Some characterization problems associated with the bivariate exponential and geometric distributions

It is highly desirable that any multivariate distribution possessescharacteristic properties that are generalisation in some sense of the corresponding results in the univariate case. Therefore it is of interest to examine whether a multivariate distribution can admit such characterizations. In the exponential context, the question to be answered is, in what meaning— ful way can one extend the unique properties in the univariate case in a bivariate set up? Since the lack of memory property is the best studied and most useful property of the exponential law, our first endeavour in the present thesis, is to suitably extend this property and its equivalent forms so as to characterize the Gumbel's bivariate exponential distribution. Though there are many forms of bivariate exponential distributions, a matching interest has not been shown in developing corresponding discrete versions in the form of bivariate geometric distributions. Accordingly, attempt is also made to introduce the geometric version of the Gumbel distribution and examine several of its characteristic properties. A major area where exponential models are successfully applied being reliability theory, we also look into the role of these bivariate laws in that context. The present thesis is organised into five Chapters

A Study on the Reversed Lack of Memory Property and its Generalizations

In this thesis, the concept of reversed lack of memory property and its generalizations is studied.We we generalize this property which involves operations different than the ”addition”. In particular an associative, binary operator ” * ” is considered. The univariate reversed lack of memory property is generalized using the binary operator and a class of probability distributions which include Type 3 extreme value, power function, reflected Weibull and negative Pareto distributions are characterized (Asha and Rejeesh (2009)). We also define the almost reversed lack of memory property and considered the distributions with reversed periodic hazard rate under the binary operation. Further, we give a bivariate extension of the generalized reversed lack of memory property and characterize a class of bivariate distributions which include the characterized extension (CE) model of Roy (2002a) apart from the bivariate reflected Weibull and power function distributions. We proved the equality of local proportionality of the reversed hazard rate and generalized reversed lack of memory property. Study of uncertainty is a subject of interest common to reliability, survival analysis, actuary, economics, business and many other fields. However, in many realistic situations, uncertainty is not necessarily related to the future but can also refer to the past. Recently, Di Crescenzo and Longobardi (2009) introduced a new measure of information called dynamic cumulative entropy. Dynamic cumulative entropy is suitable to measure information when uncertainty is related to the past, a dual concept of the cumulative residual entropy which relates to uncertainty of the future lifetime of a system. We redefine this measure in the whole real line and study its properties. We also discuss the implications of generalized reversed lack of memory property on dynamic cumulative entropy and past entropy.In this study, we extend the idea of reversed lack of memory property to the discrete set up. Here we investigate the discrete class of distributions characterized by the discrete reversed lack of memory property. The concept is extended to the bivariate case and bivariate distributions characterized by this property are also presented. The implication of this property on discrete reversed hazard rate, mean past life, and discrete past entropy are also investigated.

Tide And River Runoff Interactions In The Cochin Estuary

Hydrodynamic characteristics of an estuary resulting from interaction of tide and river runoff are important since problems regarding flood, salinity intrusion, water quality, ecosystem and sedimentation are ubiquitous. The present study focuses on such hydrodynamic aspects in the Cochin estuary. Most of the estuaries that come under the influence of Indian Summer Monsoon and for which the salinity is never in a steady state at any time of the year are generally shallow and convergent, i.e. the width decreases rapidly from mouth to head. In contrast, Cochin estuary is wider towards the upstream and has no typical river mouth, where the rivers are joining the estuary along the length of its channel .Adding to the complexity it has dual inlets and the tidal range is 1 m which is lower than other Indian estuaries along west coast. These typical physical features lead to its unique hydrodynamic characteristics. Therefore the thesis objectives are: I) to study the influence of river runoff on tidal propagation using observations and a numerical model ii) to study stratification and property distributions in Cochin estuary iii) to understand salinity distributions and flushing characteristics iv) to understand the influence of saltwater barrage on tides and salinity v) To evaluate several classification schemes for the estuary

Identification of models using failure rate and mean residual life of doubly truncated random variables

In this paper, we study the relationship between the failure rate and the mean residual life of doubly truncated random variables. Accordingly, we develop characterizations for exponential, Pareto 11 and beta distributions. Further, we generalize the identities for fire Pearson and the exponential family of distributions given respectively in Nair and Sankaran (1991) and Consul (1995). Applications of these measures in file context of lengthbiased models are also explored

Characterizations of some continuous distributions using partial moments

Inthis paper,we define partial moments for a univariate continuous random variable. A recurrence relationship for the Pearson curve using the partial moments is established. The interrelationship between the partial moments and other reliability measures such as failure rate, mean residual life function are proved. We also prove some characterization theorems using the partial moments in the context of length biased models and equilibrium distributions

Dynamics of some Environmentally Significant Pesticides in a Tropical Waterway - A Toxicological Approach

Industrial pollutants, consisting of heavy metals, petroleum residues, petrochemicals, and a wide spectrum of pesticides, enter the marine environment on a massive scale and pose a very serious threat to all forms of aquatic life. Although, earlier, efforts were directed towards the identification of pollutants and their major sources, because of a growing apprehension about the potential harm that pesticides can inflict upon various aquatic fauna and flora, research on fundamental and applied aspects of pesticides in the aquatic environment has mushroomed to a point where it has become difficult to even keep track of the current advances and developments. The Cochin Estuarine System (CES), adjoining the Greater Cochin area, receives considerable amounts of domestic sewage, urban wastes, agricultural runoff as well as effluent from the industrial units spread all along its shores. Since preliminary investigations revealed that the most prominent of organic pollutants discharged to these estuarine waters were the pesticides, the present study was designed to analyse the temporal and spatial distribution profile of some of the more toxic, persistent pesticides ——— organochlorines such as DDT and their metabolites; HCH-isomers; a cyclodiene compound," Endosulfan and a widely distributed, easily degradable, organophosphorus compound, Malathion, besides investigating their sorptional and toxicological characteristics. Although, there were indications of widespread contamination of various regions of the CBS with DDT, HCH-isomers etc., due to inadequacies of the monitoring programmes and due to a glaring void of baseline data the causative factors could not identified authentically. Therefore, seasonal and spatial distributions of some of the more commonly used pesticides in the CES were monitored systematically, (employing Gas Chromatographic techniques) and the results are analysed.

Characterizations of bivariate models using dynamic Kullback–Leibler discrimination measures

In this paper, the residual Kullback–Leibler discrimination information measure is extended to conditionally specified models. The extension is used to characterize some bivariate distributions. These distributions are also characterized in terms of proportional hazard rate models and weighted distributions. Moreover, we also obtain some bounds for this dynamic discrimination function by using the likelihood ratio order and some preceding results.