We introduce a one parameter probability model bounded on (0, 1) support

called One Parameter Minimax distribution which is a special case of both the

Kumaraswamy distribution and Beta distribution. Its statistical properties are

systematically explored; we provide explicit expressions for its moments, quantile

function, reliability function and failure rate. The method of maximum likelihood

estimation was used in estimating its parameter. The proposed model can be used to

model data sets with increasing failure rate

Adejumo,, A. O.

Oguntunde, P.E.

Covenant University Repository

Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015                                                                          A Note on the Minimax Distribution  Oguntunde, P. E1, Adejumo, A. O2  1Department of Mathematics,  Covenant University, Ota, Ogun State, Nigeria  2Department of Statistics,  University of Ilorin, Kwara State, Nigeria  Abstract: We introduce a one parameter probability model bounded on (0, 1) support called One Parameter Minimax distribution which is a special case of both the Kumaraswamy distribution and Beta distribution. Its statistical properties are systematically explored; we provide explicit expressions for its moments, quantile function, reliability function and failure rate. The method of maximum likelihood estimation was used in estimating its parameter. The proposed model can be used to model data sets with increasing failure rates.  Keywords: Beta distribution, Increasing failure rate, Kumaraswamy distribution, Minimax distribution.  1.0 Introduction According to [4] and [6], “Despite the many alternatives and generalizations, it remains fair to say that the beta distribution provides the premier family of continuous distributions on bounded support (which is taken to be (0, 1))”. Let X denote a non-negative continuous random variable that follows a beta distribution with parameters „a‟ and „b‟, we write;  ,X Beta a b , the probability density function is given by;       11 1baa bf x x xa b   ;  0,1x      (1) Where , 0a b   are shape parameters Besides lot of other attractive properties, the shape of beta densities could be unimodal, uniantimodal, constant, increasing or decreasing depending on the values of parameters „a‟ and „b‟ relative to 1 ; See [1] for details. It is good to note that the cumulative density function of the beta distribution involves an incomplete beta function; this makes the distribution not to be very tractable. [5] introduced the Kumaraswamy distribution and many authors have explored some of the properties of this distribution. Noticeably is the work of [2] who described the distribution as a Minimax distribution sharing many desirable properties with the beta distribution. The Minimax distribution has been 1 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015  described as a viable alternative to the beta distribution and its basic properties reflects that it is more tractable compared to the beta distribution. For a random variable X having Minimax distribution with parameters „a‟ and „b‟, the probability density function (pdf) is given by;    11 1ba af x abx x   ;  0,1x   (2) The corresponding cumulative density function (cdf) is given by;    1 1baF x x     ;  0,1x  (3) Where , 0a b   are shape parameters We refer readers to [7] and [8] for some details on the Minimax distribution. This article seeks to extend the notable work of [3] by exploring a special case of both the beta distribution and the Minimax distribution when we assume one of the shape parameters ' 'a  to equal 1, we also investigate some basic statistical properties of the resulting model. The rest of this article is organized as follows; section two discusses how the new model was derived including derivations of some of its basic properties, section three discusses the estimation of the model parameter using the method of maximum likelihood estimation, followed by a concluding remark.  2.0 The Proposed Model Substituting 1a   in both Equation (1) and (2) gives the pdf of the proposed model to be;    11bf x b x    ;  0,1x  (4) Thus, we easily derive its cdf to give;    1 1bF x x     ;  0,1x  (5) Where 0b   is a shape parameter  2.1 The new model is a valid probability density function To proof that the model in Equation (4) is a valid pdf, we show that integrating Equation (4) with respect to x from 0 to 1 equals 1. That is,  1101 1bb x dx    Let  1u x  ; then, dx du   Therefore,  1 11 10 01b bb x dx b u du       After simple calculations,   1101 1bb x dx   This completes the proof. We also provide graphical representations of the pdf of the model and at various values of parameter ' 'b . For brevity purpose, we present plots for the pdf at ' 1'b  , ' 4 'b   and at ' 20'b   in Figure 1, 2 and 3 respectively.  2 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015            Fig. 1: The pdf of the proposed model when parameter 1b                             Fig. 2: The pdf of the proposed model when parameter 4b       3 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015                         Fig. 3: The pdf of the proposed model when parameter 20b    We observe in Figure 1 that the shape of the model could be “constant” and both plots in Figure 2 and 3 indicate that as the value of „x‟ increases, the curve decreases. Hence, we can also say the shape of the model could be “decreasing”. A plot for the cdf of the model is as shown in Figure 4;  Fig. 4: The cdf of the model when parameter ' 4 'b    2.2 Some properties of the new model In this section, we provide some desirable properties of the new model, beginning with its moments.   2.2.1 Moments The rth moment for a random variable X denoted by  is given by; rr E X         4 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015  Hence, the rth moment for the new model in Equation (4) is given by;   1101brr b x x dx   After simple calculations, we obtain the rth moment as;  1,r bB r b      (6) We can simply re-write Equation (6) as;     1 11rr bb r                  (7) Note that,    1 !k k    where; 0k   is an arbitrary constant. The mean, 1  is easily gotten as; 11Meanb              (8) The variance is given by;      23, 2,Var X bB b bB b           21 2bVar Xb b       (9) With the expression given in Equation (6), other higher-order moments can be obtained.  2.2.2 The Quantile Function and Median The quantile function Q  is an alternative to the pdf as it is a way of prescribing a probability distribution and it can be derived as the inverse of the cdf. With this understanding, we give the expression for the quantile function of the new model as;    11 1 bQ u u          (10) The median of the model can be obtained by substituting 0.5u   into Equation (10). Therefore, we have;  11 1 0.5 bMedian     This can be simplified to give;  11 0.5 bMedian          (11) For simulation purposes, random variables from the new model can be obtained using the expression;  11 1 bX u          (12) where  0,1U Uniform   2.2.3 Reliability Analysis The reliability (survival function) for the model in Equation (4) is derived by;    1S x F x     1 1 1bx          1bS x x    (13) The corresponding plot for the survival function of the model is as shown in Figure 5; 5 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015   Fig. 5: The Survival Function of the model when parameter ' 4 'b    The probability that a system having age ' 'x  units of time will survive up to ' 'x t  units of time for 0, 0x b   is given by; ( )( | )( )S x tS t xS x     1( | )1bbx tS t xx      (14) The corresponding hazard function is thus given by;    1f xh xF x   111 1 1bbb xx        1bh xx   (15) We obtain plots for the hazard function of the model at various parameter values but the plot at 4b   is as shown in Figure 6;    6 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015   Fig. 6: The Hazard Function of the model when 4b   Figure 6 shows that the proposed model has an increasing hazard rate.   3.0 Estimation In an attempt to estimate the shape parameter of the proposed model, we employ the use of the method of the Maximum Likelihood Estimation (MLE). Let 1 2, ,..., nX X X  be a random sample of „n‟ independently and identically distributed random variables each distributed according to the model defined in Equation (4), the likelihood function L is given by;    11 21, ,..., | 1nbniL x x x b b x       111nbniib x    Let  1 2log , ,..., |nl L x x x b     1ln 1 ln 1niil n b b x     Differentiating  with respect to „b‟ and equating the result to zero gives;  1ln 1niil nxb b      Hence;  1ln 1niinxb     1ln 1niinbx   (16)  4.0 Conclusion This article presents a model called One parameter Minimax distribution which was obtained as a special case of both the Beta distribution and the Kumaraswamy distribution. The resulting model is bounded on a (0, 1) support and its shape of the model could be “constant” or “decreasing” (depending on the value of the parameter). Expressions for the rth moment, mean, variance, quantile function, survival function and failure rate are provided. The method     7 Covenant Journal of Physical and Life Sciences (CJPL) Vol. 3 No 1. June, 2015  of maximum likelihood estimation (MLE) was proposed in estimating the only parameter in the model. The model shows an increasing hazard rate, thus, we propose the use of the distribution in modeling situations where risk is low at the beginning but increases with time.  References  Johnson, N. L., Kotz, S., and Balakrishnan, N. “Continuous Univariate Distributions”, Vol. 2, 2nd Edition, New York: Wiley, 1994 Jones M. C. “The Complementary Beta Distribution”, Journal of Statistical Planning and Inference, 104, 329-337, 2002 Jones M. C. Kumaraswamy‟s distribution: “A beta-type distribution with some tractability advantages”. Statistical Methodology, Vol. 6, 70-81, 2009 Kotz, S., and Van Dorp, J. R. “Beyond Beta; Other Continuous Families of Distributions with Bounded Support and Applications”, World Scientific Press, New Jersey, U.S.A., pp: 289, 2004 Kumaraswamy, P. “A generalized probability density function for double-bounded random processes”. Journal of Hydrology, Vol. 46, 79-88, 1980 Nadarajah, S., and Gupta, A. K. “Generalizations and Related Univariate Distributions,” in “Hanbook of Beta Distribution and Its Applications, eds. A.K. Gupta, and S. Nadarajah”, Dekker, New York, U.S.A., pp. 97-163, 2004 Oguntunde, P. E., Babatunde, O. S., Ogunmola, A. O. “Theoretical Analysis of the Kumaraswamy-Inverse Exponential Distribution” International Journal of Statistics and Applications, Vol. 4, No. 2, 113-116, 2014 Oguntunde, P. E, Odetunmibi O. A, Okagbue H. I, Babatunde O. S, Ugwoke P. O. “The Kumaraswamy-Power Distribution: A Generalization of the Power Distribution” International Journal of Mathematical Analysis, Vol. 9, no. 13, 637-645, 2015.                                                               8 

A Note on the Minimax Distribution

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A Note on the Minimax Distribution

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