Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS019) p.4301

Transcription

1 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 0, Dublin (Session CPS RELIABILITY STUDIES OF BIVARIATE LOG-NORMAL DISTRIBUTION Pusha L.Guta Deartment of Mathematics and Statistics Universit of Maine Orono, Maine, U.S.A Ke words and hrases: failure rate, hazard comonents, series and arralel sstems, monotonicit Abstract In this aer, we stud the bivariate log-normal distribution from a reliabilit oint of view. The monotonicit of the hazard rates of the univariate as well as the conditional distributions is discussed. The robabilit distributions, in the case of series and arallel sstems, are derived and the monotonicit of their failure rates is discussed.. INTRODUCTION It has been observed that, in man ractical roblems, the log-normal distribution is widel used to model ositive random variable that exhibits skewness. The log-normal distribution is commonl used in medicine and economics, where the basic rocess under consideration leads to a skewed distribution. Zou et al. (009 comment that the log-normal distribution ma be used to aroximate right skewed data arising in a range of scienti c enquiries, see also Limert et al. (00. In order to test the equalit of means of health care costs in a aired design, Zhou et al. (00 showed that a bivariate log-normal distribution is aroriate to model the outatient costs, six months before and after the Medicaid olic change in the state of Indiana. Hawkins (00 discussed an examle dealing with 56 assa airs of cclosorin from blood samles of organ translant reciients obtained b two di erent methods. In this connection, assuming the bivariate log-normal distribution, Zhou et al. (00 resented ve tests for the equalit of two log-normal means. In this aer, we are interested in studing the class of bivariate log-normal distributions from a reliabilit oint of view. More seci call, we stud the association between the variables and obtain conditions for which this class of distributions is T P ( totall ositive of order or RR ( reverse rule of order. This enables us to stud the deendence roerties of the model. We stud the hazard comonents of the hazard gradient in the sense of Johnson and Kotz (975 and their monotonic structure. In this connection, the conditional distribution of X given Y > is found to be log-skew normal, a class not widel studied in the literature. An association measure (x; de ned b Oakes (989, is investigated for this class of bivariate distributions. Some of the results resented here are general and would be useful in studing the association in other classes of bivariate distributions.

2 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 0, Dublin (Session CPS THE MODEL A random variable (X; Y is said to have bivariate log-normal distribution if (ln X; ln Y has a bivariate normal distribution. It is clear that the marginal and conditional distributions are univariate log-normal. For the bivariate normal distribution, it is well known that the joint densit is T P according as > (<0; see for examle Joe (997. We now state the following result Theorem. Suose the random variable (X; Y has the T P (RR roert. Let h(: be an increasing function. Then (h(x; h(y has the T P (RR roert. Proof. See Shaked ( Bivariate Log-normal Distribution Suose (X; Y has a bivariate log-normal distribution so that (ln X; ln Y has a bivariate normal distribution. It is evident that the conditional distribution of X given Y or of Y given X x is univariate log-normal. Therefore, their reliabilit functions can be easil obtained b using the formulas for log-normal distribution and aroriate values of the arameters. The monotonicit of the failure rate and of the mean residual life function will follow the same attern as of the univariate log-normal distribution. To stud the monotonicit of the failure rate of the conditional distribution of X given Y as a function of ; we emlo the following result due to Shaked (977. Lemma. If f(x; is T P (RR ; the conditional failure rate of X given Y is decreasing(increasing in : Using the above result, we conclude that the failure rate of the conditional distribution of X given Y is decreasing ( increasing in according as > (<0: We now discuss the conditional distribution of X given Y > : Conditional distribution of XjY > Suose (X; Y BV LN( ; ; ; ; : Then the conditional densit of X given Y > is given b

3 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 0, Dublin (Session CPS f XjY > (xjy P (X > xjy Z R f(x; vdvp (Y > xv exf ( [( ln x ( ln x ( ln v +( ln v ]gdv P (Y > x ( ln x [ ( ln [ +((ln x ] ] ( ln : Secial case When 0 and, the above reduces to f XjY > (xjy > f X (x[ ( ln ln x ]P (Y > ; (3. where f X (x is the marginal densit function of a standard log-normal distribution. Thus X given Y > has a log-skew normal distribution. Note that this result is true for standard as well as for non standard bivariate log-normal vectors, see Arnold (009. Hazard comonents and their monotonicit In order to stud the hazard comonents and their monotonicit, it is enough to consider the standard bivariate log-normal model. The hazard rate of the conditional distribution of X given Y > is given b h (x; d dx ln F (x; (3. Z f X (x[ ( f(x; vdvf (x; ln ln x ]F (x; ;. Using equation (3., the distribution of X given Y > is log-skew normal, its hazard rate is of the te U: In a similar manner, we can obtain the conditional densit of Y given X > x and the hazard comonent h (x; : The monotonicit of the conditional hazard of X given Y > as a function of can be determined b emloing the following result due to Shaked (977. Lemma 3. If f(x; is T P (RR ; the conditional failure rate of X given Y > is decreasing( increasing in 3

4 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 0, Dublin (Session CPS As noticed before, the bivariate log-normal distribution is T P (RR according as > (<0: Thus, in the case of bivariate log-normal distribution h (x; is decreasing ( increasing in according as > (<0: Similar statement can be made regarding the other hazard comonent. 4. SERIES AND PARALLEL SYSTEMS In this section, we shall obtain the densit functions of T Min(X; Y and T Max(X; Y : Also we stud the monotonicit of the failure rates of T and T : We know that for an bivariate vector (X; Y, the densit functions of T and T are given b and f T (t f X (tp (Y > tjx t + f Y (tp (X > tjy t (4. f T (t f X (tp (Y < tjx t + f Y (tp (X < tjy t; (4. see Guta and Guta (00. For the bivariate log-normal distribution, using (4., it can be veri ed that f T (t t (ln t [ ( + t (ln t [ ( + ] (4.3 + ]: For the standard bivariate log-normal distribution, it reduces to f T (t r t (ln t[ (ln t ]: (4.4 + Hence T has a log-skew normal distribution. earlier, the failure rate of T is of the te U: Using the result established It can be veri ed that the turning oint of T (t is given b the solution of the equation where ( distribution. ln t + r N ( ln t r 0 N( ln t; (4.5 ( + and r N (: is the failure rate of the standard normal For examle, for ; the maximum value of T (t is achieved at t : Similarl, using (4., it can be veri ed that 4

5 Int. Statistical Inst.: Proc. 58th World Statistical Congress, 0, Dublin (Session CPS f T (t t (ln t [( + t (ln t [( + (4.6 + ]: For the standard bivariate log-normal distribution, the above exression reduces to U: f T (t r t (ln t[(ln t ]: (4.7 + Again, T has a log-skew normal distribution and its failure rate is of the te References [] Arnold, B.C. (009. Flexible univariate and multivariate models based on hidden truncation. Journal of Statistical Planning and Inference, 39, [] Guta, P.L. and Guta, R.C. (00. Failure rate of the minimum and maximum of a multivariate normal distribution. Metrika, 53, [3] Guta, R.C. and Warren, W. (00. Determination of change oint of nonmonotonic failure rates.communications in Statistics. Theor and Methods, 30 (8&9, [4] Hawkins, D.H. (00. Dignostics of conformit of aired quantitative mesurements. Statistics in Medicine,, [5] Joe, H.(997. Multivariate Models and Deendence Concets. Chaman and Hall, New York. [6] Johnson, N.L. and Kotz, S. (975. A vector valued multivariate hazard rate. Journal of Multivariate Analsis 5, [7] Oakes, D. (989. Bivariate survival models induced b fralities. Journal of the American Statistical Association 84, [8] Shaked, M. (977. A famil of concets of deendence for bivariate distributions. Journal of the American Statistical Association 7, [9] Zhou, X.H.,Li, C., Gao, S. and Tierne, W.M. (00. Methods for testing equalit of means of health care costs in a aired design stud. Statistics in Medicine, 0, [0] Zou, G.Y.,Taleban, J. and Huo, C.Y. (009. Con dence interval estimation for log-normal data with alication to health economics. Comutational Statistics and Data Analsis, 53,