Patrick Maué
email: patrick.maue@uni-muenster.de

Dumitru Roman
email: dumitru.roman@sintef.no

The ENVISION Environmental Portal and Services Infrastructure

Keywords: Environmental Services, Environmental Models, Portals, Ontologies

The ENVISION Portal is a Web-enabled infrastructure for the discovery, annotation, and composition of environmental services. It is a tool to create Web sites dedicated to particular domain-specific scenarios such as oil spill drift modeling or landslide risk assessment. The underlying architecture based on pluggable user interface components is briefly discussed, followed by a presentation of the components resulting from the first iteration of the implementation. A walkthrough explains how to create a scenario website and populate it with the user interface components required for one specific scenario. The paper concludes with a discussion of open challenges identified during the implementation.

Introduction

Geospatial workflows such as the processing and analysis of environmental data to support decision making are traditionally realized with Geographic Information Systems (GIS). The benefits of distributing tasks into the Web have been acknowledged by the geospatial community. Since the INSPIRE directive entered into force, the number of geospatial Web services embedded in Spatial Data Infrastructures (SDI) has grown considerably. Standards by the Open Geospatial Consortium 3 (OGC) have been specified to enable seamless integration of these Web service into existing geospatial workflows. The standards specify Web service interfaces, data models, and encodings for feature-and raster-based data, or data coming from in-situ environmental sensors. Even though there has been considerable uptake of these Web services in the geographic information community, they are playing only a small role for environmental modeling infrastructures.

Environmental computer models, which, for example, predict future weather conditions or interpolate temperature observations, are still rarely seen on the Web. Migrating existing models into the Web, and the development of Webbased user interfaces for environmental models, is a challenging task. But the outcome of this process is of large interest not only for the environmental modeling community [START_REF] Granell | Service-oriented applications for environmental models: Reusable geospatial services[END_REF]. Initiatives such as SEIS ("Shared Environmental Information System") and SISE ("Shared Information Space for Europe") call for a better availability of environmental information for the end users, both public authorities and citizens. Making environmental computer models available on the Web is also the vision of the "Model Web" [START_REF] Geller | Looking Forward: Applying an Ecological Model Web to assess impacts of climate change[END_REF]. The Model Web has been defined within GEOSS ("Global Earth Observation System of Systems") [START_REF] Christian | Planning for the Global Earth Observation System of Systems (GEOSS)[END_REF] as the potential solution to enhance the interoperability of environmental computer models. It is a vision of a multidisciplinary network of models, data sources, processes, and sensors, which seamlessly interact through the well-defined Web service interfaces specified by the OGC. The interoperable computer models are supposed to be available to researchers, managers, policy makers, and the general public. Environmental service infrastructures are one first step towards the idea of havings Models as services in the Model Web [START_REF] Roman | Model as a Service (MaaS)[END_REF].

Several open issues contribute to the lack of environmental services on the Web. The existing models are often complex monolithic systems implemented decades ago in programming languages incompatible with modern distributed architectures. The variability of the input parameters is challenging, and the computer model's graphical user interface required to specify these parameters cannot be simply migrated into the Web. The required data sources are often locked away due to licensing issues and the often proprietary formats impair their reusability for other applications. Another open issue is the modeling expert's fear of losing control over the model execution. In a workshop with geologists working with the modeling infrastructure developed in the research project SWING, the experts were in general convinced about the benefits of migrating existing models into the Web [START_REF] Urvois | D1.3 -Experience Report[END_REF]. But they also mentioned the importance of communicating how the model has been executed, e.g. by not also only delivering the resulting maps but also the results of intermediate steps and the input parameter and input data.

The ENVISION ("Environmental Services Infrastructure with ontologies") project 4 started early 2010 as follow-up project to SWING [START_REF] Andrei | SWING : A Geospatial Semantic Web Service Environment[END_REF]. The research aims to enable the migration of computer models into the Web through the conceptual specification and implementation of an environmental services infrastructure. The results include tools for a semantically enhanced and multilingual discovery of existing environmental services and the adaptive composition of environmental models as services. The Web-enabled creation and publication of scenario Web sites built on these models supports decision making in the environmental domain even for users with little technical skills. The scenario Web sites visualize the results of the environmental models using interactive map viewers, diagrams, and other scenario-dependent components. A pluggable user interface based on Portal frameworks ensures a high flexibility for the configuration of these Web sites. This paper introduces the underlying architecture of Portals and the ENVISION user interface components.

The presented research builds on two scenarios to validate the implemented infrastructure for environmental service. Risk assessment for landslide hazards is subject of the first scenario; the focus lies on the integration of real-time data coming from in-situ sensors deployed in the pilot area (the French department Guadeloupe). In the second pilot, the impact of oil spills in the Norwegian Sea on the development of the cod population will be investigated. For both scenarios a set of OGC-compliant Web services coming from various organizations are re-used.

In the ENVISION project, we are following the SEIS principles by enabling end-users to participate in environmental decision making processes through the planned environmental services infrastructure. The provided open source technologies will support the easy creation of new environmental models by coupling existing services to compositions, and to publish these models in domain-specific scenario Web sites which enable both, the general public and the environmental experts, to interact with the models and integrate them into decision-making processes.

This paper gives the reader a first impression about the ENVISION Portal. We start with a short overview of the general architecture, and then walk through the steps required to setup an environmental model and publish it on a scenario Web site. Before we give an outlook in the conclusion, we discuss open challenges which emerged during the first iteration in the development of the platform.

Architecture

The primary target of this research has been the specification and implementation of a Web-based portal which supports users in the discovery, annotation, and composition of new environmental models exposed as Web services. The portal enables the creation of domain-specific Web sites. These Web sites enable end users to interact with the models, preferably on maps augmented with interface components suitable for changing and calibrating certain aspects of the model. This approach requires a pluggable user interface, which makes it easy for the scenario Web site designers to select from a library of components useful for one particular computer model.

The user interface is based on Portlets based on the standard "JSR 286: Java Portlet Specification" [START_REF] Heppner | Java Portlet Specification 2[END_REF]. The Portlets are the individual building blocks of a Portal. The latter is in its broadest sense an entry point to access certain domain-specific information. The purpose of portal software is similar to other Content Management Systems: it provides a framework for Web sites in need for personalization, authentication, and the management of the presentation layer of information systems [START_REF] Heppner | Java Portlet Specification 2[END_REF]. The Portal is provided by the Portal Container, which manages the initialization and rendering of the portlets as well as Inter-Portlet Communication (IPC). Several open source implementations of Portal Containers compliant to JSR 286 exist, with Apache Jetspeed 5 and Liferay 6 being probably the most popular choices. The Portlets are delivered through Portlet Repositories. The portal software linked to these repositories then manages the installation and updating of the deployed portlets. The portal consists of several portal pages, and each page may be composed of multiple Portlets. The Portlets itself communicate with back-end Web services, which perform most of the tasks required in environmental services infrastructures. The Portals are composed of Portlets. The communication between portlets is realized through IPC events as defined in the JSR 286 standard. If, for example, a user selects a sensor in a portlet showing a map, another portlet on the same page automatically updates a displayed chart with the data of this specific sensor. A pluggable user interface relies on loosely coupled, but well defined interfaces. It should be easy to switch between different implementations for certain portlets. Loose coupling also decreases the risk of functional dependencies between the components. One portlet has to remain fully functional (but not necessarily useful) if other portlets are not present on the same page. The map's notification for the timeline to update itself therefore has to go through this interface. This technique can potentially work across pages and even different user sessions, making it for example possible to update maps on other users' browsers by the model designer if, for example, the model has been re-computed due to new input data.

Portlets for the Scenario Web sites

As illustrated in Figure 1, a Scenario Web site comprises Portlets used, for example, for interacting with a map (MapViewer Portlet) or a chart visualizing a time series of observation data (ChartViewer Portlet). Additional portlets, e.g. displaying static content explaining the purpose of the scenario, are shipped with the Portal Container software. This also includes Portlets for community building (e.g. Wikis, Forums, Chats), aggregators (display of RSS feeds, embedding of remote videos, Blogs), and many more.

The MapViewer Portlet (see Figure 4) as central entry point for the end users is built on OpenLayers 7 , an open source javascript framework to visualize geospatial data on a base map. The library comes with built-in support for OGC Web Feature Services (WFS) and OGC Web Mapping Services (WMS, including WMS Time), and rudimentary functionality to interact with an OGC Sensor Observation Service (SOS). It lacks support for the Web Coverage Service (WCS), which means that coverages (in our case in the NetCDF format 8 ) have to be channeled through an OGC WMS. The WMS transforms the coverage into a binary image which can be visualized by the client. In ENVISION, some of the services do not necessarily have an OGC-compliant interface. The composed workflows, for example, are at the moment deployed on standard workflow engines which only expose a Web service interface based on the W3C SOAP and WSDL standards. Hence, the Open Layers client had to be extended to also support these service interfaces.

Other crucial portlets for the scenario Web sites include the uncertaintyenabled ChartViewer Portlet (see also Figure 4) and the Notification Portlet. The former expects the sensor data encoded in the OGC Observation&Measurement [START_REF] Cox | Observations and Measurements -Part 1 -Observation schema[END_REF] format to be augmented with uncertainty information. In this case, we rely on the UncertML standard [START_REF] Williams | UncertML : an XML schema for exchanging uncertainty[END_REF]. In-situ sensors produce measurements within a certain confidence interval. If a gaussian distribution for the measurements can be assumed, the observation would include the mean as well as the upper and lower confidence intervals. The chartviewer renders the mean as expected observation value, and the confidence intervals to illustrate the uncertainty of this value. The notification portlet enables users to subscribe to certain events which can be associated to the models and sensor services visualized in the MapViewer Portlet. The nature of the event is modelled in domain ontologies. Complex Event Processing [START_REF] Luckham | The Power of Events: An Introduction to Complex Event Processing in Distributed Enterprise Systems[END_REF] is applied to extract the events from the data streams and populate the ontologies with event instances. The underlying architecture is quite complex, and explained in more detail in [START_REF] Michels | Semantics for notifying events in the affecting environment[END_REF].

Portlets for the ENVISION Portal

The ENVISION Portal is separated from the scenario Web sites and comes with its own set of portlets required to manage and publish environmental models. The Portal is (for now) composed of three pages for the discovery, annotation, and composition of environmental models. The Resource Management Portlet acts as a bridge between the different pages and manages the user collection of Web services, Ontologies, and Composition Drafts. The user collection is shared across different Portals, the user's email address acts as unique identifier of the collection. The resources are stored in a repository compliant to the Content Repository API for Java [START_REF] Nuescheler | Content Repository for JavaTM Technology API Version 2[END_REF], which supports the storage and retrieval of arbitrary documents in a tree-like structure.

The annotation consists of a set of portlets enabling the user to link existing Web service with the ontologies. The annotation procedure is based on the methodology developed for the SWING project. We distinguish between the ontology querying tool (using the Ontology Query Portlet) and the actual semantic annotation with the visualized ontology. Querying supports natural language queries (expecting the ontologies to be grounded with a rich set of documents) which enables users unaware of the specifics of the ontology to identify the correct concepts. The annotation screen then enables the user to select the entities in the data model (e.g. an attribute of an feature type coming from an OGC WFS) and link them to the discovered ontology concept.

The Discovery Portlet is backed by a standard OGC catalogue (in this case we are using the Geonetworks9 implementation). The Discovery Portlet is a standard catalogue client with support for spatial and keyword-based queries. In the next step, it will be enhanced with semantic query capabilities. The user will be able to browse the ontologies and select the concepts and relations which best reflect the required data. The semantic query is encoded within a standard OGC CSW query. A semantic catalogue adapter is then responsible to forward the standard OGC query constraints to the existing catalogue, and the semantic part to a reasoning engine.

The Composition Editor Portlet allows user for coupling OGC Web services via workflows. The core processing of an environmental model (e.g. the interpolation method) are separated from the Web services delivering the input data (e.g. sensor data streams, feature collections, or coverages). With the composition the input data services are linked to the processing services, the result is a workflow document encoded in the Business Process Modelling Execution Language (BPEL) [START_REF] Evdemon | Web Services Business Process Execution Language Version 2.0[END_REF] which can be deployed on standard workflow engines such as Apache ODE 10 . The workflows are specified using the Busness Process Modelling Notation (BPMN) [START_REF] Allweyer | BPMN -Business Process Modeling Notation[END_REF], and translated to executable BPEL specifications. The BPMN editor is based on the Oryx Editor11 , a popular Web-based open source BPM diagram editor for the Web [START_REF] Decker | Oryx An Open Modeling Platform for the BPM Community[END_REF].

The portlets are all backed up by Web services implementing the processing, execution, and resource management. In particular, the execution of the geospatial workflows can be challenging. The amount of the transported data can be massive, and the processing algorithms can be resource-intensive. A distributed execution infrastructure based on Peer-to-Peer technology is currently implemented in ENVISION to ensure efficient and semantically-aware execution of the workflows [START_REF] Tsalgatidou | Interoperability Among Heterogeneous Services: The Case of Integration of P2P Services with Web Services[END_REF].

Setting up a Scenario Web site

The following walkthrough explains the steps required to compose a new environmental model from existing Web services and to publish it with a scenario Web site. The scenario is based on a simple use case developed for the ENVI-SION project. A WPS has been implemented to model the risk of drought in urbanized areas in France. A range of sensors measuring the current groundwater levels serve as input (via an SOS interface) for the classification. The features representing the urbanized areas come from a WFS loaded with the Corine landcover dataset. The activities described here also illustrated in a video 12 .

The targeted user of the ENVISION Portal is the domain expert with knowledge about setting up an environmental model. This does not necessarily include any ICT skills. The Portal is supposed to enable experts with particular expertise in one particular environmental domain (e.g. oil spills) to create a Web site. He might know how to program a model, and has in depth knowledge about the expected outcomes of models and the impact of the input parameter. He most probably has no experience in Web services, OGC standards, data formats, or Ontologies. The end users, which are the target audience of the Scenario Web sites, might have neither experience in environmental models nor ICT. Here, the focus lies on a simple user experience, which enables any user to interact with the models and understand the potential impact of changes in our environment.

Importing a resource

In this example we assume that we know the location of the SOS and the WFS. The Resource Manager Portlet allows for directly importing Web services and Ontologies into the user's personal collection. After selecting the service type and version, the importing procedure will request the service capabilities and store a resource descriptor in the user collection. This service itself cannot be used yet for the annotation and composition. The annotation procedure requires RDF-based service models which can be extended with the references pointing to the ontologies. The composition relies on WSDL service descriptions due to the implementation of the workflows in BPEL. Before moving to the next step, we select the "Translate Resource" action in the resource portlet. This triggers three subsequent actions: translation, cross-linking, and proxying. The first results in a set of WSDL and RDF files for each feature type for the WFS, observed property for the SOS, or process for the WPS. Cross-linking injects references to the other representations in each of these files and the original capabilities. Many of the components in the ENVISION architecture require access to the other representations: the semantic-aware optimization of the BPEL compositions has for example only access to the WSDL files, but requires the RDF-based service models to reason on the services. Proxying will register the original capabilities file to the semantic proxy, which ensures that the semantic enhancements are also available to standard OGC clients [START_REF] Maué | Injecting semantic annotations into ( geospatial ) Web service descriptions[END_REF]. The following Figure 2 shows the Annotation Page, which contains the Resource Manager Portlet as well as the annotation portlets.

Annotating a Web service

In the next step, we semantically annotate the RDF-based service models resulting from the translation. The annotation procedure itself does not put any constraints on the ontologies (besides being encoded in RDF). The ontologies are grounded, which means each concept and relation is automatically enriched with a rich set of textual documents to enable natural language queries. In this example, we are going to annotate the feature type clc00_c112 of the WFS. According to the CORINE documentation, this code represents urbanized areas in Europe. After the ontology has been grounded through the "Prepare Ontology" action in the Resource Manager Portlet, the ontology is loaded into the Annotation Portlet. The user inputs the natural language query "populated areas" to find the appropriate concept in the ontology. Due to the grounding, the Query Portlet is able to infer that the concept UrbanAreas matches the user query. The user selects the most relevant result, which results in the next step in an annotation linking the feature type clc00_c112 to the domain concept UrbanAreas through a rule injected into the service description.

Discovering a Web service

A WFS and an SOS have been imported in the personal collection, but a Web service capable of processing and reclassifying the input features according the real-time observations is still missing and unknown to the user. The discovery tool in the ENVISION portal allows for searching for Web services registered to an OGC-compliant catalogue. The catalogue itself might be configured to federate queries to other catalogues, making it possible to cover a potentially very large set of relevant Web services. Discovery of geospatial data traditionally requires the specification of standard keyword based queries as well as the spatiotemporal constraints (e.g., a set of geographic coordinates representing the region the data should cover, and the time interval the data should be valid for). In the case of a processing service only the keyword-based query is useful. The user enters the search term "drought", which results in a list of Web services which have been computed to be relevant for this query. He selects the WPS doing the classification, and adds it to the personal collection visualized in the resource portlet.

Composing an environmental model

With the composition portlet, the modeler is able to compose services by specifying the control and data flow between the services. The compositions are specified using the BPMN graphical notation used for representing processes. The composition portlet wraps the open source Web-based BPMN Oryx editor. It has been extended with support for including OGC services as part of process compositions and a module that generates executable BPEL code that can be deployed and executed in a BPEL engine. In combination with the Resource Manager Portlet, the composition portlet offers an integrated modeling environment where the user selects services from the resource module, imports them as tasks in the composition, specifies the necessary control and data flow in BPMN, and generates executable BPEL code from the BPMN composition. The following Figure 3 provides a screenshot of the resource portlet and the composition portlet. The composition portlet depicts the composition of an SOS service (GetElevation) and an WFS service (GetWaterLevel) together with their data and control flows. Once the composition is finished, it is transformed into an executable BPEL composition which is stored in the user collection. As final step, the user deploys the composition to the runtime infrastructure via the Deployment Service.

Accessing the environmental model

The Scenario Web site has been published and consists of the MapViewer Portlet showing the results of a composition computing the risk of water shortage for certain regions in France. The MapViewer Portlet has been configured to show the regions exceeding a certain risk of drought, as well as the groundwater sensors. The MapViewer Portlet supports basic map interaction modes such as panning and zooming. Besides that the user is able to toggle layers, display a legend, and select the objects presented on the map. Selecting one of the groundwater sensors displays a timeline of the groundwater observations in the ChartViewer Portlet. The OpenLayer-based MapViewer Portlet also supports the animation of spatial data with a temporal dimension (using the time extension of the OGC WMS). The following Figure 4 shows the result of an oil spill drift model within the MapViewer Portlet.

Open Challenges

In the following section we highlight some of the open research challenges which came up during the first iteration of the implementation, and which will be further investigated in the following years.

Interacting with environmental models on the Web Environmental models are complex software systems whose initial core has been implemented decades ago, and are since then gradually improved and extended. The underlying implementation is based on sometimes ancient software architectures realized with programming languages such as Fortran or Cobolt. This makes it challenging to split up these models into individual reusable software components which can then be migrated to Web services. In addition, invoking models requires rich user interaction. In the oil spill scenario, for example, the domain experts need to be able to constantly adapt the input parameters to calibrate the models. To observe the results, sophisticated data anlyzing tools such as computing and visualizing the cross profile of a submerged oil slick are applied. Making Web-enabled rich user interface components which are generic enough to support different models is an issue which requires a thourough analysis of the end user requirements. Standardized Web service interfaces for environmental models The environmental models are transformed either as a whole into Web services or split up into individual and reusable components. These componenets are later coupled again through Web service compositions. In the end, the new Web services have to be made accessible to generic spatial data clients to enable the integration into existing geospatial workflows. Web service standards from the W3C or the OGC (e.g. the Web Processing Service) ensure flexibility regarding the supported operations, but also impair reusability for clients expecting well-defined interfaces. Supporting both -reusability and flexibility -requires a compromise which reflects the best of both worlds. Integration of generic Web services and domain-specific file formats Even though OGC standards have found wide-spread adaption within some communities, they are rarely seen on a global scale. Talking about Web service usually implies the use of WSDL service descriptions and the SOAP standards for encoding the messages. In contrary to OGC services, these Web service don't have predefined semantics for the operations and the parameters. Finding ways to semantically annotate generic Web services with the methodology described above will be further investigated in the next year. But in the environmental community, even the concept of a generic Web service itself is not common. Here, much data is made available as files, often in unique formats. Making this data available to the BPEL workflows requires the implementation of proxy services which translate the Web service requests and return the appropriate files. Design-time mediation for the composition Support for the specification of mappings between data structures is a necessity when one needs to specify data passing in compositions. Being able to map between the output of a task and the input of another task is part of the actual specification of the composition and necessary for the generation of executable code for the composition. Mappings both at the syntactic and semantic layers are needed, as well as techniques to semi-automatically suggest possible mappings to the user. Efficiency of runtime mediation The mappings required for linking Web services can range from simply assignments to complex rules including arithmetic or spatial transformations. In the case of environmental data, the number of data entities which have to be processed during the mediation can significantly slow down the execution process. Caching results and the specification of the mediation rules in low-level languages are currently investigated to ensure performant execution. Inspecting the executed compositions Severe trust issues still impair the uptake of Web services by environmental modeling experts. The Web services are considered to be black boxes, which produce results hard to understand without knowing the internal processes of the model [START_REF] Urvois | D1.3 -Experience Report[END_REF]. The Scenario Web sites will include Portlets allowing for inspecting the provenance of the models, by retracing the individual steps in the workflows and inspecting the intermediate results. Integration of existing ontologies Ontologies come in different flavors. Their applicability for annotating the environmental services depends on the chosen ontology language, the invested effort for its engineering, and the authors' experience in the domain. The technical integration of remote ontologies requires a translation into formats which can be processed by the reasoning engines deployed in ENVISION. Ontology Maintenance Ontologies usually represent one particular perspective on a domain and have been engineered with one particular application in mind. They hardly ever suit the needs of other domain-specific applications.

Integrating existing ontologies relies on techniques to adapt the ontologies for other applications while staying consistent with the original source. Users of the ENVISION Portal will be able to import ontologies into their personal collections. The annotation interface will give them means to dynamically add new concepts or rename existing concepts. Creating new ontologies If no suitable ontologies exist, the users have to be able to simply create new ontologies without having to know how to write the actual ontology code. A simple concept suggestion Portlet has been created in the first year, which supports the user through data mining techniques to come up with new ontologies. In the next step, users will align these concepts to a foundational ontology by answering automatically generated questions.

The new ontologies are published through the Resource Management Portlet, which enables other users to reuse them for other tasks.

This list only includes practical challenges which have to be addressed to move towards the vision of the Model Web. It is obviously far from complete, other research issues such a handling the uncertainty of environmental data or incorporating the real-time aspects of sensor data are also planned to be investigated.

Conclusion

This paper presented the ENVISION Portal, a Web-based infrastructure for the discovery, annotation, and composition of environmental services. We have presented a brief discussion of the underlying architecture, and listed a subset of the user interface components developed in the first iteration of the implementation. The implementation of the Portal is part of the research conducted in the ENVISION project, which is scheduled to end in 2012. The open challenges identified in this paper are part of the project's research agenda.

The targeted user community of the ENVISION Portal are scientists and environmental modeling experts willing to share their models to the public. The Portal supports the publication of Scenario Web sites, which are domainspecific portals making the models accessible to users lacking knowledge about the inner working of environmental computer models. We don't claim that the presented infrastructure will be able to cover all possible environmental models. In fact, probably only very few will be ever made available as Web services. The inherent complexity of model algorithms, the sophisticated requirements for the user interfaces, and open questions regarding trust and licensing rights are and will be a major obstacle for the distribution of models to the public. Nonetheless we believe that the ENVISION Portal is one important step towards the vision of shared environmental information systems.

Fig. 1 .

 1 Fig. 1. High-level overview of architecture

Fig. 2 .

 2 Fig. 2. The Annotation Page with the Resource Portlet

Fig. 3 .

 3 Fig. 3. The Composition Page with the Oryx Workflow Editor

Fig. 4 .

 4 Fig. 4. A Scenario Web site with an Oil Spill Model in the MapViewer.

See http://www.envision-project.eu

See http://portals.apache.org/jetspeed-2

See http://www.liferay.com

See http://openlayers.org/

See http://www.unidata.ucar.edu/software/netcdf/

See http://geonetwork-opensource.org/

See http://ode.apache.org/

See http://bpt.hpi.uni-potsdam.de/Oryx/

Screencasts are available through the project Web site at http://www. envision-project.eu

Acknowledgements

The presented research has been funded by the European project ENVISION (FP7-249120).


Mirko Filetti
email: filetti@tu-cottbus.de

Albrecht Gnauck

A Concept of a Virtual Research Environment for Long-Term Ecological Projects with Free and Open Source Software

Keywords: Long-term ecological research, data management, virtual research environment, software tools

The management of data and data resources created by different research activities are heavily influenced by various research philosophies and sampling strategies. Within long-term environmental research (LTER) projects data on flows of individuals, chemical substances and other biotic and abiotic materials are collected by different project partners and institutions. This leads not only to different data bases, but also to incomparable data sets. Therefore, a virtual research environment (VRE) for research projects concerning environmental management should be worked out. The facilities of data sharing, interactive data collaboration and data storage as well as the communication within a project team by metadata are in the focus of a VRE which have to be optimised by WEB 2.0 and other collaboration tools. From this background the FOSS application "GeoNetwork -Opensource" (GNOS) is aimed to be used as a central component for data management in a VRE.

Introduction

Annually, around ten billion Euros will be invested for information on the public sector by European countries. Approximately 50% of this information consists on spatial data [START_REF]Interministerielle Ausschuss für Geoinformationswesen (IMAGI): Geoinformation und moderner Staat[END_REF]. But much often, wrong data management strategies concerning quantity, quality, actuality and availability of data leads to high redundancies of collected data sets. Therefore, for a good scientific practice the following aspects of data management should be of interest:

1. A data management concept is needed; 2. Primary data must be sorted with the ability to re-find; 3. Data have to be stored in defined standards. [START_REF] Rueppel | An internet-based spatial decision Support system for Environmerntal data[END_REF]. Data security, data privacy, copy rights have to be guaranteed; 5. Open access for the public; 6. All steps of data processing have to be transparent.

According to the Deutsche Forschungsgemeinschaft [START_REF]Sicherung guter wissenschaftlicher Praxis[END_REF] primary research data should be stored for minimal 10 years on solid storages in the institution that creates the data. Within long-term environmental research (LTER) projects long data sets on flows of individuals, chemical substances and other biotic and abiotic materials are collected by different project partners and institutions. Therefore, an IT-based web 2.0 management concept for the whole project is needed to supply the scientific work with collaboration tools inside and giving representative presentations to the public. The use of free and open source software (FOSS) is an attractive alternative to high cost commercial and often proprietary software solutions because of low cost, short timely software terms, independency and security.

From this background the FOSS application "GeoNetwork -Opensource" [START_REF]GeoNetwork Opensource: Homepage -GeoNetwork Opensource[END_REF] is aimed to be used as a central component for data management in a VRE. Based on international standards, GNOS is a data and information management system, which includes interoperability, metadata, data harvesting, and geographical references as well as user groups with different authorisation levels. The development of GNOS has been initiated by the UN in 2001, which has been continued by several partners. In the meanwhile, basic GNOS applications are successfully implemented and modified in many environmental projects of different governmental and non-governmental organisations worldwide.

General working steps of long-term research projects

The data resources within a multidisciplinary research project like diagrams, pictures, photographs, measured data curves, visual observations and others should be available for collaboration and permanent long term access. As a first attempt, data sets have to be stored in repositories which should be identified by metadata. The data sets keep their different formats, but they can be searched and collected with their metadata assets by a search engine. For this reason, a modified use of the GNOS application is necessary to guarantee data sharing facilities.

In this context, some questions have to be answered concerning usability of original data, retrieval by metadata and authorised access by specific policies. The following general working steps should be considered:

1. A general database has to be set up to store the data of subprojects and to record them with metadata. The interoperability is warranted with all external and internal data repositories.

2. Technical processes have to be adapted and implemented to reference and to maintain research data. Linkages with and references to different data systems and repositories outside of the research project considered are intended as well.

3. The storage system has to be designed. This will be happen by implementation of interoperability tools and/or interfaces for retrieval and referencing.

4. Long term storage of data and archiving tools have to be set up within the project under consideration or outside the project to save and to store research data over longer time horizons for further usage.

Thereby, it is not necessary to develop a completely new data storage and retrieval system. Adaptations and modifications of an available GNOS application have to be adapted to the requirements of a specific research project.

Virtual Research Environment

A virtual research environment (VRE) can be defined as a set of web applications and tools, systems and processes interoperating to facilitate or enhance any research activity within and outside of institutional boundaries. The key issue of a VRE is the development and implementation of an information and data sharing concept [START_REF] Rueppel | An internet-based spatial decision Support system for Environmerntal data[END_REF] where data sharing can be done by different media [START_REF] Carusi | Virtual Research Environment Collaborative Landscape Study[END_REF]. Further technological problems and challenges are given by software and hardware aspects. In practice, for a successful use of a VRE it is necessary to have clear ownerships of data, a confirmed research project plan with data policies among the collaborators, clear research objectives and responsibilities, and an adequate personal resource for the IT management. But, the most important point is that VREs need to be more considered as community building projects than as technology projects. VRE"s give benefits scientific disciplines at all levels of research. There is neither an "out of the box-solution" nor a "one size fits all realizations" approach that will meet the demands of all research activities. Research results will come out very quickly by using a VRE environment. Also new research directions will be supported as well. The range of applications of VRE is very broad. For long-term research, sustainability is required with the same long-term commitment as other parts of the infrastructure of the project lifecycle.

Most VRE"s have an international dimension. Therefore it is necessary to attend legal, ethical and other policies and frameworks that govern the sharing of data and other resources. The VRE"s offer benefits not only for the project itself but also for improvements of general co operations and best practice solutions. Arguments for national and international funding schemes of VRE"s are formulated by Carusi and Reimer [START_REF] Carusi | Virtual Research Environment Collaborative Landscape Study[END_REF]. Research funds should be used to organize general networking procedures and interdisciplinary research. They support expensive research infrastructures and the productivity of researchers. It can be expected that an increasing speed of information and communication lead to a faster dissemination of research results, to a better preservation of research outputs, and to a new quality of research outputs.

VRE services

Depending from the project goals within a VRE environment several services are available. The external view on a VRE should be a unique monolithic system with fast access. But, neither a total self-development of a VRE is possible (because of time, knowledge and resource conflicts) nor is a software tool known, that fits all the requirements for such a VRE. The challenge for an IT-developer is to find best practice FOSS-solutions for all services needed and patch them optionally all together to a "stand alone" VRE-application that is according to all (international) standards and to take care for sustainability (support from the FOSS community).

An overview on some services contained in a VRE is presented in table 1. Between definition of services and selection of applications a lot of work is hidden. For testing and evaluating each software part, several extensions, different versions and the complete VRE system including the web server software and hardware several years have to be spent.

Search engine optimization (SEO) ranking in top search engines like Google

In table 2 selected State-of-the-Art of FOSS applications are presented for some core services and their relations to other applications. The CMS Joomla [6] will be recommended as a second core component. It is a very popular CMS with a world-wide developer community and over 7,300 components for all kinds of extensions and hundreds of bridges to third party software.

GeoNetwork Opensource

The issues of data management, data sharing and data storage are getting more and more important on the different research fields. Only software systems like GeoNetwork Opensource [START_REF]GeoNetwork Opensource: Homepage -GeoNetwork Opensource[END_REF] are suitable for an open access onto scientific data within a worldwide net. This software is of low cost, with a worldwide developer scene, independent from commercial dependencies, and follows a unique administrator scheme. The development of GNOS has been initiated by the UN in 2001, which has been continued by several partners as Food and Agriculture Organisation (FAO), UN Office for the Coordination of Humanitarian Affairs (UNOCHA), Consultative Group on International Agricultural Research (CSICGIAR), The UN Environmental Programme (UNEP) and the European Space Agency (ESA). In the meanwhile, basic GNOS applications are successfully implemented and modified in many environmental projects of national and international governmental and non-governmental organisations worldwide. Figure 1 shows the start-up page of a fresh GNOS installation.

Fig. 1. Start-up page of GeoNetwork Opensource with map-viewer and hit list

For the facilities of data management, which should be realised in such a project (collaborative processing, aggregation, sharing, publication and long tern archiving) there is obviously no need to initiate a completely new development of GNOS or to create a new application. But it is necessary to modify the available GNOS application, which it fits to the specific requirements of the different sub-projects and workflows. For this reason, a communication tool should be developed which fulfils the actual data requirements, and connects all long-term data partners within a research project.

Consideration of standards

The implementation of a research project on information and storage systems has to be in congruence with national and international standards. Based on GNOS the some international standards for metadata and harvesting which should be introduced in a project work are presented in table 3: The Directive addresses 34 spatial data themes needed for environmental applications, with key components specified through technical implementing rules. This makes INSPIRE a unique example of a legislative "regional" approach [START_REF]European Commission INSPIRE: Homepage, Inspire Directive[END_REF].

OAI-PMH (Open Archives Initiative Protocol for Metadata Harvesting).

The Open Archives Initiative Protocol for Metadata Harvesting (OAI-PMH) is a lowbarrier mechanism for repository interoperability. Data Providers are repositories that expose structured metadata via OAI-PMH. Service Providers then make OAI-PMH service requests to harvest that metadata. OAI-PMH is a set of six verbs or services that are invoked within HTTP" [START_REF]Open Archives Initiative: Interoperability through Metadata Exchange[END_REF].

Implementation of GNOS

From the statements given above some requirements of an appropriate hardware environment will be derived. Such an environment enables the implementation of the project objectives. GNOS is mainly built on XML/XSLT and offers excellent opportunities for a structured and platform independent programming. Fast processors, large storage devices as well as redundant data provision and back-up services are needed. The system can run on Windows or LINUX including the implementation of an Apache Web-Server, a MySQL database, Tomcat and Java. Generally a fast multicore server is recommended for GeoNetwork. The running database application and the integrated map-server require sufficient RAM and processing power. LINUX or Windows can be used as operational systems, according to other basic conditions. The hard drive capacity should be calculated by the expected data volume. If only the metadata will be stored, no big capacities are needed. But typically the GNOS-Server is the central storage unit even for the big raw data. The GNOS database can be built with Postgress, MySQL or with a GNOS integrated third party database, which is not so powerful. For different web 2.0 components JAVA and Tomcat needs to be installed on the webserver. As an alternative to the provided third party open source map-server it is possible to use the enterprise ESRI ArcIMS (Map-Server).

For system scaling and reliability of the service cloud computing-and virtualisation technologies can be used. Depending on the last mentioned parameters the following system requirements will be recommended:  cloud-server (optional).

 Fast (

Functions

The core function of GeoNetwork is the metadata referenced search for local and distributed spatial data in a repository by keywords and time-and location filters (what, when where?). The raw data for the repertory can be up-/downloaded in different data types and formats (like maps, PDF, Excel, Word, JPG, etc.) by using of a web interface. An interactive map server provides a fast intuitive search for spatial data with the ability to use layer technology an own or third party (web) maps like Google Maps (see figure 2). Own online composed maps from GNOS can be downloaded as PDF directly.

Fig. 2. Search with keyword by location and time

The metadata model is implemented by a template machine with predefined metatemplates according to the international norms and standards. Templates can be new generated or customised to the own requirements. Internal and external thesauri catalogues provides sensitive search functions and suggestions for categorisation while the data input process (controlled vocabulary). A fine granulated user-and group management grants controlled access to each data set. The worldwide exchange of data and the synchronisation of the metadata on distributed (GeoNetwork) server architectures is realised by a standardised harvesting interface. Further core functions of GeoNetwork are given by up-and downloading of data and documents (for instance maps, PDF, Excel-Sheets), by an interactive map-viewer with own maps and layer technology, by map-and layer export procedures as PDF-files, by exchange and synchronization of metadata with standard harvesting technologies, by privacy and access control on each dataset, by an internal/external Thesauri catalogues for keywords, and by international (multilingual) translation procedures.

Long-term data management

Water quality management deals with diverse tasks as river basin management, nature conservation, and pollution control. All of these tasks are covered by experts from different scientific, engineering and social-economic disciplines. The resulting statements for water management are based on data sets with different origins and sampling intervals. The management of freshwater ecosystems requires long-term observations of water quantity and water quality indicators like water flow, DO, BOD, algal biomass, nutrients, water plants and others. In table 4 some groups of data for a sustainable water management are given which have to be combined by GNOS to establish a unified research and management information system. The development of optimal management strategies can only be achieved by using powerful informatic tools like a virtual research environments (VRE), that have a collaborative focus also. The most important point is that VREs need to be more considered as community building projects than as technology projects. To combine data sets from different project partners within a research project is often not only a difficult, but also a complicated task store, to handle and to analyse these data sets. Longterm storage and retrieval of such data is mostly impossible because of the inconsistent spatio-temporal structure of the data sets. The objectives of most research projects don"t aim a data warehouse oriented design to evaluate the data sets or the collection of data in a single database and cause the usage of GeoNetwork Opensource. On the base of GNOS an interoperable metadata database will be implemented to aggregate, store, and share datasets from the different projects. In addition a high flexible content management system like Joomla, Dupral or Typo3 or collaboration tool like BSCW is needed to include several enhancements and a single sign in system like LDAP and Shibboleth to build a monolithic VRE in one piece. The exchange of data with harvesting technologies becomes very popular in the last years. Each VRE might have a standard interface for harvesting like the OAI-PMH (Open Archives Initiative Protocol for Metadata Harvesting). The main challenges and problems for the development and success are not only from technical nature but also effected by anthropogenic factors: space, time, funding, isolation, procrastination, poor motivation, trust, commitment, working style, ownership, data access, difficulty of learning software and technology, lack of appropriate skills and ready access to technical support and extensive training needs, rapid advantage of technology. Research activities in the future can be established in the socioeconomic field, in visualisation of data and structures and network structures with harvesting strategies. New useful fields for VREs from the IT-branch are also cloud computing and virtualisation techniques and exploring the semantic web technologies with ontologies.

  web)server with multicore processor;  > 500 GB Hard drive;  > 4 GB RAM;  Operation System: Windows Server 2003/2008, Linux;  Add on: MySQL, PHP, Tomcat, Java SDK;  ArcIMS (GIS webserver) (optional);

Table 1 .

 1 Services for a Virtual Research Environment (VRE)

	VRE service	Characteristic
	Access management	Single sign in for different applications
	Communication	Web 2.0 elements like messaging, chat, forum, wiki
	Data analysis	Data analysis tools, statistical methods
	Data visualization	Visualization of information and datasets
	Data warehousing	Complex data storage and data analysis
	Decision support	Aggregated data for decision makers
	E-Learning	Platform for students with E-Learning procedures
	Event calendar	Internal and external community events
	Group management	Groups-and rights management, organization for teams
	Map and spatial data	Map-server, case area maps
	Metadata management	Information about data
	Mobile access	Optimized layout of webpages, augmented reality, access
		control
	Monitoring	Real time monitoring of sensor data etc.)
	Project management	Project organization tools like tasks, milestones, workflow,
		reports
	Project website	Flexible content management system (CMS)
	Repository	Data repository and data storage, compression, indexing
	Search engine	Global and local comfortable search engines
	Social web	Facebook, Twitter integration, etc.

Table 2 .

 2 Selected FOSS for VRE Services

	Service	FOSS	Relations
	Project website, community	Joomla, Typo3, Dupral	LDAP, Shibboleth
	Access management	LDAP, Shibolet
	Project collaboration, communication	BSCW, Joomla components	LDAP, Shibboleth
	and workflows
	Repository search with meta data	GNOS, (BExIS, Pangea)	LDAP, Shibboleth
	Documentation and best practice	Mediawiki	Bridge for Joomla
	Monitoring	Joomla component "Art Data"	LDAP, Shibboleth
	Data analysis/Data warehousing	Infobright.org, noSQL, Joomla	Stand alone on
		component "Art Data"	local PC, data
			Bridge for Joomla
	Decision support	Infobright.org, Joomla compo-
		nents,
	E-Learning	Moodle	Bridge for Joomla
	Webserver-software	Linux, Apache, MySQL,
		Tomcat, PHP, XML/XSLT
	Social web	Joomla-Facebook/Twitter	Bridge for Joomla
		integration
	Mobile access	Optimized Homepage	Bridge for Joomla
	SEO	Joomla component

Table 3 :

 3 Considered metadata standards for GeoNetwork Opensource

	Standard	Description

Table 4 .

 4 Data types of different origin for water management

	Origin of data	Types of data
	Hydrology	Morphology, water level, water flow
	Ecology	Biodiversity, biomass, individuals
	Land use	Land cover, patches, farming
	Socio economy	Anthropogenic uses, tourism, industry
	Energy	Energy production,
	Administration	Planning, statistics
	Politics	Environmental law, decision support


Christophe Chesneau
email: christophe.chesneau@unicaen.fr

Hassan S Bakouch
email: hnbakouch@yahoo.com

Tassaddaq Hussain

A new class of probability distributions via cosine and sine functions with applications

Keywords: Cosine-sine transformation, Estimation, Goodness-of-fit statistics, Hazard rate function, Statistical distributions. 2000 MSC: 60E05, 62E15, 62F10

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A complete simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some of goodness-of-fit tests.

Introduction

The statistical literature contains a plethora of (probability) distributions for modeling different real life random phenomenon in several areas such as engineering, actuarial, medical sciences, demography, economics, finance and insurance. Since no particular distribution is appropriate for modeling every phenomenon, the list of new distributions with a high degree of flexibility is growing every year. Some of the recent univariate continuous distributions can be found in [START_REF] Hamedani | On characterizations and infinite divisibility of recently introduced distribution[END_REF][START_REF] Hamedani | Characterizations and Infinite Divisibility of Certain 2016 Univariate Continuous Distributions[END_REF], [START_REF] Hamedani | Characterizations and Infinite Divisibility of Certain 2016-2017 Univariate Continuous Distributions II[END_REF] and [START_REF] Tahir | Compounding of distributions: a survey and new generalized classes[END_REF]. As a matter of fact, the statistical literature has a lack of distributions which are based on trigonometric functions; most of them are based on algebraic functions. Among the distributions using trigonometric distributions, let us mention the von Mises distribution (see [START_REF] Evans | Statistical Distributions[END_REF]"von Mises Distribution" Ch. 41,[189][190][191]), the distributions introduced by [START_REF] Fisher | Statistical Analysis of Directional Data[END_REF] to analyze circular data, the beta-type distributions using some trigonometric functions proposed by [START_REF] Nadarajah | Beta Trigonometric Distribution[END_REF], the circular Cauchy distribution introduced by [START_REF] Kent | Maximum likelihood estimation for the wrapped Cauchy distribution[END_REF] and the sine square distribution explored by [START_REF] Al-Faris | Sine Square Distribution: A New Statistical Model Based on the Sine Function[END_REF]. However, the increased interest of data analysis, and directional data analysis in particular, motivates the development of new approaches. The one explored by [START_REF] Kumar | A New Distribution Using Sine Function-Its Application to Bladder Cancer Patients Data[END_REF] provides a modem alternative. They introduced a new class of distributions obtained by compounding a baseline distribution with the sine function called the SS transformation. For any cumulative distribution function (cdf) F (x), it is defined by G(x) = sin π 2 F (x) . Taking F (x) as the cdf of an exponential distribution, [START_REF] Kumar | A New Distribution Using Sine Function-Its Application to Bladder Cancer Patients Data[END_REF] propose a new distribution to solve a parametric estimation problem inherent to bladder cancer patients data, and it is shown that it has a better fit compared to several well-known distributions.

Based on the motivations above, we propose a generalization of the SS transformation, using both cosine and sine functions, which can give many of new trigonometric distributions, as shall be pointed out. Several tuning parameters increasing the degree of flexibility of the original SS transformation.

We exhibit the associated probability density function (pdf) and the hazard rate function (hrf). A decomposition of the cdf is proposed. By considering two different sets of parameters, we introduce two particular CS transformations, called the CS1 and CS2 transformations. New distributions are derived by considering well-known distributions (uniform, exponential, normal, arcsine. . . ). Some of their mathematical properties are studied. A special focus is given when the baseline distribution is the exponential distribution. For this case, we discuss a parametric estimation via the maximum likelihood estimation method. A simulation study is performed to evaluate the performances of these estimates using different sample sizes and two criteria: the average bias the average mean square error. Then we show the applicability of a specific cosine-sine exponential distribution by considering four data sets. They are fitted to our distribution and other recent distributions based on some wellknown goodness-of-fit statistics (A * 0 , W * 0 , AIC, AICc, HQIC, CAIC). Moreover, it demonstrates a better flexibility, which is illustrated among the compared distributions.

The rest of the paper is organized as follows: Section 2 presents the general form of our cosine-sine transformation. New distributions are derived in Sections 3 and 4 using particular parametrization. In Section 5, the performance of the estimates of the parameters of two types of cosine-sine exponential distributions is investigated using a simulation study. In Section 6, four real data sets are fitted to a specific cosine-sine exponential distribution, with comparison to other recent distributions based on some of goodness-of-fit statistics. The proofs of our main results are given in Appendix.

CS transformation

In the present study, using a cdf F (x), we introduce the following cosine-sine transformation :

G(x) = (α + γ) sin π 2 F (x) α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) . (1)
Remark that the last term in the denominator can be expressed as :

cos π 2 F (x) sin π 2 F (x) = 1 2 sin (πF (x)
). We will call the transformation (1) as the CS transformation for frequently used purposed.

Theorem 1. If α ≥ 0, β ≥ 0, γ ≥ 0, θ ≥ 0, α + γ > 0 and α + β > 0, the function G(x) given by (1) possesses the properties of a cdf.

The proof of Theorem 1 is given in Appendix.

Note that, taking α > 0, β = 0, γ = 0 and θ = 0, G(x) becomes the SS transformation of [START_REF] Kumar | A New Distribution Using Sine Function-Its Application to Bladder Cancer Patients Data[END_REF]. Further specific choices of parameters, yielding new cdfs, will be studied later. Observe that G(x) ∼ π(α+γ) 2(α+β) F (x), when F (x) → 0, showing a first look to the flexibility of the CS distribution. The following results follow from Theorem 1. Let f (x) be a pdf associated to F (x).

• The pdf associated to G(x) is given by

g(x) = π(α + γ) β + α cos π 2 F (x) + θ sin π 2 F (x) 3 f (x) 2 α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) 2 .
Hence g(x) is a weighted pdf of the form g(x) = cw(x)f (x), where w(x) is the weight

w(x) = β+α cos( π 2 F (x))+θ(sin( π 2 F (x))) 3 (α+β cos( π 2 F (x))+γ sin( π 2 F (x))+θ cos( π 2 F (x)) sin( π 2 F (x)))
2 and c the normalization constant:

c = π(α+γ)

2

.

• The hrf associated to G(x) is given by

h(x) = π(α + γ) β + α cos π 2 F (x) + θ sin π 2 F (x) 3 f (x) 2q(x) , where q(x) = α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) × α + β cos π 2 F (x) -α sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) .
From [START_REF] Al-Faris | Sine Square Distribution: A New Statistical Model Based on the Sine Function[END_REF] and existing transformations, one can construct a wide variety of distributions. For instance, from a cdf H(x), one can use the cdf F (x) into (1) using

• the dilation transformation : F (x) = H(αx), α > 0,

• the power transformation introduced by [START_REF] Gupta | Generalized Exponential Distributions[END_REF] :

F (x) = (H(x)) α , α > 0,
• the DUS transformation introduced by [START_REF] Kumar | A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data[END_REF] :

F (x) = 1
e-1 (e H(x) -1), • the quadratic rank transmutation map (QRTM) introduced by [START_REF] Shaw | The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map[END_REF], yielding the transformation :

F (x) = (1 + λ)H(x) -λ(H(x)) 2 , λ ∈ [-1, 1].
All these possibilities give new families of distributions using cosine and sine functions, with a potential interest in probability and statistics.

Proposition 1 below shows that the CS transformation can be expand as a non trivial sum of powers of the function sin π 2 F (x) . Proposition 1. Let us set ν = α + β + γ + θ. We have the following expansion:

G(x) = α + γ ν +∞ k=0 k =0 m=0 m q=0 -m r=0 k- s=0 +∞ t=0 a k, ,m,q,r,s,t sin π 2 F (x) q+r+2t+1 , with a k, ,m,q,r,s,t = k m m q -m r k - s (q + s)/2 t β ν k- θ ν m γ ν -m (-1) q+r+s+t .
In the rest of the study, we focus our attention on two special cases of our CS transformation, both using a particular parametrization on the parameters (α, β, γ, θ).

The CS1 transformation

An interesting special case of our CS transformation is with the parametrization α > 0, β = 0, γ = 0 and θ ≥ 0. Hence the cdf G(x) (1) only depends on the parameters α and θ. Using trigonometric formulas, we have

G(x) = α sin π 2 F (x) α + θ cos π 2 F (x) sin π 2 F (x) = 2α sin π 2 F (x) 2α + θ sin (πF (x)) . (2)
An alternative expression of G(x) is given by using the survival function S(x) defined by S(x) = 1 -F (x) and standard trigonometric formulas:

G(x) = 2α sin π 2 (1 -S(x)) 2α + θ sin (π(1 -S(x))) = 2α cos π 2 S(x) 2α + θ sin (πS(x))
.

Again, note that, taking θ = 0, we obtain the SS transformation of [START_REF] Kumar | A New Distribution Using Sine Function-Its Application to Bladder Cancer Patients Data[END_REF]. We thus have more flexibility in our transformation thanks to the additional parameter θ, opening new perspective in parametric estimation for instance. We will call the transformation (2) as the CS1 transformation for frequently used purposed. Using again trigonometric formulas, the associated pdf is

g(x) = 2πα α cos π 2 F (x) + θ sin π 2 F (x) 3 f (x) (2α + θ sin (πF (x))) 2 . (3)
Hence, the associated hrf is

h(x) = πα α cos π 2 F (x) + θ sin π 2 F (x) 3 f (x) 2q(x) ,
where

q(x) = α + θ cos π 2 F (x) sin π 2 F (x) α -α sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) = 1 4 (2α + θ sin (πF (x))) 2α 1 -sin π 2 F (x) + θ sin (πF (x)) .
Table 1 presents new cdfs characterizing new distributions arising from the CS1 transformation by considering well-known cdfs for F (x).

Distribution cdf F (x) cdf G(x) of the CS1 transformation Uniform(a, b) x -a b -a 1 [a,b] (x) + 1 ]b,+∞[ (x) 2α sin π(x-a) 2(b-a) 2α + θ sin π(x-a) b-a 1 [a,b] (x) + 1 ]b,+∞[ (x) Triangular(a) 1 - (x -a) 2 a 2 1 [0,a] (x) + 1 ]a,+∞[ (x) 2α cos π(x-a) 2 2a 2 2α + θ sin π(x-a) 2 a 2 1 [0,a] (x) + 1 ]a,+∞[ (x) Exponential(λ) (1 -e -x λ )1 [0,+∞[ (x) 2α cos π 2 e -x λ 2α + θ sin πe -x λ 1 [0,+∞[ (x) Rayleigh(σ 2 ) 1 -e -x 2 2σ 2 1 [0,+∞[ (x) 2α cos π 2 e -x 2 2σ 2
2α + θ sin πe

-x 2 2σ 2 1 [0,+∞[ (x) Normal(µ, σ 2 ) Φ(x) = x -∞ 1 √ 2πσ 2 e - (t-µ) 2 2σ 2 dt 2α sin π 2 Φ(x) 2α + θ sin (πΦ(x)) Gumbel(µ, β) e -e µ-x β 2α sin π 2 e -e µ-x β 2α + θ sin πe -e µ-x β Logistic(µ, s) 1 1 + e -x-µ s 2α sin   π 2 1+e - x-µ s   2α + θ sin π 1+e - x-µ s Cauchy(x 0 , a) 1 π arctan x -x 0 a + 1 2 √ 2α sin 1 2 arctan x-x 0 a + cos 1 2 arctan x-x 0 a 2α + θ cos arctan x-x 0 a Arcsine 2 π arcsin( √ x)1 [0,1] (x) + 1 ]1,+∞) (x) α √ x α + θ x(1 -x) 1 [0,1] (x) + 1 ]1,+∞) (x) Pareto(xm, k) 1 - xm x k 1 [xm,+∞[ (x) 2α cos π 2 xm x k 2α + θ sin π xm x k 1 [xm,+∞[ (x)
Table 1: Some cdfs using the CS1 transformation Proposition 2 below presents a decomposition of the expectation of a function of a random variable having the cdf given by (2). Proposition 2. Let X be a random variable having the cdf given by (2) and d(x) be a function on R. Then we have

E(d(X)) = +∞ k=0 k =0 b k, u , where b k, = 2πα θ k (2α + θ) k+2 (k + 1) k (-1) and u = +∞ -∞ α cos π 2 F (x) + θ sin π 2 F (x) 3 (sin (πF (x))) d(x)f (x)dx,
provided that u exist.

The proof of Proposition 2 is given in Appendix. Naturally, E(d(X)) can be useful to determine moments (by taking d(x) = x r , r ≥ 0), characteristic function (by taking d(x) = e itx , t ∈ R). . .

Note that u can be expressed as

u = 1 0 α cos π 2 x + θ sin π 2 x 3 (sin (πx)) d(F -1 (x))dx.
The complexity in the calculus of u mainly depends on the nature of d(x), F (x) and f (x).

Introduction of the CS1 E (α, θ, λ) distribution. Let us now focus our attention on the CS1 transformation in the case where F (x) is the cdf of the exponential distribution with parameter λ > 0. We call this distribution as the CS1 E (α, θ, λ) distribution for frequently used purposed. As noted in Table 1, using trigonometric formulas, the associated cdf is

G(x) = 2α sin π 2 1 -e -x λ 2α + θ sin π 1 -e -x λ 1 [0,+∞) (x) = 2α cos π 2 e -x λ 2α + θ sin πe -x λ 1 [0,+∞) (x). ( 4
)
Note that, taking θ = 0 and λ = 1 ν , we obtain the SS E (ν) distribution introduced by [START_REF] Kumar | A New Distribution Using Sine Function-Its Application to Bladder Cancer Patients Data[END_REF]. Using the same mathematical arguments, the associated pdf and hzf are, respectively, given by

g(x) = 2πα α sin π 2 e -x λ + θ cos π 2 e -x λ 3 e -x λ λ 2α + θ sin πe -x λ 2 1 [0,+∞) (x) (5)
and

h(x) = 2πα α sin π 2 e -x λ + θ cos π 2 e -x λ 3 e -x λ λ 2α + θ sin πe -x λ 2α 1 -cos π 2 e -x λ + θ sin πe -x λ 1 [0,+∞) (x). (6)
To show the flexibility of this distributions, we provide some graphs. The associated cdf, pdf and hrf are presented in Figures 1, 2 and3, where various shapes are observed and showing symmetries and skewness. Applicability of the CS1 E (α, θ, λ) distribution will be explored in Section 6.

The CS2 transformation

Let us now consider another particular case of the CS transformation. Taking α = 0, β > 0, γ > 0 and θ = 0, G(x) becomes the following function :

G(x) = γ sin π 2 F (x) β cos π 2 F (x) + γ sin π 2 F (x) . (7)
An alternative expression of G(x) is given by using the survival function S(x) defined by S(x) = 1 -F (x) and standard trigonometric formulas:

G(x) = γ sin π 2 (1 -S(x)) β cos π 2 (1 -S(x)) + γ sin π 2 (1 -S(x)) = γ cos π 2 S(x) β sin π 2 S(x) + γ cos π 2 S(x)
.

In this case, observe that the associated pdf and hrf have the forms :

g(x) = πγβf (x) 2 β cos π 2 F (x) + γ sin π 2 F (x) 2 (8)
and

h(x) = πγf (x) 2 β cos π 2 F (x) + γ sin π 2 F (x) cos π 2 F (x)
.

Table 2 presents new cdfs characterizing new distributions arising from the CS2 transformation by considering well-known cdfs for F (x).

Distribution cdf F (x) cdf G(x) of the CS2 transformation Uniform(a, b) x -a b -a 1 [a,b] (x) + 1 ]b,+∞[ (x) γ sin π(x-a) 2(b-a) β cos π(x-a) 2(b-a) + γ sin π(x-a) 2(b-a) 1 [a,b] (x) + 1 ]b,+∞[ (x) Triangular(a) 1 - (x -a) 2 a 2 1 [0,a] (x) + 1 ]a,+∞[ (x) γ cos π(x-a) 2 2a 2 β sin π(x-a) 2 2a 2 + γ cos π(x-a) 2 2a 2 1 [0,a] (x) + 1 ]a,+∞[ (x) Exponential(λ) (1 -e -x λ )1 [0,+∞[ (x) γ cos π 2 e -x λ β sin π 2 e -x λ + γ cos π 2 e -x λ 1 [0,+∞[ (x) Rayleigh(σ 2 ) 1 -e -x 2 2σ 2 1 [0,+∞[ (x) γ cos π 2 e -x 2 2σ 2 β sin π 2 e -x 2 2σ 2 + γ cos π 2 e -x 2 2σ 2 1 [0,+∞[ (x) Normal(µ, σ 2 ) Φ(x) = x -∞ 1 √ 2πσ 2 e - (t-µ) 2 2σ 2 dt γ sin π 2 Φ(x) β cos π 2 Φ(x) + γ sin π 2 Φ(x) Gumbel(µ, β) e -e µ-x β γ sin π 2 e -e µ-x β β cos π 2 e -e µ-x β + γ sin π 2 e -e µ-x β Logistic(µ, s) 1 1 + e -x-µ s γ sin   π 2 1+e - x-µ s   β cos   π 2 1+e - x-µ s   + γ sin   π 2 1+e - x-µ s   Cauchy(x 0 , a) 1 π arctan x -x 0 a + 1 2 γ sin 1 2 arctan x-x 0 a + cos 1 2 arctan x-x 0 a (γ -β) sin 1 2 arctan x-x 0 a + (γ + β) cos 1 2 arctan x-x 0 a Arcsine 2 π arcsin( √ x)1 [0,1] (x) + 1 ]1,+∞) (x) γ √ x β √ 1 -x + γ √ x 1 [0,1] (x) + 1 ]1,+∞) (x) Pareto(xm, k) 1 - xm x k 1 [xm,+∞[ (x) γ cos π 2 xm x k β sin π 2 xm x k + γ cos π 2 xm x k 1 [xm,+∞[ (x)
Table 2: Some cdfs using the CS2 transformation Proposition 3 below presents a decomposition of the expectation of a function of a random variable having the cdf given by (2). Proposition 3. Let X be a random variable having the cdf given by (7) and d(x) be a function on R. Then we have

E(d(X)) = +∞ k=0 k =0 m=0 k- s=0 +∞ t=0 c k, ,m,s,t v s,t , where c k, ,m,s,t = πγβ 2(α + β) 2 (k + 1) k m k - s m/2 t β (α + β) (-1) m+s+t α k- (α + β) k- and v s,t = +∞ -∞ sin π 2 F (x) s+2t d(x)f (x)dx, provided that v s,t exist.
The proof of Proposition 3 is given in Appendix. This can be useful to determine moments, characteristic function. . . Note that v s,t can be expressed as v s,t = 1 0 sin π 2 x s+2t d(F -1 (x))dx. The difficulty in the calculus of v s,t mainly depends on the nature of d(x), F (x) and f (x).

Introduction of the CS2 E (β, γ, λ) distribution. Let us now focus our attention on the CS2 transformation in the case where F (x) is the cdf of the exponential distribution with parameter λ > 0 :

F (x) = 1 -e -x λ 1 [0,+∞) (x)
. We call this distribution as the CS2 E (β, γ, λ) distribution for frequently used purposed. As noted in Table 2, using trigonometric formula, we obtain

G(x) = γ sin π 2 1 -e -x λ β cos π 2 1 -e -x λ + γ sin π 2 1 -e -x λ 1 [0,+∞) (x) = γ cos π 2 e -x λ β sin π 2 e -x λ + γ cos π 2 e -x λ 1 [0,+∞) (x). ( 9
)
Similar mathematical arguments yields the associated pdf and hrf :

g(x) = πγβe -x λ 2λ β sin π 2 e -x λ + γ cos π 2 e -x λ 2 1 [0,+∞) (x) (10)
and

h(x) = πγe -x λ 2λ β sin π 2 e -x λ + γ cos π 2 e -x λ sin π 2 e -x λ 1 [0,+∞) (x). ( 11
)
In order to illustrate the nature of the CS2 E (β, γ, λ) distribution, graphs of some cdfs, pdfs and hrfs are presented in Figures 4, 5 and6. They show various shapes of those functions.

Maximum Likelihood Estimation and simulation

Maximum likelihood estimation:

Let X 1 , X 2 , . . . , X n be a random sample from the CS1 E (α, θ, λ) distribution with parameter vector Θ = (α, θ, λ) and x 1 , x 2 , . . . , x n are the observed values, then joint probability function of X 1 , X 2 , . . . , X n as a log-likelihood function can be expressed as

(Θ) = n log 2πα λ + n i=1 log α sin π 2 e -x i λ + θ cos π 2 e -x i λ 3 - 1 λ n i=1 x i -2 n i=1 log 2α + θ sin πe -x i λ .
The associated nonlinear log-likelihood equations ∂ (Θ) ∂Θ = 0 are given by

∂ (Θ) ∂α = n α + n i=1 sin π 2 e -x i λ α sin π 2 e -x i λ + θ cos π 2 e -x i λ 3 -4 n i=1 1 2α + θ sin πe -x i λ = 0, ∂ (Θ) ∂θ = n i=1 cos π 2 e -x i λ 3 α sin π 2 e -x i λ + θ cos π 2 e -x i λ 3 -2 n i=1 sin πe -x i λ 2α + θ sin πe -x i λ = 0 and ∂ (Θ) ∂λ = - n λ + n i=1 πx i e -x i λ α cos π 2 e -x i λ -3θ cos π 2 e -x i λ 2 sin π 2 e -x i λ 2λ 2 α sin π 2 e -x i λ + θ cos π 2 e -x i λ 3 -2 n i=1 πx i e -x i λ cos πe -x i λ λ 2 2α + θ sin πe -x i λ = 0.
By solving these three equations simultaneously, we obtain the maximum likelihood estimators of the parameters.

Simulation study

A general form to generate a random variable X from the CS1 E (α, θ, λ) and CS2 E (β, γ, λ) distributions is to generate the values x from the proposed models with parameters θ, α, λ and β, γ, λ by using the Mathematica 8.0 computational package. A complete simulation analysis of the CS1 and CS2 distributions was carried out by generating 1000 samples for each of triplet (θ, α, λ) and (β, γ, λ) with n = 25, 50, 100. The analysis computes the following values: • Average bias of the simulated estimates:

1

n n i=1 (Θ -Θ),
where Θ = (α, θ, λ) are the MLEs of the CS1 E (α, θ, λ) distribution and Θ = ( β, γ, λ). are the MLEs of the CS2 E (β, γ, λ) distribution, respectively.

• Average mean square error (MSE) of the simulated estimates:

1 n n i=1 (Θ -Θ) 2 .
Table 3 concerns the CS1 E (α, θ, λ) distribution. It shows the average bias and average MSE of the estimates for different values of α, θ, λ. A downward bias for α and λ and upward bias for θ is observed when α, θ and λ less than one, similar results are also observed in MSE of the estimates. However when the true value of parameter λ and θ becomes greater than one and α less than one the downward biases for α and λ is worth mentioning. Moreover, if all the parameters exceeds one a downward bias is only for α and θ and λ upward bias. In addition the biases and MSE decreases as sample size increases.

Similarly Table 4 portrays the average bias and average MSE of the CS2 E (β, γ, λ) parameters estimates. During simulation study, it is noticed that MLE γ fluctuate between 0.0009 to 6 while the β may also vary from 0.25 to 35. However, the λ remains unchanged. Moreover, it is also noted that when parameters are greater than one a downward bias in γ and β is observed and when all parameters values are less than one an upward bias can be seen in all estimates, also bias and MSE becomes smaller as sample size increase. However, for λ > 1 and for γ, β < 1 a downward bias with decreasing attitude in both bias and MSE is observed.

Evaluation tests and real data examples

In statistical literature, a plethora of distributions exist for life testing experiments. Some of them are suitable in increasing/decreasing failure rate scenario, others are appropriate for bathtub and upside down bath tub shapes and some are for both features. In this context, we studied those distributions which are very strong in their respective area, namely transmuted modified inverse Rayleigh (TMIR) defined by [START_REF] Khan | Transmuted Modified Inverse Rayleigh Distribution[END_REF], transmuted inverse Weibull (TIW) introduced by [START_REF] Khan | Characterizations of the transmuted inverse Weibull distribution[END_REF], new modified Weibull (MW) defined by [START_REF] Lai | A modified Weibull distribution[END_REF], generalized linear failure rate (GLFR) defined by [START_REF] Sarhan | Generalized Linear Failure Rate Distribution[END_REF] and single parameter sine transformed distributions proposed by [START_REF] Kumar | A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data[END_REF]. The pdfs of the competing models are expressed in Table 5.

Name Pdf Authors Domain TMIR g(x) = α + 2θ x ( 1 x ) 2 e -α x + θ x 2 1 + λ -2λe -α x + θ x 2
Khan and King (2015) [START_REF] Khan | Transmuted Modified Inverse Rayleigh Distribution[END_REF] x > 0, α, θ > 0, |λ| ≤ 1

TIW g(x) = αθ α x -α-1 e -( θ x ) α 1 + λ -2λe -( θ x ) α Khan et al. (2014) [11] x > 0, α, θ > 0, |λ| ≤ 1 GLFR g(x) = θ(α + λx)(1 -e -(αx+0.5λx 2 ) ) θ-1 e -(αx+0.5λx 2 )
Sarhan and Kundu (2014) [START_REF] Sarhan | Generalized Linear Failure Rate Distribution[END_REF] x > 0, α, θ, λ > 0 MW g(x) = θ(α + λx)x α-1 e λx-θx α e λx Lai et al. ( 2003) [START_REF] Lai | A modified Weibull distribution[END_REF] x > 0, α, θ, λ > 0 ST g(x) = π 2 βe -βx sin π 2 e -βx Kumar et al. (2015) [START_REF] Kumar | A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data[END_REF] x > 0, θ > 0

Table 5: The compared distributions to the CS distribution

Evaluation tests

In order to demonstrate the proposed methodology, we consider four different real-world data sets, representing various failure rate pattern like increasing, decreasing, and bathtub shape, compared via the Akaike Information Criterion (AIC), the Corrected Akaike Information criterion (AICc), the Hannan-Quinn Information criterion (HQIC) and the Consistent Akaike Information criterion (CAIC) which are used to select the best model among several models. The definitions of AIC, AICc, HQIC and CAIC are given below:

AIC = 2k -2 ( Θ), AICc = AIC + 2k(k + 1) n -k -1 , HQIC = -2 ( Θ) + 2k ln(ln(n)), CAIC = -2 ( Θ) + 2kn n -k -1 .
Moreover, perfection of competing models is also tested via the Kolmogrov-Simnorov(K-S), the Anderson-Darling (A * 0 ) and the Cramer Von Misses (W * 0 ) statistics. The mathematical expressions for the statistics above are given below

KS = max i m -z i , z i - i -1 m , A * 0 = 2.25 m 2 + 0.75 m + 1 -m - 1 m m i=1 (2i -1) ln(z i (1 -z m-i+1 )) , W * 0 = m i=1 z i - 2i -1 2m 2 + 1 12m
,

where k denotes the number of parameters, n denotes the number of observations, m denotes the number of classes and z i = cdf (x i ), the x i 's being the ordered observations. It is also worth mentioning that we have not only confine our interest in the information criterion and statistics but also adopted the graphical displays, so that the reader can gain a perspective of the various meanings and associated interpretations. Moreover, in many applications there is a qualitative information about the failure rate shape, which can help in selecting a specified model. A hazard may be considered as a dangerous event that can lead to an emergency or disaster. A hazard analysis can be performed statistically based on the hazard rate (HR), also known as chance function, failure rate, intensity function, or risk rate, among other names. A nice property of the HR is that it allows us to better characterize the behavior of statistical distributions, and to differentiate models with very similar cdfs. For example, the HR may have several different shapes, such as increasing (IHR), constant, decreasing (DHR), bathtub (BT), inverse bathtub (IBT) approaching a non-null constant, or IBT approaching zero. In this regard, a device called the total time on test (TTT) plot is useful see [START_REF] Almalki | Statistical Analysis of Lifetime Data Using New Modified Weibull Distributions[END_REF]. Usually the TTT plot is drawn by plotting T ( i n ) = i r=1 y r:n + (n -i)y i:n n r=1 y r:n against i n , where i = 1, . . . , n and y r:n , r = 1, . . . , n are the order statistics of the sample. A TTT curve may be concave (convex) is related to the IHR (DHR) class. A concave (convex) and then convex (concave) TTT curve is related to a BT (IBT) HR. Finally, a TTT curve expressed by a straight line corresponds to the exponential distribution. The TTT plots for the current data sets are displayed in Figure 10, which reveal increasing hrfs in both cases. Therefore, these plots indicate the appropriateness of the new family to fit these data, since its special models can present increasing, decreasing, bathtub and upside-down bathtub hazard functions.

Figure 7: Theoretical aspects of TTT plots

Real data examples with statistical analyses

In order to check the competency we consider four data sets. Their descriptions and our statistical analyses are given below.

Data I: Generally it is observed that the brake pads of vehicles have a nominal lifetime, which is the number of miles or kilometers driven before the pads are reduced to a specified minimum thickness. To study the lifetime distribution, a manufacturer selected a random sample of 98 vehicles sold over the preceding 12 months to a specific group of dealers. Only cars that still had the initial pads were selected. For each car the brake pad lifetime (x) could have then been observed by following the cars prospectively. Table 6 contains the life times of 98 vehicles given by [START_REF] Lawless | Statistical Models and Methods for Lifetime data 2 nd Edition[END_REF].

Statistical analysis: The TTT plot of this data indicates an increasing failure rate pattern of the failure rate. Moreover, analysis of break pad data shows that the proposed model is the only suitable model from every aspects of data selection including the increasing hazard function. As we can see in Table 7, the proposed model have not only the minimum values of the test statistics and higher p -value but also have least loss of information by showing least AIC, AICc, HQIC and CAIC. CDF plot also confirm this suitability behavior. Data II: This data set contains the times to failure of 48 devices by [START_REF] Almalki | Statistical Analysis of Lifetime Data Using New Modified Weibull Distributions[END_REF] and reference therein. It can be found in Table 6.

Statistical analysis: The data are known to have a bathtub-shaped failure rate as portrayed in TTT plot. In the analysis of the data the proposed model again shows a promising behavior with high p -value and lower Information criterion values which is portrayed in Table 8. Moreover CDF plot also shows that data is adequately modeled by the proposed distribution.

Data III: 21 advanced lung cancer patients, taken from a study discussed by [START_REF] Lawless | Statistical Models and Methods for Lifetime data 2 nd Edition[END_REF], who were randomly assigned the chemotherapy treatments termed as "standard". Survival times t, measured from the start of treatment for each patient, are recorded in Table 6.

Statistical analysis: Clearly the TTT plot of the current data set reveals decreasing hazard function pattern. Although the proposed distribution again shows minimum Log-Likelihood and higher p -value yet Sine Transformed shows AIC and HQIC value a little bit smaller than the proposed distribution which seems to be lager number of parameters plenty for the proposed distribution. But the proposed again achieved the adequate model fitting behavior by showing minimum AICc and CAIC which is portrayed in Table 9. Moreover CDF plot also shows the appropriateness of the proposed model.

Data IV:

The data in Table 6 show the number of cycles to failure for a group of 60 electrical appliances in a life test. The failure times have been ordered for convenience. This data set is reported by [START_REF] Lawless | Statistical Models and Methods for Lifetime data 2 nd Edition[END_REF].

Statistical analysis:

The TTT plot indicates that it follows the bathtub shape failure rate pattern. The analysis of the data is shown in Table 10 which clearly shows that proposed model is the most appropriate model for such data set, which is consolidated by CDF plot too. (some trigonometric formulas have been used to develop u(x)). Since f (x) ≥ 0, sin π 2 F (x) ∈ [0, 1], cos π 2 F (x) ∈ [0, 1], α ≥ 0, β ≥ 0, γ ≥ 0, θ ≥ 0, α + γ > 0 and α + β > 0, we have u(x) ≥ 0 and v(x) > 0, so G (x) ≥ 0. .

• Let us now investigate lim

By putting these equalities together, we end the proof of Proposition 3.

Figure 1 :

 1 Figure 1: Some cdfs G(x) = G(x, α, θ, λ) (4) associated to the distribution CS1 E (α, θ, λ).

Figure 2 :

 2 Figure 2: Some pdfs g(x) = g(x, α, θ, λ) (5) associated to the distribution CS1 E (α, θ, λ).

Figure 3 :

 3 Figure 3: Some hrfs h(x) = h(x, α, θ, λ) (6) associated to the distribution CS1 E (α, θ, λ).

Figure 4 :

 4 Figure 4: Some cdfs G(x) = G(x, β, γ, λ) (9) associated to the distribution CS2 E (β, γ, λ).

Figure 5 :

 5 Figure 5: Some pdfs g(x) = g(x, β, γ, λ) (10) associated to the distribution CS2 E (β, γ, λ).

Figure 6 :

 6 Figure 6: Some hrfs h(x) = h(x, β, γ, λ) (11) associated to the distribution CS2 E (β, γ, λ).

Figure 8 :

 8 Figure 8: TTT plots of the selected data sets.

• 2 F 2 F 2 F 2 F (x) sin π 2 F (x) -sin π 2 F 2 F 2 F 2 F (x) 2 + θ cos π 2 F (x) 2 = (α + γ) π 2 f (x) β + α cos π 2 F (x) + θ sin π 2 F (x) 3 and v(x) = α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) 2

 222222222222222322222 Let us prove that G (x) ≥ 0. We haveG (x) = u(x) v(x)almost everywhere, whereu(x) = (α + γ) π 2 f (x)× cos π (x) α + β cos π (x) + γ sin π (x) + θ cos π (x) -β sin π (x) + γ cos π (x) -θ sin π

sin π 2 F 2 F.. 2 F (x) + θ sin π 2 F (x) 3 f (x) 1 ( 1 - 2 F (x) + θ sin π 2 F-cos π 2 F-sin π 2 Fπ 2 F 2 F 2 F (x) m sin π 2 F. 1 ) 2 F

 22223112222222212 (x) = (α + γ) sin (0) = 0 and, using α+β > 0, lim x→-∞ α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) = α + β cos (0) + γ sin (0) + θ cos (0) sin (0) = α + β > 0, we have lim x→-∞ G(x) = 0. On the other hand, note that lim x→+∞ (α + γ) sin π 2 F (x) = (α + γ) sin π 2 = α + γ and, since α + γ > 0, sin π (x) q+r Using the equation: cos 2 (x) = 1 -sin 2 (x) and the generalized binomial series expansion, we get [H(x)] k = Combining the equalities above, we end the proof of Proposition 1.Proof of Proposition 2. Let g(x) be the pdf given by (3). Then we haveE(d(X)) = +∞ -∞ d(x)g(x)dx. We can write g(x) = 2πα (2α + θ) 2 α cos π H(x)) 2 ,where H(x) = θ 2α+θ (1 -sin (πF (x))). Since H(x) ∈ (0, 1), the geometric series expansion givesg(x) = 2πα (2α + θ) 2 α cos π 1)[H(x)] k .On the other hand, the binomial series expansion gives[H(x)] k = θ k (2α + θ) k k =0 k (-1) (sin (πF (x))) .21By putting these equalities together, we end the proof of Proposition 2.Proof of Proposition 3. Let g(x) be the pdf given by (8). Then we haveE(d(X)) = +∞ -∞ d(x)g(x)dx. We can write g(x) = πγβ 2(α + β) 2 f (x) 1 (1 -H(x)) 2 ,where H(x) = β α+β 1 (x) + α α+β 1 (x) . Since H(x) ∈ (0, 1), the geometric series expansion givesg(x) = πγβ 2(α + β) 2 f (x) +∞ k=0 (k + 1)[H(x)] k .On the other hand, the binomial series expansion gives[H(x)] k = (x) α k- (α + β) k-1 -sin π β) (-1) m+s α k- (α + β) k-cos π (x) sUsing the equation: cos 2 (x) = 1 -sin 2 (x) and the generalized binomial series expansion, we get[H(x)] k = m+s+t α k- (α + β) k-sin π (x)s+2t

Table 3 :

 3 Average bias and MSE values from simulation of the CS1 E (α, θ, λ) distribution

	Parameter	Sample Size Bias(α)	Bias( θ)	Bias( λ) MSE(α)	MSE( θ)	MSE( λ)
		n = 25	-0.2856	17.0311	-0.2198	0.0427	290.9392	0.0496
		n = 50	-0.2641	16.30518	-0.2048	0.0392	266.3227	0.0461
	α = 0.7221, θ = 0.3498, λ = 0.4123	n = 100	-0.2145	16.1199	-0.2032	0.0348	261.0002	0.0449
		n = 25	-0.0017	15.0352	-1.1195	0.0164	246.9911	1.3995
		n = 50	-0.0012	15.0164	-1.1100	0.0153	240.6821	1.3115
	α = 0.5212, θ = 1.2131, λ = 1.5317	n = 100	-0.0011	15.0108	-1.1094	0.0125	238.2420	1.2990
		n = 25	-0.4764	-3.0304	3.8484	0.2329	9.1843	14.8262
		n = 50	-0.4272	-3.0182	3.8487	0.1874	9.1678	14.8010
	α = 1.3247, θ = 3.7904, λ = 2.3683	n = 100	-0.4271	-3.0171	3.8478	0.1873	9.1166	14.7576
	Parameter	Sample Size Bias(γ) Bias( β) Bias( λ) MSE(γ) MSE( β) MSE( λ)
		n = 25	-1.4062	-0.3042	-0.2323	2.1054	0.1719	0.6437
	β = 0.5096, γ = 1.8465, λ = 0.7846	n = 50	-0.9001	-0.1929	-0.2054	0.6439	0.1583	0.3744
		n = 100	-0.2839	-0.1508	-0.1609	0.3578	0.1378	0.3719
		n = 25	0.9139	1.3176	0.0479	3.1477	9.2488	0.0473
	β = 0.5476, γ = 0.5324, λ = 0.5634	n = 50	0.7406	1.2017	0.0299	2.4056	5.2761	0.0194
		n = 100	0.5277	0.6437	0.0079	1.0018	4.0083	0.0091
		n = 25	-1.9584	-2.9835	0.0200	3.9079	8.9075	0.2275
	β = 3.6543, γ = 2.4537, λ = 1.6533	n = 50	-1.9228	-2.9809	0.0152	3.8261	8.8341	0.1233
		n = 100	-1.0185	-2.0011	0.0135	3.0921	8.0086	0.0451

Table 4 :

 4 Average bias and MSE values from simulation of the CS2

E (β, γ, λ) distribution

Table 6 :

 6 Data sets

	Distribution	α	θ	λ	-( Θ)	KS	p -value	A * 0	W * 0	AIC	AICc	HQIC	CAIC
	CS1 E	0.0408	2.2947	14.6839	459.4411	0.0554	0.763	0.5552	0.0565	924.8822	921.1271	928.0189	920.8822
	GLFRD	0.0001	0.9733	0.0004	466.9339	0.1986	0.031	5.6182	1.0648	939.8678	936.1127	943.0045	935.8678
	MW	0.2058	0.0518	0.0255	479.4219	0.1659	0.089	3.8553	0.5709	964.8439	961.0888	967.9806	960.8439
	TMIR	25.4898	2.9973	-0.0004	536.7692	0.4141	0.000	23.6871	4.9882	1079.538	1075.783	1082.675	1075.538
	TIW	2.5011	41.7168	-0.7856	465.9444	0.9465	0.000	68.2524	15.0603	937.8888	934.1337	941.0255	933.8888
	ST	-	0.0085	-	506.0077	0.3231	0.000	14.5046	2.8601	1014.015	1012.056	1015.061	1012.015

Table 7 :

 7 Summary results for data set I

	Distribution	α	θ	λ	-( Θ)	KS	p -value	A * 0	W * 0	AIC	AICc	HQIC	CAIC
	CS1 E	0.6240	3.0357	22.0911	218.6197	0.1351	0.578	0.7528	0.0816	443.2393	439.7393	445.3607	439.2393
	GLFRD	0.0040	0.5231	0.0003	222.1862	0.1486	0.516	0.8760	0.1403	450.3724	446.8724	452.4937	446.3724
	MW	0.2700	0.2002	0.0099	227.0751	0.2069	0.277	1.7361	0.2523	460.1501	456.6501	462.2715	456.1501
	TMIR	1.7012	0.0010	-0.9599	289.3668	0.5344	0.000	22.1076	3.1491	584.7336	581.2336	586.855	580.7336
	TIW	0.3999	30.0910	0.9100	249.0317	0.8681	0.000	34.8866	6.3921	504.0634	500.5634	506.1848	500.0634
	ST	-	0.01255	-	229.2952	0.16024	0.463	1.6232	0.1966	460.5904	458.6737	461.2975	458.5904

Table 8 :

 8 Summary results for data set II

	Distribution	α	θ	λ	-( Θ)	KS	p -value	A * 0	W * 0	AIC	AICc	HQIC	CAIC
	CS1 E	26.3247	25.7904	94.3683	116.8890	0.1103	0.794	0.3030	0.0353	239.778	236.9209	240.4581	235.778
	GLFRD	0.0081	0.8028	3.41×10 -6	117.8188	0.1109	0.792	0.4220	0.0589	241.6376	238.7805	242.3177	237.6376
	MW	0.2099	0.2329	0.0037	123.7618	0.3101	0.161	1.5287	0.2511	253.5236	250.6665	254.2037	249.5236
	TMIR	26.3251	14.7931	0.0004	118.5053	0.2502	0.304	1.5675	0.2849	243.0105	240.1534	243.6906	239.0105
	TIW	0.8849	13.0655	-0.9999	118.0363	0.9139	0.000	18.0276	3.7296	242.0726	239.2154	242.7526	242.7526
	ST	-	0.0054	-	118.4882	0.1721	0.569	0.6439	0.0879	238.9764	237.1669	239.2031	236.9764

Table 9 :

 9 Summary results for data set III

Appendix

Proof of Theorem 1. Let us now investigate the sufficient conditions for G(x) to be a cdf.

0, θ ≥ 0, α + γ > 0 and α + β > 0, we have G(x) ≥ 0. On the other hand, observe that (α + γ) sin π 2 F (x) ≤ α + γ sin π 2 F (x) ≤ α + β cos π 2 F (x) + γ sin π 2 F (x) + θ cos π 2 F (x) sin π 2 F (x) . Hence G(x) ≤ 1.