Theoretical Properties of Weighted Generalized. Rayleigh and Related Distributions

Transcription

1 Theoretical Mathematics & Applications, vol.2, no.2, 22, ISSN: (print, (online International Scientific Press, 22 Theoretical Properties of Weighted Generalized Rayleigh and Related Distributions Xueheng Shi, Broderic O. Oluyede 2 and Mavis Pararai 3 Abstract In this paper, a new class of weighted generalization of the Rayleigh distribution is constructed and studied. The statistical properties of these distributions including the behavior of hazard or failure rate and reverse hazard functions, moments, moment generating function, mean, variance, coefficient of variation, coefficient of sewness, coefficient of urtosis are obtained. Other important properties including entropy (Shannon, beta and generalized which are measures of the uncertainty in these distributions are also presented. Mathematics Subject Classification: 62E, 62F3 Keywords: Weighted Distribution, Weighted Generalized Rayleigh Distribution, Generalized Rayleigh Distribution Georgia Institute of Technology, Atlanta, GA 3332, xhshi@gatech.edu 2 Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 346, boluyede@georgiasouthern.edu 3 Indiana University of Pennsylvania, Indiana, PA 575, pararaim@iup.edu Article Info: Received : March 7, 22. Revised : April 25, 22 Published online : June 3, 22

2 46 Properties of Weighted Generalized Rayleigh Distributions Introduction The use and application of weighted distributions in research related to reliability, bio-medicine, ecology and several other areas are of tremendous practical importance in mathematics, probability and statistics. These distributions arise naturally as a result of observations generated from a stochastic process and recorded with some weight function. The concept of weighted distributions has been employed in wide variety applications in reliability and survival analysis, analysis of family data, meta-analysis, ecology, and forestry. Several authors have presented important results on length-biased distributions and on weighted distributions in general. Rao [] unified the concept of weighted distributions. Vardi [4] derived the nonparametric maximum lielihood estimate (NPMLE of a lifetime distribution in the presence of length bias and established convergence to a pinned Gaussian process with a simple covariance function under mild conditions. Rao [] identified the various sampling situations that can be modeled by what he called weighted distributions, extending the idea of the methods of ascertainment upon estimation of frequencies by Fisher. Patil and Rao [7], [8] investigated the applications of the weighted distributions. Statistical applications of weighted distributions, especially to the analysis of data relating to human population and ecology can be found in Patil and Rao [7]. For additional and important results on weighted distributions, see Gupta and Keating [4], Oluyede [6], Patil and Ord [9], Zelen and Feinleib [5] and references therein. Consider the following general class of weight functions given by w(x x e tx F i (xf j (x, ( where F (x F (x and F (x is the survival or reliability function and cumulative distribution function (cdf of the random variable X. Now setting t ; j i ; t i j ; ( t, i i, j n i; t i and t j in the weight function, one at a time, implies probability weighted moments, moment-generating functions, moments, order statistics, proportional hazards and proportional reversed hazards, respectively. Let X be a non-negative random variable (rv with its natural probability density function (pdf f(x;, where the natural parameter is Ω (Ω is the

3 X. Shi, B.O. Oluyede and M. Pararai 47 parameter space. Suppose a realization x of X under f(x; enters the investigator s record with probability proportional to w(x; β, so that the recording (weight function w(x; β is a non-negative function with the parameter β representing the recording (sighting mechanism. Clearly, the recorded x is not an observation on X, but on the rv X w, having a pdf f w (x;, β w(x, βf(x;, (2 ω where ω is the normalizing factor obtained to mae the total probability equal to unity by choosing < ω E[w(X, β] <. The random variable X w is called the weighted version of X, and its distribution is related to that of X and is called the weighted distribution with weight function w. The generalized Rayleigh distribution (GRD is considered to be a very useful life distribution. It presents a flexible family in the varieties of shapes and is suitable for modeling data with different types of hazard rate function: increasing, decreasing and upside down bathtub shape (UBT. The main objective of this article is to explore the properties of weighted generalized Rayleigh distribution (WGRD. This article is organized as follows. Section 2 contains the some basic definitions, utility notions and useful functions including the weighted generalization. The probability density function (pdf, cumulative distribution function (cdf, hazard function and reverse hazard function of the WGRD are derived. In section 3, moments and related measures are derived. Measures of uncertainty are presented in section 4, followed by concluding remars. 2 Basic Utility Notions In this section, some basic utility notions and results on the WGRD are presented. Suppose the distribution of a continuous random variable X has the parameter set {, 2,, n }. Let the pdf of the rv X be given by f(x;. The corresponding cdf is defined to be F (x; x f(t; dt. (3 The hazard function of X can be interpreted as the instantaneous failure rate or the conditional probability density of failure at time x, given that the

4 48 Properties of Weighted Generalized Rayleigh Distributions unit has survived until time x, see Shaed and Shanthiumar [3]. The hazard function h(x; is defined to be h(x; P (x X x + x lim x x[ F (x; ] F (x; F (x; f(x; F (x;, (4 where F (x; is the survival or reliability function. The concept of reverse hazard rate was introduced as the hazard rate in the negative direction and received minimal attention, if any, in the literature. Keilson and Sumita [5] demonstrated the importance of the reverse hazard rate and reverse hazard orderings. Shaed and Shanthiumar [3] presented results on reverse hazard rate. See Chandra and Roy [2], Bloc and Savits [] for additional details. We present a formal definition of the reverse hazard function of a distribution function F. See Ross [2], Chandra and Roy [2] for additional details. The reverse Hazard function can be interpreted as an approximate probability of a failure in [x, x + dx], given that the failure had occurred in [, x]. Definition 2.. Let (a, b, a < b <, be an interval of support for F. Then the reverse hazard function of X (or F at t > a is denoted by τ F (t and is defined as τ(t; d dt log F (t; f(t; F (t;. (5 Some useful functions that are employed in subsequent sections are given below. The gamma function is given by Γ(x The digamma function is defined by where Γ (x t x e t dt. (6 Ψ(x Γ (x Γ(x, (7 t x (log te t dt is the first derivative of the gamma function. The second derivative of the gamma function is Γ (x t x (log t 2 e t dt.

5 X. Shi, B.O. Oluyede and M. Pararai 49 Definition 2.2. The n th -order derivative of gamma function is given by: Γ (n (s z s (log z n exp( z dz. (8 This derivative will be used frequently in this paper. The lower incomplete gamma function and the upper incomplete gamma function are γ(s, x respectively. x t s e t dt and Γ(s, x x t s e t dt, (9 2. Weighted Generalized Rayleigh Distribution First consider the following two parameter probability density function (pdf given by f(x;, Γ( exp ( x The corresponding moments are given by Let c Γ( E(X m x m for x, > and >. ( Γ ( exp ( x dx. ( and z x, then x z, dx z dz, and E(X m c c m+ ( m m z ( exp( z z dz z m+ exp( z dz m+ Γ( m+ Γ( m Γ( m+ Γ(. Now consider the following weighted (length-biased version of the pdf given above in equation (: f l (x;, xf(x E(X x 2 Γ( 2 exp ( x for x, >, and >. (2

6 5 Properties of Weighted Generalized Rayleigh Distributions With 2 we obtain the usual Rayleigh probability density function (pdf. The weighted pdf corresponding to f l (x with weighted function w(x; m x m and for x, >, and > is given by Note that g w (x;,, m xm f l (x E(X m x m+ m+2 exp ( x lim g w(x;,, m, and lim g w (x;,, m. (3 x x. The cdf corresponding to the weighted generalized Rayleigh distribution (WGRD g w (x;,, m is given by: Let c have m+2 G W (x;,, m x y and z, then y z g w (y;,, m dy. (4 and dy z dz. We x ( ( G W (x;,, m c m+ m+ z exp( z z dz c m+2 x Since the lower incomplete gamma function is defined as we have: γ(s, x x G W (x;,, m c m+2 z m+2 exp( z dz. (5 t s exp( t dt, (6 x m+2 z m+2 exp( z dz m+2 γ( m+2, x γ( m+2, x. (7

7 X. Shi, B.O. Oluyede and M. Pararai 5 The survival function is given by Ḡ W (x;,, m G W (x;,, m The corresponding hazard function is given by h GW (x;,, m g w(x;,, m Ḡ W (x;,, m ( x m+ exp( x m+2 m+2 /( x m+ exp( x [ Γ( m+2 m+2 γ( The reverse hazard function τ GW (x;,, m is given by τ GW (x;,, m g w(x;,, m G w (x;,, m x m+ exp( x m+2 γ( m+2, x xm+ exp( x m+2 γ( m+2 γ( m+2, x. (8 γ( m+2, x, x]. (9, x. (2 We study the behavior of the hazard function of the WGRD via the following lemma, due to Glaser [3]. Lemma 2.3. Let f(x be a twice differentiable probability density function of a continuous random variable X. Define η(x f (x f(x, where f (x is the first derivative of f(x with respect to x. Furthermore, suppose the first derivative of η(x exist.. If η (x <, for all x >, then the hazard function is monotonically decreasing (DHR. 2. If η (x >, for all x >, then the hazard function is monotonically increasing (IHR. 3. Suppose there exist x such that η (x >, for all < x < x, η (x and η (x < for all x > x. In addition, lim x f(x, then the hazard function is upside down bathtub shape (UBT.

8 52 Properties of Weighted Generalized Rayleigh Distributions 4. Suppose there exist x such that η (x <, for all < x < x, η (x and η (x > for all x > x. In addition, lim x f(x, then the hazard function is bathtub shape (BT. Now consider the weighted distribution discussed above. We compute the quantity η GW (x;,, m(x g w(x;,,m g w(x;,,m, and apply Glaser [3] result. Note that η GW (x;,, m g W (x;,, m g W (x;,, m [ ] x m+ exp( x The derivative η G W (x is given by η G W (x d ( dx cx m+ exp( x x (m +. x m + x 2 + m + + x x ( x 2. (2 If, then η G W (x > ; and the hazard function is monotonically increasing. If < <, then η G W (x m+ x 2. There exists x x 2 (m+ such that η G W (x >, for all < x < x, η G W (x and ( ( η G W (x < for all x > x. In addition, lim x g w (x, consequently, the hazard function is upside down bathtub shape (UBT. 3 Moment Generating Function and Moments Under the density function f l (x xf(x E(x E(e tx e tx c l x exp x 2 Γ( 2 x exp(, we have x 2 Γ( 2 exp( x dx [ ( ] x t dx, (22

9 X. Shi, B.O. Oluyede and M. Pararai 53 where c l and dx 2 Γ( 2. Let z x ( z ( t dz, so that t, where t <, then we have x z ( t E(e tx c l c l ( t 2 z ( t z exp( z ( t dz z 2 exp( z dz (23 Γ( 2 2 Γ( 2 ( t 2. (24 ( t 2 We also present the raw moments, mean, variance, coefficients of variation, sewness and urtosis for the WGRD. Now, under the weighted generalized Rayleigh pdf g w (x;,, m, the n th raw moment is given by: E(X n where c c x n x m+ exp m+2 ( ( x dx c x n+m+ exp x dx, (25. Let z x, then E(X n c c z m+n+ m+2 ( exp z ( z dz z m+n+2 exp( z dz [ m+n+2 ] Γ( m+n+2 n Γ( m+n+2. (26 Let Γ and Γ j Γ( m+j+2, then E(X n can be rewritten as n Γ n Γ. The variance σ 2, coefficient of variation (CV, coefficient of sewness (CS, and coefficient of urtosis (CK are given below. The variance is σ 2 2 Γ( m+4 2 Γ 2 ( m+3 Γ 2 ( m+2. (27

10 54 Properties of Weighted Generalized Rayleigh Distributions The coefficient of variation (CV is given by CV [ Γ( m+2 m+4 Γ( Γ2 ( m+3] /2 /Γ( m+2 Γ( m+3 m+2 /Γ( Note that (Γ Γ 2 Γ 2 2 Γ. (28 µ 3 3 Γ( m+5 3 Γ( m+3 2 Γ( m and σ 3 [ 3 Γ( m+2 m+4 Γ( Γ2 ( m+3] 3 2 Γ 3 ( m+2. The coefficient of sewness (CS is given by CS µ 3 σ 3 3 Γ 3 ( m+3 Γ 3 ( m+2, Γ2 Γ 3 3Γ Γ Γ 2 + 2Γ 3. (29 [Γ Γ 2 Γ 2 ] 3 2 The coefficient of urtosis (CK is defined as CK µ 4 computed. In fact, σ 4 which can be readily µ 4 4 Γ( m+6 4 Also σ 4 is 4 ( Γ4 4 Γ Γ 3 Γ Γ 2 Γ( m Γ2 Γ 2 Γ 3 σ 4 ( σ 2 2 ( 2 Γ 2 Γ 3 Γ( m+5 3 Γ4 Γ 4 2 Γ 2 ( m+3 2 Γ( m Γ 4 ( m+3 Γ 4 ( m+2. ( Γ 2 Γ so that the coefficient of urtosis (CK is given by ( 4 Γ4 4Γ 3Γ + 6Γ2 Γ 2 3Γ4 Γ Γ CK 2 Γ 3 Γ 4 (Γ 2 Γ Γ Γ 4 Γ 4 ( Γ2 Γ Γ 2 2, (3 Γ 4Γ 3 4Γ Γ 3 Γ 2 + 6Γ 2 Γ 2 Γ 3Γ 4 (Γ 2 Γ Γ 2 2. (32

11 X. Shi, B.O. Oluyede and M. Pararai 55 Figure : Plot of coefficient of variation with fixed m or The graphs of the coefficients of variation (CV, sewness (CS and urtosis (K are given in Figure, Figure 2 and Figure 3 when the parameters, and are fixed. 4 Some Measures of Uncertainty for WGRD The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. In this section, we present some useful and important measures of uncertainty including Shannon entropy, β entropy, and generalized entropy for the WGRD. 4. Shannon Entropy Shannon entropy of a random variable X is a measure of the uncertainty

12 56 Properties of Weighted Generalized Rayleigh Distributions Figure 2: Plot of coefficient of sewness with fixed m or and is given by E F ( log(f(x, where f(x is the pdf of the random variable X. Under the WGRD, Shannon entropy is given as follows: H(g w E [ log g w (X] [log g w (x] g w (x;,, m dx. (33 Note that where c m+2 log g w (x log, so that ( x m+ exp x m+2 log c + (m + log x x, (34 E [ log g w (X] [log g w (x] g w (xdx [log c + (m + log x x ] c x m+ exp ( x dx

13 X. Shi, B.O. Oluyede and M. Pararai 57 Figure 3: Plot of coefficient of urtosis with fixed m or c (m + + c c log c x m+ exp( x log x dx x m+ exp( x dx x +m+ exp( x log x dx. Let z x, then x z, dx z, x m+ m+ z m+ and log x log( z log + log z, so that c (m + Also, note that c log c x m+ exp x m+ e x c (m + m+2 log x dx 2 (m + log + ( x dx c log c m+ c m+2 log c z m+2 (log(z e z dz (m + Γ( Γ. (35 m+ z e z z m+2 e z dz ( z dz log ( m+2 log log Γ, (36

14 58 Properties of Weighted Generalized Rayleigh Distributions and ( c x+m+ exp x dx c c c m++2 m+2 x +m+ exp( x dx m++ z m++ exp( z Γ ( m+2 Therefore, Shannon Entropy H(g w is given by E [ log g w (X] z m+2 m+2 exp( z dz m++2 Γ( ( z dz Γ Γ. (37 (m + log + (m + 2 log log [log g w (x] g w (x dx (m + Γ( Γ log + log Γ + Γ Γ (m + Γ( Γ log + log Γ + Γ Γ. ( β-entropy In this section, we present β-entropy for the WGRD, which is given by H β (g w [ β β [ ] gw(x β dx c β x β(m+ exp ( ] βx dx, (39 where c m+2. Let z βx, then x ( z, dx z ( dz β β

15 X. Shi, B.O. Oluyede and M. Pararai 59 and x β(m+ ( β(m+ β g β w(xdx c β z β(m+. We have: ( β β(m+ cβ ( β β(m++ cβ ( β β(m++ Γ β β z β(m+ ( exp( z z ( β dz z β(m++ exp( z dz ( β(m + + ( β(m + + Γ Γ β ( m+2 β β(m++ Therefore, β-entropy for the WGRD is given by: for β. H β (g w β β [ [ ] gw(xdx β β β Γ β ( m+2 β β(m++. (4 ( ] β(m + + Γ, (4 4.3 Generalized Entropy In this section, we present the generalized entropy, which is defined by I GW (α v αµ α α(α α,, (42 where v α x α df (x and F (x is the cumulative distribution function (cdf for the random variable X. Since we have for α,. I GW (α v α α Γ( m+α+2, (43 [ α Γ( m+α+2 α(α α(α α(α [ Γ( m+α+2γ α ( m+3 Γ α ( m+2 ( Γα Γ α Γ α α Γ α ( m+3 ] Γ α ( m+2 ], (44

16 6 Properties of Weighted Generalized Rayleigh Distributions The mean logarithmic deviation (MLD index is given by I GW ( lim α I(α log µ v The Theil index is given by log Γ( m+3 log + log Γ ( m + 3 I GW ( lim α I(α µ v log µ Γ( m+3 m+2 /Γ( Γ( m+3 m+2 /Γ( log µ ( m + 3 log log Γ ( m + 2 log Γ. (45 ( m log Γ. (46 5 Concluding Remars Some properties of the generalized Rayleigh distribution and weighted generalized Rayleigh distributions are presented. The probability density function, cumulative distribution function, the failure rate function or hazard function and the reverse hazard function are presented. The behavior of the hazard function was also established, indicating that this generalization covers a wide range of possibilities as a lifetime model. Entropy measures, including Shannon entropy, β entropy, and generalized entropy for the WGRD are also derived. Acnowledgements. The authors wish to express their gratitude to the referees and editor for their valuable comments.

17 X. Shi, B.O. Oluyede and M. Pararai 6 References [] H.W. Bloc and T. H. Savits, The Reverse Hazard Function, Probability in the Engineering and Informational Sciences, 2, (998, [2] N.K. Chandra and D. Roy, Some Results on Reverse Hazard Rate, Probability in the Engineering and Information Sciences, 5, (2, [3] R.E. Glaser, Bathtub and Related Failure Rate Characterizations, Journal of American Statistical Association, 75, (98, [4] R.C. Gupta and J.P. Keating, Relation for Reliability Measures under Length Biased Sampling, Scandinavian Journal of Statistics, 3, (985, [5] J. Keilson and U. Sumita, Uniform Stochastic Ordering and Related Inequalities, Canadian Journal of Statistics,, (982, [6] B.O. Oluyede, On Inequalities and Selection of Experiments for Length- Biased Distributions, Probability in the Engineering and Informational Sciences, 3, (999, [7] G.P. Patil and C.R. Rao, Weighted Distributions and Size-Biased Sampling with Applications to Wildlife and Human Families, Biometrics, 34, (978, [8] G.P. Patil and C.R. Rao, Weighted distributions: a survey of their application, P.R. Krishnaiah (Ed., Applications of Statistics, North Holland Publishing Company, 977, [9] G.P. Patil and J.K. Ord, Weighted distributions, in Encyclopedia of Biostatistics, P. Armitage and T. Colton, eds, 6, Wiley, Chichester, (997, [] C.R. Rao, On Discrete Distributions Arising Out of Methods of Ascertainment,in Classical and contagious Discrete Distributions, G.P. Patil, ed., Pergamon Press and Statistical Publishing Society, Calcutta, (965,

18 62 Properties of Weighted Generalized Rayleigh Distributions [] C.R. Rao, Weighted Distributions Arising Out of Methods of Ascertainment, in A Celebration of Statistics, A.C. Atinson and S.E. Fienberg, eds, Springer-Verlag, New Yor, Chapter 24, , 985. [2] S.M. Ross, Stochastic Processes, Wiley, New Yor, 983. [3] M. Shaed and J.G. Shanthiumar, Stochastic Orders and Their Applications, New Yor, Academic Press, 994. [4] Y. Vardi, Nonparametric Estimation in the Presence of Bias, Annals of Statistics, (2, (982, [5] M. Zelen and M. Feinleib, On the Theory of Chronic Diseases, Biometria, 56, (969, 6-64.