differences of cdf VI Parameter estimation We present the problem of parameter estimation of a location and scale extension of ASLG ( ) If Z ~ ASLG( ), then Y Z is said to be the location ( ) and scale extension ( ) of Z and has the pdf {( ( y ) / ) }exp( ( y ) / ) f Y ( y;,, ) ; y R,, R, 0 () (6 ) ( exp( ( y ) / )) We denote it by Y ~ ASLG(,, ) In particular, for or, ASLG(,, ) BLG 6 Method of moments Let m, m and m are respectively the first three sample raw moments for a given random sample ( y, y,, yn) of sie n drawn from ASLG(,, ) distribution in equ () Then the moment estimates of the three parameters, and are obtained by simultaneously solving the following equations derived by equating first three population moments with corresponding sample moments m m () (0 7 ) m () 5(6 ) 0 (0 7 ) m (5) 5(6 ) Substituting the value of from equ () in equ () and solving for 5( m m )(6 ) (60 7 ) Finally, putting these values of and m m m ( m m ) we get, in equ (5), we get the following equ in 5 ( 6 6 (6 ) (6) 7 ) (6 ) ( ) m m 60 7 (7) / Page

9 While the exact solution of the equ (7) is not tractable, it can be numerically solved to estimate Once is estimated rest of the two parameters namely, and can be directly estimated from equ () and (6) respectively Remar 6 It may be noted that the equ (7) may return multiple solutions for and hence leading to more than one set of feasible estimates for the parameters This situation may be resolved by retaining the best set (according to some criterion eg, lielihood or entropy) among the feasible solutions 6 Maximum lielihood estimation For a random sample of sie n ( y, y,, yn) drawn from ASLG(,, ) distribution in equ (), the lielihood function is given by log L log n i f Y ( y;,, ) n (6 ) The MLE s of n log[( ( y i ) / ) ] log[( exp( ( y i ) / ))] i n i ( )/ (8) y i, and are obtained by numerically maximiing log L with respect to, and The variance-covariance matrix of the MLEs can be derived by using the asymptotic distribution of MLEs as, log log log E L E L E L log log log E L E L E L (9) log log log E L E L E L When the exact expressions for various expectations above are cumbersome, in practice they are estimated as log E L log L ˆ log log, E L L ˆ, ˆ VII Real life applications: comparative data fitting Here we have considered data set of the heights of 00 Australian athletes (Coo and Weisberg, 99) and compared the proposed distribution ASLG(,, ) with the logistic distribution L (, ), sew logistic distribution SL(,, ) of Wahed and Ali (00), the new sew logistic distribution L (,,, ) of Charaborty et al (0), alpha-sew-normal distribution ASN(,, ) of Elal-Olivero (00), alpha-sew- Laplace distribution ASL(,, ) of Harandi and Alamatsa (0) and alpha sew normal slash distribution ASNS(, q,, ) of Gui (0) The MLE of the parameters are obtained by using numerical optimiation routine In the present wor, the GenSA-pacage, Version-0, of the software R is used Aaie Information Criterion (AIC) and Beysian Information Criterion (BIC) are used for model comparison Further, since L(, ) and ASLG(,, ) are nested models the lielihood ratio (LR) test is used to discriminate between them The LR test is carried out to test the following hypothesis H 0 : 0, that is the sample is drawn from L (, ), against the alternative H : 0, that is the sample is drawn from ASLG(,, ) From the Table it is seen that the proposed alpha sew logistic distribution ASLG(,, ) provides best fit to the data set under consideration in terms of all the criteria, namely the log-lielihood, the AIC as well etc S Page

10 as the BIC The plots of observed (in histogram) and expected densities (lines) presented in Figure 9, also confirms our finding LR test result: The value of LR test statistic is 5058 which exceed the 95% critical value, 8 Thus there is evidence in favors of the alternative hypothesis that the sampled data comes from ASLG(,, ) not from L(, ) Table : MLE s, log-lielihood, AIC and BIC for heights (in cm) of 00 Australian Athletes data Parameters Loglielihood q AIC BIC Distributions L (, ) SL (,, ) L (,,, ) S ASN(,, ) ASL(,, ) ASNS(, q,, ) ASLG(,, ) Figure Plots of observed and expected densities for heights of 00 Australian athletes Remar 7 The variance-covariance matrix of the MLEs of the parameters of ASLG(,, ) is obtained by using (9) as Variace Covrance ( ˆ, ˆ, ˆ) VIII Concluding remars In this article a new alpha sew logistic distribution which includes both unimodal as well as bimodal logistic distributions is introduced and some of its basic properties are investigated A truncated version of the proposed distribution is seen to have bathtub shaped failure rate function Closeness of the alpha sew normal and alpha sew logistic distributions is briefly explored The numerical results of the modeling of a real life data set considered here has shown that the proposed distribution provides better fitting in comparison to the logistic and sew logistic distribution of Wahed and Ali (00), alpha-sew-normal distribution ASN(,, ) of Elal-Olivero (00), alpha sew logistic distribution ASL(,, ) of Harandi and Alamatsa (0), the new multimodal sew logistic distribution of Charaborty et al (0) and alpha sew normal slash distribution of Gui (0) 5 Page

11 References [] Aalini, A (985) A class of distributions which includes the normal ones Scand J Stat : 7-78 [] Balarishnan, N (99) Handboo of Logistic Distribution Marcel Deer New Yor [] Burnham, K P, Anderson, D R (00) Model Selection and Multimodel Inference: A Practical Information Theoretic Approach nd edn SpringerNew Yor [] Charaborty, S, Haaria, P J (0) A survey of the theoretical developments in univariate sew normal distributions Assam Statist Rev 5(): - 6 [5] Charaborty, S Haaria, P J, Ali, M M (0) A new sew logistic distribution and its properties Pa J Statist 8(): 5-5 [6] Charaborty, S Haaria, P J, Ali, M M (0a) A multimodal sewed extension of normal distribution: its properties and applications Statistics-A Journal of Theoretical and Applied Statistics, [7] Charaborty, S Haaria, P J, Ali, M M (0b) A multimodal sew-laplace distribution: its properties and applications Pa J Statist, 0(): 5-6 [8] Coo, R D, Weisberg, S (99) An Introduction to Regression Analysis Wiley New Yor [9] Elal-Olivero, D (00) Alpha sew normal distribution Proyecciones Journal of Mathematics 9: - 0 [0] Gui, W (0) A generaliation of the slashed distribution via alpha sew normal distribution Stat Methods Appl DOI 0007/s [] Harandi, S S, Alamatsa, M H (0) Alpha-sew-Laplace distribution, Statist Probab Lett 8():77-78 [] Huang, W J, Chen, Y H (007) Generalied sew Cauchy distribution, Statist Probab Lett 77:7-7 [] Johnson, N L, Kot, S, Balarishnan, N (00) Continuous Univariate Distributions (Volume ) A Wiley-Interscience Publication John Wiley & Sons Inc New Yor [] Nadarajah, S (009) The sew logistic distribution AStA Adv Stat Anal 9: 87-0 [5] Nadarajah, S, Kot, S (006) Sew distributions generated from different families Acta Appl Math 9: - 7 [6] Nadarajah, S, Kot, S (007) Sew models I Acta Appl Math 98: - 8 [7] Prudniov, A P, Brychov, Y A and Marichev, Y I (986) Integrals and Series, (Volume - ), Gordon and Breach Amsterdam [8] Wahed, A S, Ali, M M (00) The sew logistic distribution J Stat Res 5: Page