An attempt has been made to assess the feasibility of application of biological data to evaluate and monitor water pollution of the river Kapila, near Nanjangud, Karnataka. Two pollution index factors, one at the generic level and another at species level of the Algae, have been computed. Significant correlation between biological and some physico-chemical factors has been established. The theme that algae serve as tools of pollution and that their index scores at the species level is a more reliable parameter for the evaluation of water quality has been established. Â© 1984, Taylor & Francis Group, LLC. All rights reserved

Ramaswamy, S. N.

Somashekar, R. K.

University of Mysore - Digital Repository of Research, Innovation and Scholarship (ePrints@UoM)

59 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   mySCIENCE  VII (1-2), 2012, 59-72    ©University of Mysore Research Article                                                            http://myscience.uni-mysore.ac.in  A NOTE ON SOME RELIABILITY PROPERTIES OF EXTREME VALUE, GENERALIZED PARETO AND TRANSFORMED DISTRIBUTIONS  S. Ravi1* and Ahmad Mirza Aghazadeh Attari Department of Studies in Statistics University of Mysore, Manasagangotri, Mysore 570006, India *E-mail: ravi@statistics.uni‐mysore.ac.in (Received August 2012; Accepted September 2012)  Abstract In this note, we study a few reliability properties of extreme value, generalized Pareto and some transformations of these distributions having positive support. These distributions are used in modelling series and parallel systems when the number of components is large.  Keywords: Reliability Properties, Extreme Value Distributions, Generalized Pareto Distributions. Introduction           Suppose that  denote independent and identically distributed (iid) random variables (rvs) representing life times of components with common distribution function (df)  Suppose further that these life times correspond to a series system and that the number of components tends to be very large. We know that  is the life time of such a system and one is often interested in                                                                60 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   knowing reliability properties of such systems when the number of components becomes larger and larger. Here  denotes minimum and later  denotes maximum. The dual of this situation is when the system is parallel and in this set‐up the system life time is given by  We refer to Barlow and Proschan (1975) wherein Chapter 8 discusses approximate distributions for these life times under suitable normalizations. In the next section, we study reliability properties of the extreme value distributions and some transformations of these defined on positive support. In the third and last section, we recall some definitions and look at some reliability properties for generalized Pareto distributions (gPds) and some transformed distributions of these having positive support. Reliability properties of extreme value distributions (EVDs) and their transformations           If  and   then   and    where  is survival function. Note that  and  denote, respectively, the life times of parallel and series systems of  components whose life times are              If the number of components increases indefinitely, then it is easily seen that the limiting dfs of  and  are degenerate or do not exist. In order to get approximate limiting dfs for these two useful random variables, one can normalize these rvs. If there exist normalizing constants  and  and non‐constant dfs  and  such that   and   then it is known that  and  have to be extreme value limit laws for the maximum and minimum respectively and these give approximations to the 61 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   normalized life times of parallel and series systems as the number of components increase indefinitely. It is well known that  has to be only one of three types of extreme value dfs, viz.,    being a parameter, with respective pdfs,   Here, we say that two dfs  and  are of the same type if  for all  for constants  and . And,  has to be only one of the following three types of dfs:   with respective pdfs, 62 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari               It is easy to see that results for minima can be obtained by transformation from corresponding results for maxima. So, except in a few cases, we do not state results for minimum. The Gumbel law and its transformations           Since the support of the Gumbel law is the entire real line, we study the positive Gumbel law in this subsection. Let  denote the positive Gumbel rv, with df                                The survival function (sf) and the probability density function (pdf) of  respectively are                                   The failure rate or hazard rate (hr)  and cumulative hazard rate (chr)  of  are, respectively, 63 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari                                                   The reversed hazard rate (rhr)  and cumulative reversed hazard rate (crhr)  of  are, respectively,             Now to verify if  is IFR (DFR), it is enough to show that the first derivative of the hazard rate is greater than (less than) zero. We have   and hence conclude that the positive Gumbel law is IFR.           Block and Savits (1998) have shown with proof that every distribution that has a DFR (decreasing failure rate) function on its interval of support also has a DRHR (decreasing reversed hazard rate) function. In particular, the exponential distribution and the Weibull and gamma distributions with shape parameters less than 1 are all DRHR distributions. These authors have also shown that the IFR Weibull, gamma, and log‐normal distributions are also DRHR distributions. We have similar results for positive Gumbel law                          which implies that the IFR positive Gumbel law is also DRHR. 64 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari             Block and Savits (1998) also show, by an example, that not all non‐negative rvs have a DRHR distribution. Using the following relation between  and                        we conclude that if hr is decreasing, then rhr is also decreasing. See Finkelstein (2002) for more details.           The mean residual (remaining) lifetime (MRL) and mean reversed residual (inactivity) lifetime MRRL (MIT) functions of T are respectively given by             The MRL and fr, and MRRL and rhr satisfy the relations,   The Gumbel law for minima           If  has the Gumbel law, then  has the Gumbel law associated with the minima of normalized iid rvs, with df                                 We define the positive Gumbel minima rv  having df 65 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari               Its sf is  and its pdf is  Its hr is given by  which is increasing in  showing that  is IFR. The chr of  is given by     The rhr of  is     which is decreasing in  since      showing that  is DRHR. The crhr of  is given by  Thus the positive Gumbel minima rv  is IFR and DRHR.           The MRL and MRRL (MIT) of  are in order, as follows   which is a decreasing function in  And the MRRL (MIT) of  is             We have following relations between MRL and fr and MRRL and rhr of the positive Gumbel minima distribution:                             66 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   The Fr chet laws           The hr of the Fr chet law for maxima is given by   and its rhr is given by                                      It is trivial that the rhr,  is decreasing as in  and hence the Fr chet law is DRHR. We shall now study the IFR / DFR behaviour of the Fr chet law. We have   using Lemma 5.9 in page 77 of Barlow and Proschan (1975) and the following lemma.   67 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari              And the rhr of the Fr chet law for maxima is given by   so that the Fr chet law for maxima is DRHR always as  is decreasing in   Reliability properties of generalized Pareto distributions (gPds) and their transformations           The gPds are limit laws of linearly normalized conditional excesses over a high threshold as the threshold tends to infinity. These have been used to model exceedances over high thresholds and were first introduced and studied by Balkema and de Haan (1974). It is well known that the extreme value laws are related to the gPds by the relation  where  is an extreme value law and  is the corresponding gPd. 68 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari             Corresponding to the three types of extreme value laws, there are three possible types of gPds, namely,    being a parameter, with respective pdfs             One can define gPds, replacing the extreme value distributions by their counterparts, which are limit laws of normalized minima of iid rvs. For more details about extreme value dfs and gPds, we refer to Resnick (1987), Kotz and Nadarajah (2000), Coles (2001) and Castillo et al. (2004). We next study some reliability properties of gPds and some of their transformations. The gPds and some transformations           Note that the reliability properties of the exponential law which is a gPd is well known. We look at reliability properties of gPd  when  is the Gumbel law for minima, namely,  We have the df, sf and the pdf of this gPd respectively given by 69 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari               The fr of this gPd is given by   and hence  is concave, so that  is PF  and  is IFR.           Also the rhr of  is   which implies that  is decreasing and hence that  is DRHR. On the gPd laws           More generally, if  is a gPd, then its sf is   and its pdf is  so that its fr is given by   70 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari             So, we can conclude that  is IFR if  is IRHR, and conversely,  is DRHR if  is DFR. Similarly, the rhr of  is given by   so that  is DRHR if  DRHR and conversely,  is IRHR if  is IRHR. The gPd law associated with positive Gumbel maxima distribution           Define   with     Then its df, sf and pdf are respectively given by   so that  is concave or  is PF  Thus the fr given by   71 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   is increasing and  is IFR.           The rhr given by   is decreasing in  and hence  is DRHR as   The gPd law associated with Gumbel minima distribution           Defining     with   we have the df, sf, and pdf of  respectively given by   is decreasing in  using Lemma 5.9 in page 77 of Barlow and Proschan (1975) as  is convex.           The rhr   and hence  is DRHR.  72 /  A Note on some Realiability Properties of Extreme Value, Generalized…. S. Ravi and Ahmad Mirza Aghazadeh Attari   The Pareto gPd law           Defining     with     the df, sf, pdf, fr and rhr of the Pareto law  are respectively given by             Hence the Pareto law is DFR and DRHR.           Similar to the above discussions, it is possible to study reliability properties of gPd laws associated with other distributions.  References Block Henry W., Savits Thomas H., Singh Harshinder (1998). The Reversed Hazard Rate Function,  Probability in the Engineering and Informational Sciences, 12, pp. 69-90. Barlow Richard E, Proschan Frank (1975). Statistical Theory of Reliability and Life Testing, Probability Models, Holt, Rinehart and Winston, INC. Castillo Enrique, Hadi Ali S., Balakrishnan and N., Sarabia Jos e Maria (2004). Extreme Value and Related Models with Applications in Engineering and Science, A John Willey and Sons, Inc., Publication. Coles Stuart (2001). An Introduction to Statistical Modelling of Extreme Values,  Springer-Verlag Finkelstein M.S. (2002). On the reversed hazard rate, Reliability Engineering and System Safety, 78, pp. 71-75. Kotz Samuel, Nadarajah Saralees (2000). Extreme Value Distributions:Theory and Applications, Imperial College Press. Resnick Sidney I. (1987). Extreme Values, Regular Variation, and Point Processes,  Springer-Verlag.  

Biological assessment of water pollution: A study of the river Kapila

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Biological assessment of water pollution: A study of the river Kapila

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