Romain Vuillemot
email: romain.vuillemot@gmail.com

Charles Perin
email: charles.perin@ucalgary.ca

Sport Tournament Predictions Using Direct Manipulation

. We present an advanced interface to facilitate users predicting soccer games with simple drag and drops of teams badges on the tournament bracket. We logged and plotted users activity , and performed a visual analysis of those log to identify users' behaviors .

INTRODUCTION

Millions of fans consume information related to sport every day, through media or by watching games. The most enthusiastic fans make informed decisions by betting on teams performance on games or on whole tournaments. Such prediction activity involves an heavy reasoning mental process, that usually sums up a whole body of distributed information together (e. g., statistics on players, individual players performances, teams tactics, etc.) to decide who will win and who will lose. This prediction process can also be subjective by involving human intuition, indivudal preferences on teams or players, or just randomness.

Despite the strong interest for predictions and their relevance to soccer and many other sports, few interfaces exist that support this process. When such interfaces exist, they are text-based and force users to go through a long list of steps, such as predicting each outcome for each team. This process is repetitive and never allows users to predict a big picture first, and then refine it. For instance, one might want to predict who will win the World Cup first, and then detail each games outcomes to support this claim.

Improving the support for predictions is difficult because the mental process behind a prediction is usually heavy as it requires guessing unknown values from previous observations. It is also difficult since the guess has to comply with the constrained structure of a tournament. This structure is usually twofold: first a group phase where all the teams or subsets of teams play against each others either once or twice; and a bracket phase where teams can be eliminated against a specific opponent. Improving prediction will have impact beyond sport, such as to many domains such as weather forecast or economics projections.

This paper aims at improving the predictions creation process by users. For this purpose, we designed, implemented, and evaluated an interface that supports predictions of soccer tournaments. It implements the constraints of the tournament structure and makes it possible for people to predict-and adjust the prediction-for the big picture of a tournament outcomes in a single view, and its details. Our process outline follow the steps below, which are the key contributions of this paper:

1. We defined a list of design criteria to understand current needs from soccer tournaments predictions input interfaces and the state of the art in HCI, including direct manipulation [START_REF] Shneiderman | Direct manipulation: A step beyond programming languages[END_REF].

2. We designed and implemented an advanced visual interface that complies with the design criteria by using direct manipulation principles to let users drag and drop teams towards their final position in a single view.

3. We also designed and implemented a set of novel logs visualization to make sense of the 504, 307 recoreded interaction logs from the 3, 029 visitors in order to identify and discuss interesting behaviors.

Overall, we contribute evidence that sport enthusiastics can predict sport tournaments using an advanced direct manipulation interface that is non-linear. From our analysis of interaction logs, we provide a list of strategies that people employ to perform predictions. These strategies will help inform the design of prediction interfaces. Such technique could be applied to various fields involving predictions such as economics, finance, or weather forecasts that are currently run by automated models that could benefit from more user-generated data (e. g., parameters, customization of models, constraints).

CONTEXT

The context of our work is sport tournaments which are competitions involving multiple competitors, such as sport teams or individuals. We define a prediction in a broad way as the act of determining future, unknown values. We focus on the ones that are generated in a quantitative, user-generated way, without any model or automated assistance.

Sport Tournaments

We consider sport tournaments with a very common configuration: as the combination of a ranking phase and a bracket phase (Figure 2). Usually, the ranking phase precedes the bracket phase (and is called the group stage), during which teams play against each other once or twice. Figure 2 (left) shows such a group stage occurring at the beginning of a tournament. The best-ranked teams after the group phase enter the second phase where teams are assigned an opponent and are eliminated after each round. Figure 2, right shows such a stage that looks like a bracket converging to the winner of the tournament.

Schedules

Tournaments usually have two ways of planning games for the elimination bracket. Either a draw has decided of a schedule of games ahead (e. g., winner of group stage A will meet the second best ranked team of group stage B) or a draw occurs at each round of the bracket. Not some draws are not uniform, as they often account some constraints. For example, there can be country-specific constraints where teams from the same country cannot play against each other.

Tournament Predictions

The general problem of prediction can be formulated as follows: it consists in determining a set of future values. A tournament prediction is the guess of all games outcomes, which indirectly decide of the tournament's winner. This decision in made in two consecutive steps: first the calculation of the group stage results as a ranking, and then the progressive eliminations during the bracket phase. This is where there is a divergence between the current interfaces, and the way users perform predictions. Current interfaces only implement the above structure of predictions, as the sum of a series of games. While users' mental model follows the opposite path: it focuses on the tournament's outcome first, and then focuses on finding individual games results given the outcome.

Prediction Space

A prediction space is the set of all the possible results that may occur in a tournament. The prediction space is usually large as it is the permutation of all possible games and outcomes. However, the more the tournament progresses, the smaller the prediction space is. Right before the final game, the prediction space is of size two, since all the games already happened but the last one, which has two possible outcomes.

DESIGN CRITERIA

To the best of our knowledge, there is no best practice or previous work on input interactions to support user-generated predictions. Therefore, we conducted a pre-study to capture any best practice in the building of interfaces to support tournament predictions. Informed by this study and related work in HCI, we derived a set of design criteria to support users for their predictions process.

Pre-Study

We identified 27 websites dedicated to bet and that support brackets entry in the area of sport online media. 15 websites were dedicated to soccer in particular and 12 to bets in general. We found that bet websites, are sometimes already populated with predictions suggestions (e. g., using bets odds), also allow users to enter their predictions. Figure 2 shows the typical interface for sport brackets (both for predictions and results communication) we observed. Most of the websites used standard HTML widgets as input widgets such as checkboxes, dropdown lists, and text inputs and organized on a bracket layout. To successfully enter their predictions, users have to manually input data 1) for each team by game, and 2) starting with the first round of games to end up with the final (from left to right on figure 2).

It is only looking at non sport-specific websites, such as general-purpose newspapers that we found innovating input interfaces. Bloomberg ( http://www.bloomberg.com/ visual-data/world-cup/) displays a soccer bracket to let users predict the games outcomes of the 2014 World Cup. FiveThirtyEight ( http://fivethirtyeight. com/interactives/world-cup/) also uses brackets for the same event, but as a way to communicate their model's probability of outcome for each game (with no user interaction allowed). Finally, The New York Times article 512 Paths to the White House ( http://nyti.ms/WhmfT7) communicated predictions for the 2012 U.S. elections by letting users to interact with the prediction space (i. e. the 512 possible configurations of results) which can be filtered by various scenarios depending on state-level elections results.

Design Criteria

From both the previous analysis of current systems, and our understanding of sport prediction in news articles (written in text), we derived the following set of design criteria for a prediction interface that support current needs:

C1 Partial: Allow partial predictions, especially the ones about the favorite or best teams (e. g., winner, finalist). C2 Levels of details: Allow to fill in the scores, or the number of goals scored, or only the outcomes of games. C3 Non-linear: Let the user fill in the prediction from multiple entry points (e. g., semi-final, quarter-finals), not necessarily starting with the first games of the tournament. C4 Reversible: Any action or the whole prediction can be reverted to its previous or initial state in case of mistakes or change of mind [START_REF] Shneiderman | Direct manipulation: A step beyond programming languages[END_REF]. C5 Structure: Show the structure, such as the rules of the tournament, to let the user know the connection and dependencies between each round. C6 Suggested interactions: Such as visual cues to tell the user what and how he can interact with graphical elements on the screen [START_REF] Boy | Suggested interactivity: Seeking perceived affordances for information visualization[END_REF] (especially if widgets are not used any longer). C7 Familiarity: Keep many aspects of the domain such as team badges and bracket layout to reduce the learning phase.

From our pre-study, we observed that C4 and C5 are usually implemented in prediction interfaces. The familiarity criteria C7 is also well implemented, mostly by using teams badges or logos (which are small glyphes), along with teams names as text.

DIRECT MANIPULATION FOR PREDICTIONS

Informed by the previous design criteria, we introduce an interface to better assist sport enthusiastics in making sport tournament predictions. We focused on making it is easy to discover and learn, to let users focus on the mental process of predicting rather than being distracted by the input interface.

The interface provides an overview (C1) of the prediction space (all possible results for teams) which makes visible the structure of the tournament (C5) and is used as a suggested interaction (C6) to inform users on what path a team can follow in the competition. Leveraging this overview, users can start with the winning team as well as with the semi-finalist or teams quickly eliminated, i. e. can perform a non-linear bracket filling (C3).

The technique in action is illustrated on figure 1 and figure 4.

The main interaction is illustrated below on figure 3 and consists of three steps: Step 1: The user starts dragging a team by clicking and moving the mouse over the teams' badge.

Step 2: The user keeps dragging the team badge and a visual cue (a line) shows to which stage of the tournament the team can be dropped. The visual cue is green if the team can be dropped or is red otherwise.

Step 3: In order to make a prediction, the user releases the mouse button to stop dragging the team, which snaps it to the closest game it can be attached to.

This series of step constitute an extension of the well-known DRAG-AND-DROP that uses direct manipulation [START_REF] Shneiderman | Direct manipulation: A step beyond programming languages[END_REF] of elements (called objects of interest) of the interface (in our case, soccer teams). We call this extension DRAG-AND-SNAP following [START_REF] Hudson | Adaptive semantic snaping—a technique for semantic feedback at the lexical level[END_REF] using teams possible paths as snapping constraints for teams, and visual cues for users to inform where teams can be dropped.

Related Direct Manipulation Techniques

This interaction relates to a body of similar techniques beyond [START_REF] Hudson | Adaptive semantic snaping—a technique for semantic feedback at the lexical level[END_REF] that aim at assisting users in space or time navigation when dragging elements.

DRAG-AND-POP [START_REF] Baudisch | Drag-and-pop and drag-and-pick: Techniques for accessing remote screen content on touch-and pen-operated systems[END_REF] moves potential target icons towards the user's current cursor location, thereby allowing the user to interact with these icons using comparably small hand movements. DRAG-AND-PICK [START_REF] Baudisch | Drag-and-pop and drag-and-pick: Techniques for accessing remote screen content on touch-and pen-operated systems[END_REF] makes all icons in the direction of the mouse motion come to the cursor. DRAG-AND-DRAW from The Upshot's article You Draw It: How Family Income Predicts Children's College Chances (http: //nyti.ms/1BqOX3h) lets users freely draw a line on a chart on the prediction of how family income predicts children's college chances. The horizontal axis is the parent's income percentile (from poorest to richest) and the vertical one the percent of children who attend college. DRAG-AND-UPDATE [START_REF] Kondo | Exploring time-varying information visualizations by direct manipulation[END_REF][START_REF] Vuillemot | Investigating the direct manipulation of ranking tables for time navigation[END_REF] lets users update data graphics by dragging items (e. g., countries) along their path over time. DWELL-AND-SPRING [START_REF] Appert | Dwell-and-spring: undo for direct manipulation[END_REF] uses the metaphor of springs to enable users to undo direct manipulations.

All these techniques visual feedback that either connect elements or show additional information. The DRAG-AND-SNAP technique relates the most to DRAG-AND-POP as it connects the current team to potential placeholders for the prediction. In addition, it adds an extra layer of information during the dragging phase by showing the full prediction space.

Proxy Object of Interest

When dragging an object, the user's mouse does not necessarily follow the dragged object (e. g., if the object can only move within a perimeter). From previous work, it is common to use proxy elements as a way of duplicating the object of interest. For example, DRAG-AND-POP and DRAG-AND-PICK duplicate icons to suggest them as target destination. We used a similar design in our technique in order to ensure that the team badge always follows the mouse (as a proxy) while the original badge remains on the bracket. This is visible on figure 1 where the flags are duplicated: one strictly follows the trajectory, the other one strictly follows the mouse point, and both are connected with a dotted line to show they are the same entity.

Trajectories Design

Our interface shows all the possible predictions of the tournament in the background to suggest paths to follow during drag and drops. Using paths to connect elements has already been suggested to show previous interactions [START_REF] Baudisch | Explaining transitions in the user interface[END_REF], as well as forthcoming ones. DRAG-AND-POP [START_REF] Baudisch | Drag-and-pop and drag-and-pick: Techniques for accessing remote screen content on touch-and pen-operated systems[END_REF] uses an elastic rubber band to help users understand what it connects and a visual cue to convey how far away the target is. This is similar to [START_REF] Baudisch | Explaining transitions in the user interface[END_REF], where the band also connects items using various path styles to simulate motions (e. g., motion blur or speed lines). All those designs are not suited to our case with thousands of trajectories, often packed on a small part of the screen. Thus, we use a visual design similar to a chess board's pieces trajectories as in Thinking Machine [START_REF] Wattenberg | Thinking machine[END_REF]. Each path is an arc with a jitter that makes it visually unique. This mostly aims at preventing clutter [START_REF] Wilkinson | The grammar of graphics[END_REF], but it also conveys the dynamics of pieces trajectories.

FIRST PROTOTYPE

We created a first interface implementing DRAG-AND-SNAP for the 2014 FIFA World Cup tournament (Figure 1, left). We released it one week before the start of the first game (June 12 th , 2014) to let enthusiastics predict the outcome of the tournament. The purpose of this release was to validate the design, detect any usability flaw and collect qualitative feedback for further improvement. The prototype and its source code are available at https://github.com/romsson/ worldcup14-interactive-bracket.

The prototype follows the above description of the technique and some domain specific designs (C7). The teams participating to the tournament have an initial position on the leftmost part of the screen, before they enter the group stage. The user can do a prediction by drag and dropping a team from the left to the right until it reaches a game (placeholder). A visual hint indicates the closest trajectory to the dragged team (and not the closest game); and if the users stops the drag then the team will be assigned to the game at the end of this path.

The interface also contains action buttons (See figure 4 for their location on the screen), including a reset button (C4) to start over and an auto-complete button to ease the complete filling. A bracket can be completed partially (C1) and starting at different levels (C3). As a complement to the background showing the prediction space, we displayed animated GIFs [START_REF] Perin | À table!: Improving temporal navigation in soccer ranking tables[END_REF] that explain the drag and snap interaction (C6). Finally, we used team badges and the tournament logo for familiarity (C7).

This interface only allows to predict outcomes of group stages once completed, and not the outcome games in those group stages. Thus, we do not offer the lowest level of details possible (C2) which is a trade-off to accept to keep the interface simple.

Feedback

We released the interface and advertised on forums dedicated to soccer and social media with appropriate hashtags. During the week before the world cup began, 2, 932 unique users visited the interface, with an average session duration of 38 seconds. We observed 141 Tweets, 56 Facebook shares and 11 Google+ shares. We collected qualitative feedback in an unstructured manner. Overall the users were enthusiastic, but noticed the following issues:

• They found that snapping by closest trajectory was not intuitive and difficult to use, especially when there were multiple trajectories at the initial position, making it hard to select a specific one.

• They wanted to make predictions like playing a chess game where teams are progressively moved game by game, rather than having to set the final position.

• They needed to reset individual teams rather than the whole interface at once.

Focus

SECOND PROTOTYPE

Based on the qualitative feedback we received for the first prototype, we designed and released an improved interface (Figure 4) for the 2015-2016 UEFA Champions League available at https://github.com/romsson/ ucl16-predictions.

In this version, we improved the DRAG-AND-SNAP so that teams snap to the closest valid game (placeholder) instead of the closest prediction trajectory. We also added the possibility to reset individual teams (C4) by adding a small close button on each badge to reset it to its initial position. We also changed the timing of animations and added question marks within empty placeholders and the use of modals to explain features. Those changes aim at making the technique ecologically valid (C7), e. g., as a standard website. Finally, we adapted the prediction space to the configuration of the UEFA Champion's League which slightly differs from the World Cup tournament structure: after the group stage, the games for every round are decided by draw, which increases the number of prediction paths for each team.

TECHNICAL NOTES

We implemented the two prototypes using the JavaScript toolkit dragit [START_REF] Vuillemot | Investigating the direct manipulation of ranking tables for time navigation[END_REF] which uses D3 to handle mouse events. We used D3 for the representation of the trajectories and the snapping guides using SVG. Users can interact with the prototypes with simple mouse clicks and mouse moves, and both prototypes run in any recent web browser. As performance is key with direct manipulation interfaces, we generated an image containing all the trajectories and displayed it as background; we only highlighted user predictions using a SVG overlay (drawing each trajectory lines using SVG would have greatly impaired performance).

One last technical decision was to make the prototypes stateful, which means that after the user interacts with the system, his current prediction can be recorded and retrieved wit the page's URL. This makes it possible to share a prediction via a link to the page, which can then be used to complete or modify the prediction, or to be used as a starting point by other people. In other words, this featured immediately allowed asynchronous collaborations by users around predictions.

EVALUATION METHODOLOGY

As our interfaces have been released in-the-wild, users interacted in a natural and confortable environment similar as when using a regular website; thus they behaved the way they would have normally, rather than being distracted or feeling "tested" in a formal experimental setting. Moreover, the task of prediction cannot be assessed in terms of regular metrics such as time of error. First, there is no baseline for correct answers since the results of the correct prediction (in this case the UEFA Champions League) were unknown before May 2016 (six months before this paper was submitted). Also, even if this baseline existed it is perfectly normal to be wrong. Finally, it does not make sense to make predictions for an event that already happened. As a result, we focus on understanding how sport enthusiasts predicted rather than on assessing whether or not they make more accurate predictions using the interface. The only valid baseline would be to ask users in advance what their predictions would have been and if the interface was helpful. However, we claim that a user's prediction can sometimes be progressively formed as he or she is using the interface, and thus cannot be known in advance.

To capture distant user activity, we used logging, a nonintrusing mechanism to capture in-the-wild users activity. This approach has been shown efficient to learn about users behaviors [START_REF] Brown | Finding waldo: Learning about users from their interactions[END_REF]. Enthusiasts can interact with the system without any interruption or distraction related to the experiment. We instrumented the technique with a server-side mechanism to record key interactions from each visitor. Table 1 shows the list of events we recorded. We did not record every mouse moves as it would lead to too much data and would probably capture noise over signal.

VISUAL ANALYSIS OF LOGS

We designed and implemented a set of visualization tools to explore user logs. The goal of those tools is to assist us in better translating user interactions (e.g., click, drag) into complex behaviors, and eventually identify recurring patterns or any flaws in the interaction.

Step One: Plotting all Logs

The first step was to get a big picture of the 519, 129 interaction logs we recorded from 4, 739 visitors. We cleaned up the logs by removing sessions which duration was either under 1 second or over 45 minutes, and those with less than 5 interactions. We ended up with 3, 029 unique visitors and 504, 307 interactions. Figure 5 provides an overview of all visitors by showing each of them as a circle. The size of a circle encodes the number of interactions of the visitor (max 4, 354 interactions). The horizontal axis is the size of the bracket (32 is a full bracket), and the vertical axis is the time spent on the website (in second with a logarithmic scale). We added a low opacity to circles to better show the distribution of dense areas by reducing overplotting [START_REF] Wilkinson | The grammar of graphics[END_REF]. Step Two: Grouping Visitors

We visually identified three main groups (G0, G2, G3) of visitors (as vertical stripes from left to right in Figure 5):

• G0 contains 1, 079 visitors (36% of total visitors) who did not fill the bracket (the vertical line of circles with zero prediction in Figure 5). Those visitors left the interface unchanged with no interaction and with an empty bracket.

• G2 contains 287 visitors (9% of total visitors) who performed interactions and partially completed the bracket (the circles with at least one prediction but who did not complete the bracket in Figure 5).

• G3 contains 1, 653 visitors (55% of total visitors) who completed the full bracket (the vertical line of circles with 32 predictions in Figure 5).

We focused our analysis on G3. This group does not necessarily contain users who completed the prediction themselves. Indeed, since the bracket can be shared, lots of visitors may have started their prediction with an already completed bracket. We found that in G3, 11.8% of the visitors (198 visitors) started from scratch and we call this group G3a. We identified a sub-group of G3a called G3b of 56 visitors (3% of G3) who additionally to completing the prediction, also filled a questionnaire that popped up once the prediction was complete. Among many demographics information, the questionnaire also contained a question on the user's favorite team.

Step Three: Plotting Sequences To further investigate G3a and G3b we plotted the sequences of interactions of G3a as an overlay on the interface similar to a heatmap (see Figure 6). The result is that user behaviors visually match the prediction space. There are a few scribble outside the prediction space (e. g., over the buttons) but most of the interactions remain in the center. Overall, the flow of dragging remains horizontal or leaning towards the center of the interface. Few placeholders seem to not have been the destination of drags (such as the bottom ones for round of 16 and teams in lower group stages F, G, H). An interesting result is that the placeholder themselves are not crossed: users tend to release the teams a bit before reaching the target placeholder. We tested real-time replay as alternatives of showing the sequential nature of interactions (which is temporal), but those did not provide satisfying results. Replays are difficult to follow (even with a fast-forward to reduce their duration) and since we did not record mouse drag, it does not show smooth transitions between recorded events.

Step Four: Sequences Abstraction

We decided to abstract sequences of interactions and to lay them out temporally. Figure 7 shows user interactions from G3b as lines. The horizontal axis is time (in seconds) and the vertical axis is the level of completion of the bracket (from 0 to 32). This makes it possible to visualize the complete sessions and all interactions in each session. We observed the following notable patterns:

The first one is a LADDER pattern which shows the cascading behavior of all similar interactions (in that case, moving teams to placeholders). Those interactions are regular and with no interruption. Indeed, interruptions would have been visible as horizontal lines where the user would not ave performed any interaction we record for a significant time (e. g., he might read or drink a coffee). This confirms that the interface helps users focus on their task and that users sometimes do not need any external knowledge to complete it.

The second pattern is a RE-FORMULATE pattern, which occured when a single team had been dragged once, and then dragged again. It is interesting as it shows users tend to rearrange a current prediction along the way. This confirms our early feedback from the first prototype where users required a step by step prediction process, rather than one that were too rigid once predictions were made.

The last pattern is an UNDOS pattern. This pattern is another type of cascade but appearing when the user de-selects some teams to get back to a previous state of the prediction. This pattern seems to occur when the prediction becomes moderately filled. This confirms both the usability and the need for being able to un-select individual teams that we added in the second iteration of the technique.

Those early findings pave the way for richer visualizations and interactions to dig into the wealth of log data and support them with data-driven evidence. Still, visual inspection of logs remains powerful as human behaviors are complex and difficult -if not impossible -to automatically query and retrieve such as using SQL.

QUALITATIVE RESULTS

We received 65 responses to our questionnaire. We removed incomplete submissions and kept the 56 that were completed from G3b (1 female, average age 22). Comments from the questionnaire were very supportive, e.g., "I like the site and its good to make a prediction.", and "Awesome!!! Do this every year!. We also collected some more negative comments, e.g., I really didn't find it useful. Am I missing something? These negative comments can be explained by the fact that the interface would probably require more onboarding and interaction discovery, beyond the GIFs and tutorial we provided. Such technique probably polarized users: either people got it, or they did not know how to start and got frustrated.

Looking at online forums where the technique had been advertised, we noted some domain-specific discussions and debates that were triggered by the tool. Posts sometimes including a link to a prediction that the person made, e.g., "Just tried it out [LINK TO PREDICTION], and I feel if we get to win against Valencia [...]". Some other posts did not contain a specific link but were based on using the prediction interface, e.g., "I think you would be foolish to bet against PSV winning the whole thing.", "I think we'll get 1st in the group, and with a good draw I think we can get to the quarters, and then anything can happen, I think abdennour will play a huge part in any success", "I pressed auto complete and roma wins the champions league against Chelsea in the finals."

DISCUSSION AND CONCLUSION

The feedback we collected from sport enthusiasts who used the interface validates the design of our DRAG-AND-SNAP-based interface for making tournament predictions. We observed fully completed predictions from scratch and their sharing on forums. Based on both quantitative and qualitative data, we found that sport enthusiasts can effectively use a direct manipulation interface to perform predictions and share insights.

Among the data we collected, we only analyzed successful predictions: we have discarded the unfinished ones. Exploring incomplete predictions data in more details would inform us on design issues that should be addressed to increase the rate of predictions completion.

In the future, we plan to investigate in particular the reasons for visitors drop, as these may be diverse and difficult to hypothesize. We also plan to investigate how other sports (Basketball, Baseball, ..) currently support predictions, and how applicable are our technique and interfaces to those.

In conclusion, our technique is a stepping-stone towards making more use of direct manipulation for prediction interfaces.

Our future work relies in the deeper investigation of user behaviors, particularly to automate the detection of patterns. The first challenge would be to extract the right features of user's behavior to account into the detection model. This is nontrivial since interactions are highly contextual, as they have a meaning based on previous interactions. Another important challenge is to enhance the technique with those patterns, either with new interaction features, or with the recommendation of predictions with automatic filling of the brackets by anticipation of a behavior. We are confident those challenges can be achieved with collective inputs and collaboration, and will lead to improved prediction techniques and interfaces for a broad range of domains

Figure 2 .

 2 Figure 2. Example of a soccer tournament configuration. Left: group stage (column with teams badges) that results into a ranking after games ended. Right: the elimination bracket phase (for the best group stage teams) to decide who the winner of the competition will be.

Figure 3 .

 3 Figure 3. The main three steps to use DRAG-AND-SNAP and perform a prediction for a single team at a time.

Figure 4 .

 4 Figure 4. Screenshot of the second prototype's interface, with and emphasize on the background showing possible paths for teams, and action button such as a reset button to start over, and an auto-complete button to automate the bracket's filling.

Figure 5 .

 5 Figure 5. Scatterplot of visitors. Each circle represents a visitor. The size of a circle encodes the number of interactions of the visitor. The horizontal axis is the size of the final bracket (ranging from 0 to 32). The vertical axis is the time spent using the interface (in seconds, logarithmic scale).

Figure 6 .

 6 Figure 6. Aggregation of of dragging interactions, where each line connects all the drag and drop of a team, by a visitor.

Figure 7 .

 7 Figure 7. Each line represents a visitor from G3b (who completed the prediction from scratch and filled a questionnaire). The horizontal axis represents time and the vertical axis the level of completion of the bracket (from 0 to 32). Red dots represent clicks on the reset button of teams (not the one of the whole prediction). Green rectangles represent clicks on the auto-complete button. Blue stars represent the interactions involving the self-declared favorite team of the visitor.

Table 1 .

 1 List of user interactions events we recorded during the evaluation.

		Type	Details
	webpage	init (auto) end (auto)	When the user opens the webpage When the user closes the webpage
		hover	Mouse hover on a team badge
	teams badges	drag drag (end) close	Dragging just gets started Dragging just ende The user clicks on the close button
	teams	hover	Mouse hover on a team placeholder
		click	Reset the prediction
	buttons	click	Auto-complete the prediction
		click (submit) Share the whole prediction


Hassan S Bakouch
email: hnbakouch@yahoo.com

Abdus Saboor
email: saboorhangu@gmail.com

Christophe Chesneau
email: christophe.chesneau@unicaen.fr

Anwaar Saeed
email: anwaarsaeedqureshi@gmail.com

A new lifetime model by mixing gamma and geometric distributions with fitting hydrologic data

Keywords: Gamma distribution, Geometric distribution, Order statistics, Record values, Estimation. Mathematics Subject Classification: MSC 2010: 2E10, 62E15, 62F10

In this paper, we propose a new distribution obtained by mixing gamma and geometric distributions. We discuss different shapes of the probability density function and the hazard rate functions. We study several statistical properties. The maximum likelihood estimation method is performed for estimating the parameters. We determine the observed information matrix and discuss inference. Illustrative three hydrologic data sets are given to show the flexibility and potentiality of the proposed distribution.

Introduction

An important aspect of statistics is the determination of flexible distributions to elaborate useful models for lifetime data. Among the existing approaches, new distributions can be obtained by mixing discrete and continuous distributions. Those using geometric distributions include the exponential geometric distribution [START_REF] Adamidis | A lifetime distribution with decreasing failure rate[END_REF], the exponential-power series distribution [START_REF] Chahkandi | On some lifetime distributions with decreasing failure rate[END_REF], the extended exponential geometric distribution [START_REF] Adamidis | On a generalization of the exponentialgeometric distribution[END_REF], the complementary exponential geometric distribution [START_REF] Louzada | The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart[END_REF], the Weibull-geometric distribution [START_REF] Barreto-Souza | The Weibull-Geometric Distribution[END_REF], the Weibull-power series distribution [START_REF] Morais | A compound class of Weibull and power series distributions[END_REF], the generalized exponential-power series distribution [START_REF] Mahmoudi | Generalized exponential-power series distributions[END_REF], the complementary exponentiated exponential geometric distribution [START_REF] Louzada | The complementary exponentiated exponential geometric lifetime distribution[END_REF], the extended Weibull-power series distribution [START_REF] Silva | The compound class of extended Weibull power series distributions[END_REF], the complementary extended Weibull-power series [START_REF] Cordeiro | The complementary extended Weibull power series class of distributions[END_REF], the exponentiated extended Weibull-power series distribution [START_REF] Tahmasebi | Exponentiated extended Weibull-power series class of distributions[END_REF], the G-geometric distribution [START_REF] Alkarni | A compound class of geometric and lifetime distributions[END_REF], the alternative G-geometric distribution [START_REF] Castellares | On the Marshall-Olkin extended distributions[END_REF], the generalized linear failure rate-geometric distribution [START_REF] Harandi | A complementary generalized linear failure rategeometric distribution[END_REF] and the linear failure rate-power series distribution [START_REF] Mahmoudi | The compound class of linear failure rate-power series distributions: Model, properties, and applications[END_REF]. We also refer to the review of [START_REF] Tahir | Compounding of distributions: a survey and new generalized classes[END_REF], and the references therein.

On the other hand, among the continuous distributions, the gamma distribution is one of the most commonly used in modeling life-time data. In practice, it has been shown to be very flexible in modeling various types of lifetime distributions. To the best of our knowledge, the mixing of the geometric distribution with the Gamma distribution (not reduced to the exponential one) has not ever been considered in the literature. Based on such a mixing, this paper offers a new distribution with two parameters, called the gamma-geometric (GG) distribution. The formulation and motivations of such distribution are as follows.

Let λ > 0, θ ∈ (0, 1) and θ = 1 -θ. We say that a random variable X follows the GG distribution with parameters (θ, λ), denoted by GG(θ, λ), if it has the probability density function (pdf) given by

f (x) = θλ 2 xe -λx 1 + θe -λx ( 1 -θe -λx ) 3 , x > 0. (1)
The corresponding cumulative distribution function (cdf) is given by

F (x) = 1 -θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 , x > 0. (2)
The GG distribution arises from the following stochastic representation. Let X be a random variable having the following stochastic representation:

X | {N = n} ∼ G am (2, λn), N | θ ∼ G trunc (θ), (3)
that is N is a random variable having the truncated geometric distribution with parameter θ: P (N = n) = θ θn-1 , n = 1, 2, . . . and the distribution of X conditionally to {N = n} is the gamma distribution G am (2, λn), with a conditional pdf given by f X|{N =n} (x) = λ 2 n 2 xe -λnx , x > 0. Then X follows the GG(θ, λ) distribution; using the geometric series expansions:

+∞ ∑ n=1 n 2 x n = x(1+x) (1-x) 3 , |x| < 1, the pdf of X is given by f X (x) = +∞ ∑ n=1 f X|{N =n} (x)P (N = n) = θ θ λ 2 x +∞ ∑ n=1 n 2 ( θe -λx ) n = θλ 2 xe -λx 1 + θe -λx ( 1 -θe -λx ) 3 . ( 4
)
The stochastic representation (3) can be viewed as a natural extension of the stochastic representation (1, λn), with pdf corresponding to the one of the G-geometric class proposed by [START_REF] Alkarni | A compound class of geometric and lifetime distributions[END_REF] (applied with the exponential distribution). Ratio of two independent variables. An example of simple model using the GG distribution is given by the ratio of two independent variables as described as follows. Let Y ∼ G am (2, λ) and N ∼ G trunc (θ). Suppose that Y and N are independent. Then (2, λn). This ratio representation will be useful to determine statistical properties of the GG distribution. Note that, from the ratio representation, the cdf of X given by ( 2) can be expressed directly: using the cdf of Y : F Y (x) = 1 -e -λx -λxe -λx , x > 0, and the geometric series expansions:

X | {N = n} ∼ E xp (λn) = G am
X = Y N , follows the GG(θ, λ) distribution. It is enough to note that X | {N = n} = Y /n ∼ G am
+∞ ∑ n=0 x n = 1 1-x and +∞ ∑ n=1 nx n-1 = 1 (1-x) 2 , |x| < 1, the cdf of X is given by F X (x) = P (Y ≤ xN ) = ∞ ∑ n=1 F Y (xn)P (N = n) = 1 -θe -λx ∞ ∑ n=0 ( θe -λx ) n -λxθe -λx ∞ ∑ n=1 n( θe -λx ) n-1 = 1 -θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 .
Some limit properties for f (x) are given as:

f (x) ∼ λ 2 2 -θ θ 2 x → 0, x → 0, f (x) ∼ θλ 2 xe -λx → 0, x → +∞.
Moreover, one can show that f (x) has a unique maximum on (0, +∞) given by f (x * ) where x * satisfies the equation: 4 θλx * e λx * + (λx * -1)e 2λx * + θ2 (λx * + 1) = 0. Some plots of f (x) are given in Figure 1 for several values of (θ, λ). The rest of the paper is organized as follows. In Section 2, we give some properties of the GG distribution. The estimation by maximum likelihood is discussed in Section 3 with illustrative real data examples.

Properties of the GG distribution

In this section, we propose many features and statistical properties of the GG distribution.

The survival function and hazard rate functions

The survival function (sf) of X is given by

S(x) = 1 -F (x) = θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 , x > 0, (5)
and the associated hazard rate function (hrf) of X is

h(x) = f (x) S(x) = λ 2 x 1 + θe -λx (1 -θe -λx )(1 + λx -θe -λx ) , x > 0. (6) Observe that h(x) ∼ λ 2 2 -θ θ 2 x → 0, x → 0, h(x) ∼ λ, x → +∞.
Some plots of h(x) are given in Figure [START_REF] Adamidis | On a generalization of the exponentialgeometric distribution[END_REF] for several values of (θ, λ).

Quantile function

The quantile function of X is determined by inverting the cdf F (x). The p-th quantile x p of X is the real solution of the nonlinear equation:

F (x p ) = p ⇔ θe -λxp (1 + λx p -θe -λxp ) = (1 -p)(1 -θe -λxp ) 2 .

Moments

Some key features of a distribution, like mean and variance, can be investigated through its r-th moments E(X r ). For finding E(X r ), we can use the ratio representation of X: X has the same distribution of the ratio of 2 random variables: Y /N with Y ∼ G am (2, λ) and N ∼ G trunc (θ), Y and N independent. Therefore, considering the Gamma function:

Γ(ν) = ∫ +∞ 0 x ν-1 e -x dx, ν > 0, E (Y r ) = Γ(2+r) λ r
and the polylogarithm function: Li r (x) = +∞ ∑ n=1

x n n r , r > 0, |x| < 1, we have

E(X r ) = E ( Y r N r ) = E (Y r ) E ( 1 N r ) = Γ(2 + r) λ r +∞ ∑ n=1 1 n r P (N = n) = Γ(2 + r) λ r θ θ Li r ( θ).
In particular, by taking r = 1, since Li

1 (x) = -log(1 -x), we obtain E(X) = - 2θ λ θ log(θ).
The variance of X can be explicit in some cases. For instance, if

θ = 1 2 , since Li 1 ( 1 2 ) = log(2) and Li 2 ( 1 2 ) = 1 12 [π 2 -6(log(2)) 2 ], we have V ar(X) = E(X 2 ) -[E(X)] 2 = Γ(4) λ 2 Li 2 ( 1 2
) -

[ Γ(3) λ Li 1 ( 1
)] 2 = 1 λ 2 ( π 2 2 -7(log(2)) 2 2
) .

Moment generating function

As for the moments, the moment generating function of X can be obtain via the ratio representation 2 , t < λ, and the conditional expectation, we get

Y /N . Using E(e tY ) = λ 2 (λ -t)
M (t) = E(e tX ) = E(e t Y N ) = E ( E ( e t Y N | N )) = E ( λ 2 ( λ -t N ) 2 ) = λ 2 E ( N 2 (λN -t) 2 ) = λ 2 +∞ ∑ n=1 n 2 (λn -t) 2 P (N = n) = λ 2 θ +∞ ∑ n=1 n 2 (λn -t) 2 θn-1 .

Conditional and reversed moments

The r-th conditional moments of X is given by

E(X r | X > t) = 1 S(t) ( E(X r ) - ∫ t 0 x r f (x)dx ) , t > 0,
and the r-th reversed moments of X is given by

E(X r | X ≤ t) = 1 F (t) ∫ t 0 x r f (x)dx, t > 0.
The integral term can be expressed using the expansion (4). Indeed, introducing the lower incomplete gamma function

Γ(t, ν) = ∫ t 0 x ν-1 e -x dx, ν > 0, t > 0, we have ∫ t 0 x r f (x)dx = θ θ λ 2 +∞ ∑ n=1 n 2 θn ∫ t 0 x r+1 e -nλx dx = θ θ 1 λ r +∞ ∑ n=1 θn n r Γ(nλt, r + 2). ( 7
)

Rényi entropy

An entropy plays a central role in information theory. It provides a suitable measure of randomness or uncertainty of X. For continuous distributions, Rényi entropy (see [START_REF] Rényi | On measures of entropy and information[END_REF]) can be determined as follows:

I R (γ) = 1 1 -γ log (∫ +∞ -∞ [f (x)] γ dx ) , γ > 0, γ ̸ = 1.
We have

[f (x)] γ = θ γ λ 2γ x γ e -λγx (1 + θe -λx ) γ ( 1 -θe -λx ) 3γ .
Using the generalized binomial series: (1+x

) α = +∞ ∑ k=0 ( α k ) x k , α ∈ C, | x |< 1, ( α k ) = α(α-1)(α-2)...(α-k+1) k! , we have (1 + θe -λx ) γ = +∞ ∑ k=0 ( γ k ) θk e -λkx , 1 ( 1 -θe -λx ) 3γ = +∞ ∑ ℓ=0 ( -3γ ℓ ) (-1) ℓ θℓ e -λℓx .
Hence,

[f (x)] γ = θ γ λ 2γ +∞ ∑ ℓ=0 +∞ ∑ k=0 ( -3γ ℓ )( γ k ) (-1) ℓ θℓ+k x γ e -λ(ℓ+k+γ)x .
Therefore,

I R (γ) = 1 1 -γ [ γ log(θ) + 2γ log(λ) + log ( +∞ ∑ ℓ=0 +∞ ∑ k=0 ( -3γ ℓ )( γ k ) (-1) ℓ θℓ+k ∫ +∞ 0 x γ e -λ(ℓ+k+γ)x dx )] = 1 1 -γ [ γ log(θ) + 2γ log(λ) + log ( +∞ ∑ ℓ=0 +∞ ∑ k=0 ( -3γ ℓ )( γ k ) (-1) ℓ θℓ+k Γ(γ + 1) λ γ+1 (ℓ + k + γ) γ+1 )]
.

Order statistics distributions

The order statistics are central tools in non-parametric statistics and inference. Let us now present the distributions of some fundamental order statistics related to the GG(θ, λ) distribution. Let a sample X 1 , X 2 , . . . , X n is randomly choosen from the GG(θ, λ) distribution and X 1:n ≤ X 2:n ≤ . . . ≤ X n:n are its corresponding order statistics. A pdf of X i:n is given by

f Xi:n (x) = n! (i -1)! (n -i)! f (x) n-i ∑ l=0 ( n -i l ) (-1) l [F (x)] i-1+l = n! (i -1)! (n -i)! θλ 2 xe -λx 1 + θe -λx ( 1 -θe -λx ) 3 n-i ∑ l=0 ( n -i l ) (-1) l [ 1 -θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 ] i-1+l , x > 0
The cdf of X i:n is given by

F Xi:n (x) = n! (i -1)! (n -i)! n-i ∑ l=0 ( n -i l ) (-1) l i + l [F (x)] i+l = n! (i -1)! (n -i)! n-i ∑ l=0 ( n -i l ) (-1) l i + l [ 1 -θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 ] i+l , x > 0.
A joint pdf of (X 1:n , . . . , X n:n ) is given by

f (X1:n,...,Xn:n) (x 1 , . . . , x n ) = n! n ∏ k=1 f (x k ) = n!θ n λ 2n ( n ∏ k=1 x k ) e -λ n ∑ k=1 x k n ∏ k=1 (1 + θe -λx k ) n ∏ k=1 ( 1 -θe -λx k ) 3 , 0 < x 1 < . . . < x n .
A joint pdf of (X i:n , X j:n ), i < j, is given by

f (Xi:n,Xj:n) (x i , x j ) = n! (i -1)! (n -j)! (j -i -1) [F (x i )] i-1 [F (x j ) -F (x i )] j-i-1 [S(x j )] n-j f (x i )f (x j ) = n! (i -1)! (n -j)! (j -i -1) [ 1 -θe -λxi 1 + λx i -θe -λxi (1 -θe -λxi ) 2 ] i-1 × [ θe -λxi 1 + λx i -θe -λxi (1 -θe -λxi ) 2 -θe -λxj 1 + λx j -θe -λxj (1 -θe -λxj ) 2 ] j-i-1 × [ θe -λxj 1 + λx j -θe -λxj (1 -θe -λxj ) 2 ] n-j θ 2 λ 4 x i x j e -λ(xi+xj ) 1 + θe -λxi ( 1 -θe -λxi ) 3 1 + θe -λxj ( 1 -θe -λxj ) 3 , 0 < x i < x j .

Record values distributions

Record values arise in a wide varity of real-life applications as hydrology, industry, lifetesting, economics, among the others. See, for instance, [START_REF] Ahsanullah | Record Statistics[END_REF], [START_REF] Arnold | [END_REF] and [START_REF] Chandler | The distribution and frequency of record values[END_REF]. We now present important distributions related to record values using the GG(θ, λ) distribution as baseline. Let X 1 , X 2 , . . . , be a sequence of i.i.d. random variables having the GG(θ, λ) distribution. We define a sequence of record times U (n) as follows:

U (1) = 1, U (n) = min{j; j > U (n -1), X j > X U (n-1) } for n ≥ 2. We define the i-th upper record value by R i = X U (i) , with R 1 = X 1 . A pdf of R i is given by f Ri (x) = 1 (i -1)! [-log(S(x))] i-1 f (x) = 1 (i -1)! [ -log ( θe -λx 1 + λx -θe -λx (1 -θe -λx ) 2 )] i-1 θλ 2 xe -λx 1 + θe -λx ( 1 -θe -λx ) 3 , x > 0.
A joint pdf of (R 1 , . . . , R n ) is given by

f (R1,...,Rn) (x 1 , . . . , x n ) = f (x n ) n-1 ∏ k=1 h(x k ) = θλ 2n x n e -λxn 1 + θe -λxn ( 1 -θe -λxn ) 3 × ( n-1 ∏ k=1 x k ) n-1 ∏ k=1 (1 + θe -λx k ) n-1 ∏ k=1 (1 -θe -λx k )(1 + λx k -θe -λx k ) , 0 < x 1 < . . . < x n .
A joint pdf of (R i , R j ), i < j, is given by

f (Ri,Rj ) (x i , x j ) = 1 (i -1)!(j -i -1)! [-log(S(x i ))] i-1 [log(S(x i )) -log(S(x j ))] j-i-1 h(x i )f (x j ) = 1 (i -1)!(j -i -1)! [ -log ( θe -λxi 1 + λx i -θe -λxi (1 -θe -λxi ) 2 )] i-1 × [ log ( θe -λxi 1 + λx i -θe -λxi (1 -θe -λxi ) 2
) -log

( θe -λxj 1 + λx j -θe -λxj (1 -θe -λxj ) 2 )] j-i-1 × λ 2 x i 1 + θe -λxi (1 -θe -λxi )(1 + λx i -θe -λxi ) × θλ 2 x j e -λxj 1 + θe -λxj ( 1 -θe -λxj ) 3 , 0 < x i < x j .

Residuals life functions

The residual life functions play a fundamental role in survival or reliability studies. See, for instance, [START_REF] Bryson | Some criteria for aging[END_REF], [START_REF] Hollander | Tests for mean residual life[END_REF] and [START_REF] Muth | Reliability models with positive memory derived from the mean residual life function[END_REF]. We now present some related mathematical objects with a potential of interest in the context of the GG(θ, λ) distribution.

The residual life is described by the conditional random variable R

(t) = X -t | X > t, t ≥ 0. The sf of the residual lifetime R (t) is given by S R (t) (x) = S(x + t) S(t) = e -λx (1 + λ(x + t) -θe -λ(x+t) )(1 -θe -λt ) 2 (1 -θe -λ(x+t) ) 2 (1 + λt -θe -λt ) , x > 0.
The associated cdf is given by

F R (t) (x) = 1 -e -λx (1 + λ(x + t) -θe -λ(x+t) )(1 -θe -λt ) 2 (1 -θe -λ(x+t) ) 2 (1 + λt -θe -λt ) , x > 0.
Then, the corresponding pdf is given by

f R (t) (x) = λ 2 (x + t)e -λx (1 + θe -λ(x+t) )(1 -θe -λt ) 2 ( 1 -θe -λ(x+t) ) 3 (1 + λt -θe -λt ) , x > 0.
The associated hrf is given by

h R (t) (x) = λ 2 (x + t) 1 + θe -λ(x+t) ( 1 -θe -λ(x+t) ) (1 + λ(x + t) -θe -λ(x+t) ) x > 0,
and the mean residual life is defined as

K(t) = E(R (t) ) = E(X -t | X > t) = 1 S(t) ( E(X) - ∫ t 0 xf (x)dx ) -t.
The integral term can be expressed as [START_REF] Bryson | Some criteria for aging[END_REF] with r = 1.

On the other hand, the variance residual life is given by

V (t) = V ar(R (t) ) = V ar(X -t | X > t) = 1 S(t) ( E(X 2 ) - ∫ t 0 x 2 f (x)dx ) -t 2 -2tK(t) -[K(t)] 2 .
Again, the integral term can be expressed as [START_REF] Bryson | Some criteria for aging[END_REF] with r = 2. The reverse residual life is described by the conditional random variable R (t) = t -X | X ≤ t, t ≥ 0. The sf of the reversed residual lifetime R (t) is given by

S R (t) (x) = F (t -x) F (t) = (1 -θe -λt ) 2 [ (1 -θe -λ(t-x) ) 2 -θe -λ(t-x) (1 + λ(t -x) -θe -λ(t-x) ) ] (1 -θe -λ(t-x) ) 2 [ (1 -θe -λt ) 2 -θe -λt (1 + λt -θe -λt ) ] , 0 < x ≤ t.
The associated cdf is given by

F R (t) (x) = 1 - (1 -θe -λt ) 2 [ (1 -θe -λ(t-x) ) 2 -θe -λ(t-x) (1 + λ(t -x) -θe -λ(t-x) ) ] (1 -θe -λ(t-x) ) 2 [ (1 -θe -λt ) 2 -θe -λt (1 + λt -θe -λt ) ] , 0 < x ≤ t.
Therefore, the corresponding pdf is given by

f R (t) (x) = θλ 2 (t -x)e -λ(t-x) (1 + θe -λ(t-x) )(1 -θe -λt ) 2 ( 1 -θe -λ(t-x) ) 3 [ (1 -θe -λt ) 2 -θe -λt (1 + λt -θe -λt ) ] , 0 < x ≤ t,
and the associated hrf is given by

h R (t) (x) = θλ 2 (t -x)e -λ(t-x) (1 + θe -λ(t-x) )(1 -θe -λ(t-x) ) 2 ( 1 -θe -λ(t-x) ) 3 [ (1 -θe -λ(t-x) ) 2 -θe -λ(t-x) (1 + λ(t -x) -θe -λ(t-x) ) ] , 0 < x ≤ t.
The mean reversed residual life is defined as

L(t) = E(R (t) ) = E(t -X | X ≤ t) = t - 1 F (t) ∫ t 0 xf (x)dx.
The integral term can be expressed as [START_REF] Bryson | Some criteria for aging[END_REF] with r = 1.

The variance reversed residual life is given by

W (t) = V ar(R (t) ) = V ar(t -X | X ≤ t) = 2tL(t) -[L(t)] 2 -t 2 + 1 F (t) ∫ t 0 x 2 f (x)dx,
Again, the last integral can be expressed as [START_REF] Bryson | Some criteria for aging[END_REF] with r = 2.

Estimation with hydrologic data examples

In this section, we estimate the unknown parameters of the GG distribution using the method of maximum likelihood. Moreover, three hydrologic data sets are given to show the flexibility and potentiality of the proposed distribution.

Maximum likelihood estimation

Let X 1 , X 2 , . . . , X n be a random sample of size n from the GG(θ, λ) distribution with observed values x 1 , x 2 , . . . , x n . Set Θ = {θ, λ}. The likelihood function associated to x 1 , . . . , x n is given by

L(Θ) = n ∏ i=1 f (x i ) = n ∏ i=1 ( θλ 2 x i e -λxi 1 + θe -λxi ( 1 -θe -λxi ) 3 ) = θ n λ 2n ( n ∏ i=1 x i ) e λ n ∑ i=1 xi n ∏ i=1 ( e λxi + 1 -θ ) n ∏ i=1 (e λxi -1 + θ) 3 .
The maximum likelihood estimators (MLEs) of θ and λ are obtained by maximization of L(Θ), or alternatively, the log-likelihhod defined by

ℓ(Θ) = log(L(Θ)) = n log(θ) + 2n log(λ) + n ∑ i=1 log(x i ) + λ n ∑ i=1 x i + n ∑ i=1 log(e λxi + 1 -θ) -3 n ∑ i=1 log(e λxi -1 + θ).
It follows that the MLEs are the simultaneous solutions of the equations according to (θ, λ):

∂ℓ(Θ) ∂θ = n θ - n ∑ i=1 1 e λxi + 1 -θ -3 n ∑ i=1 1 e λxi -1 + θ = 0 and ∂ℓ(Θ) ∂λ = 2n λ + n ∑ i=1 x i + n ∑ i=1 e λxi x i e λxi + 1 -θ -3 n ∑ i=1 e λxi x i e λxi -1 + θ = 0.
Since we have no analytic forms, numerical methods, as the quasi-Newton algorithm, can be applied to determine the estimators. The observed information matrix is given by

J(Θ) = ( J θ θ (Θ) J θ λ (Θ) J λ θ (Θ) J λ λ (Θ)
) ,

where

J θ θ (Θ) = - ∂ 2 ℓ(Θ) ∂θ 2 = n θ 2 + n ∑ i=1 1 (e λxi + 1 -θ) 2 -3 n ∑ i=1 1 (e λxi -1 + θ) 2 , J θ λ (Θ) = - ∂ 2 ℓ(Θ) ∂θ∂λ = - n ∑ i=1 e λxi x i (e λxi + 1 -θ) 2 -3 n ∑ i=1 e λxi x i (e λxi -1 + θ) 2 , J λ λ (Θ) = - ∂ 2 ℓ(Θ) ∂λ 2 = 2n λ 2 + n ∑ i=1 ( e 2λxi x 2 i (e λxi + 1 -θ) 2 - e λxi x 2 i e λxi + 1 -θ ) -3 n ∑ i=1 ( e 2λxi x 2 i (e λxi -1 + θ) 2 - e λxi x 2 i (e λxi -1 + θ)
) .

This matrix is a key mathematical tool to obtain approximate confidence intervals or Wald tests for θ and λ in the case of a large sample.

Illustrative hydrologic data examples

In this section, we fit the GG distribution to three hydrologic data sets and compare with the Weibull, Gumbel, Exponentiated Exponential, Generalized Gumbel, Kappa and Weibull Geometric distributions for three data sets. Most of those distributions have received great attention for fitting hydrology data, like rainfall data, precipitation data and flood data. More precisely, the densities of the compared distributions are given as follows.:

• Weibull distribution with pdf:

f (x) = k λ ( x λ ) k-1 e -( x λ ) k , λ > 0, k > 0, x > 0.
• Gumbel distribution with pdf:

f (x) = e -e -x-µ σ -x-µ σ σ , σ > 0, x, µ ∈ R.
• Exponentiated Exponential (EE) distribution [START_REF] Ahuja | The generalized Gompertz-Verhulst family of distributions[END_REF] with pdf:

f (x) = αλ ( 1 -e -λx ) (α-1) e -λx , α, λ, x > 0.
• Generalized Gumbel (GGu) distribution [START_REF] Cooray | Generalized Gumbel distribution[END_REF] with pdf:

f (x) = α ( 1 -e -e -x-µ σ ) α-1 ( e -x-µ σ ) e -e -x-µ σ σ , α, σ > 0, µ, x ∈ R.
• Kappa distribution [START_REF] Mielke | Another Family of Distributions for Describing and Analyzing Precipitation Data[END_REF] with pdf:

f (x) = αθ β ( x β ) θ-1 ( α + ( x β ) αθ ) -(α+1) α , α, θ, β, x > 0.
• Weibull geometric (WG) distribution [START_REF] Barreto-Souza | The Weibull-Geometric Distribution[END_REF] with pdf:

f (x) = αβ α (1 -p)x α-1 e -(βx) α ( 1 -p e -(βx) α ) -2 , p ∈ (0, 1), α, β, x > 0.
For goodness-of-fit we have two main test statistics, i.e., information criterion and empirical distribution. The measures Akaike information criterion (AIC) [START_REF] Akaike | A new look at the statistical model identification[END_REF], corrected Akaike information criterion (AICC) [START_REF] Hurvich | Regression and time series model selection in small samples[END_REF], Hannan-Quinn information criterion (HQIC) [START_REF] Hannan | The Determination of the order of an autoregression[END_REF], and consistent Akaike information criterion (CAIC) [START_REF] Bozdogan | Model selection and Akaikes information criterion, The general theory and its analytical extensions[END_REF] are widely used information criterion for selecting the appropriate model among different others models. The Anderson-Darling (A * ) due to Anderson and Darling [START_REF] Anderson | Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes[END_REF], the Cramérvon Mises (W * ) due to Cramér and Mises [START_REF] Cramer | On the composition of elementary errors II[END_REF] and the Kolmogorov Smirnov (KS) statistics due to Kolmogorov [START_REF] Kolmogorov | Sulla determinazione empirica di una legge di distribuzione[END_REF] with their p-values to compare the fitted models. These statistics are used to evaluate how a particular distribution with cdf, for a given data set, fits the corresponding empirical distribution. The distribution with better fit than the others will be the one having the smallest statistics and largest p-value.

Some information about the data sets are given below as: The first data set is taken from engineering department and is presented by Linhart and Zucchini [START_REF] Linhart | Model Selection[END_REF]. The data points are [START_REF] Tahmasebi | Exponentiated extended Weibull-power series class of distributions[END_REF]261,87,[START_REF] Bryson | Some criteria for aging[END_REF]120,[START_REF] Louzada | The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart[END_REF]62,47,225,71,246,[START_REF] Silva | The compound class of extended Weibull power series distributions[END_REF]42,[START_REF] Rényi | On measures of entropy and information[END_REF][START_REF] Arnold | [END_REF][START_REF] Harandi | A complementary generalized linear failure rategeometric distribution[END_REF]120,[START_REF] Cordeiro | The complementary extended Weibull power series class of distributions[END_REF][START_REF] Alkarni | A compound class of geometric and lifetime distributions[END_REF][START_REF] Louzada | The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart[END_REF]71,[START_REF] Cordeiro | The complementary extended Weibull power series class of distributions[END_REF][START_REF] Louzada | The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart[END_REF][START_REF] Cordeiro | The complementary extended Weibull power series class of distributions[END_REF][START_REF] Mahmoudi | Generalized exponential-power series distributions[END_REF]90,[START_REF] Adamidis | A lifetime distribution with decreasing failure rate[END_REF][START_REF] Mahmoudi | Generalized exponential-power series distributions[END_REF]52,95 The second data contains of annual maximum stream amounts. The data is available in U.S. Geological Survey (USGS) website (http : //nwis.waterdata.usgs.gov). The data points are 144,179,105,280,81,35,320,248,159,570,278,315,327,182,186,102,[START_REF] Anderson | Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes[END_REF]350,435,520,715,1600,660,173,239,667,44,82,70,68,69,42,[START_REF] Mahmoudi | Generalized exponential-power series distributions[END_REF]450,333,114,121,175,299,102,93,287,64,36,438,63,146,48,37,214,[START_REF] Cooray | Generalized Gumbel distribution[END_REF]161,104,115,[START_REF] Cramer | On the composition of elementary errors II[END_REF]109,128,[START_REF] Bozdogan | Model selection and Akaikes information criterion, The general theory and its analytical extensions[END_REF] The third data describes the maximum rainfall of Pakistan from 1981 to 2010. The data points are 21.7, 172.9, 69.5, 96. 5, 12.6, 265.5, 154, 28, 142.8, 14.2, 74.8, 32.5, 25, 28.5, 113.8, 25.7, 116.3, 28, 16.9, 6, 9, 17.6, 47.3, 55, 129, 72, 92, 28, 113, 194 For the first data set, Table 1 gives us estimates of the parameters of the considered models with their corresponding standard errors. Table 2 presents their goodness-of-fit statistics. Concerning the GG model, the MLEs corresponding to the data are given by θ = 0.155401 and λ = 0.0124595, and the following information criterion are obtained: AIC = 306.205, AICC = 306.649, HQIC = 307.101 and CAIC = 306.649. These values are the smallest in comparison to those obtained for the other models. On the other hand, we have A * = 0.51583, W * = 0.0892352, KS = 0.114073 with p = 0.829838, which are also the best. The superiority of the GG model, in terms of goodness-of-fit statistics, in comparison to the others, is also observed for second data set (estimates are given in Table 3 and goodness of fit statistics in Table 4) and the third data set (estimates are given in Table 5 and goodness of fit statistics in Table 6).

Figure 1 :

 1 Figure 1: Plots of the GG density function .

Table 1 :

 1 Estimates of the parameters (standard errors in parentheses) for aeroplane data

	Distributions	Estimates
	GG(θ, λ)	0.155401	0.012459
		(0.039411)	(0.002647)
	Weibul(λ, k)	54.6134	0.853587
		(12.361500)	(0.119402)
	Gumbel(µ, σ)	31.360100	40.609000
		(0.039411)	(6.566440)
	EE(α, λ)	0.809287	0.014543
		(0.188512)	(0.003720)
	GGu(α, σ, µ)	1.611330	63.122100	58.019100
		(0.361676)	(13.339200)	(12.796100)
	Kappa(α, θ, β)	1.035070	1.271930	29.742000
		(0.827705)	(0.695590)	(12.688400)
	WG( λ , a , b)	0.793447	0.007379	1.137020
		(0.281986)	(0.005622)	(0.251924)

Table 2 :

 2 Goodness of fit statistics for aeroplane data

	Distributions	Log(L)	AIC	AICC	HQIC	CAIC
	GG(θ, λ)	-151.102	306.205	306.649	307.101	306.649
	Weibul(λ, k)	-151.937	307.874	308.318	308.770	308.318
	Gumbel(µ, σ)	-161.982	327.964	328.408	328.860	328.408
	EE(α, λ)	-152.201	308.401	308.846	309.298	308.846
	GGu(α, σ, µ)	-164.116	334.233	335.156	335.577	335.156
	Kappa(α, θ, β)	-152.183	310.367	311.290	311.712	311.290
	WG( λ , a , b)	-151.278	308.557	309.480	309.902	309.480
	Distributions	A *	W *	KS	p
	GG(θ, λ)	0.51583	0.0892352	0.114073	0.829838
	Weibul(λ, k)	0.567419	0.0990543	0.153363	0.480628
	Gumbel(µ, σ)	5.90727	0.931143	0.355422	0.00102163
	EE(α, λ)	0.691402	0.123059	0.171971	0.337505
	GGu(α, σ, µ)	8.98938	1.39876	0.39446	0.000176396
	Kappa(α, θ, β)	0.477954	0.0885657	0.121557	0.767156
	WG( λ , a , b)	0.476917	0.0869025	0.125281	0.734085

Table 3 :

 3 Estimates of the parameters (standard errors in parentheses) for carnero data

	Distributions	Estimates
	GG(θ, λ)	0.336657	0.00480789
		(0.0567315)	(0.000630488)
	Weibul(λ, k)	229.722	1.05334
		(30.3587)	(0.100504)
	Gumbel(µ, σ)	133.735	132.415
		(18.0217)	(15.0402)
	EE(α, λ)	1.27283	0.00520083
		(0.234449)	(0.000861595)
	GGu(α, σ, µ)	70.6639	1229.31	2083.55
		(5.77893)	(73.4427)	(55.7163)
	Kappa(α, θ, β)	0.885474	1.88622	136.533
		(0.63983)	(0.937705)	(35.2732)
	WG( λ , a , b)	0.952363	0.00108257	1.64806
		(0.06844)	(0.00071915)	(0.224096)

Table 4 :

 4 Goodness of fit statistics for carnero data

	Distributions	Log(L)	AIC	AICC	HQIC	CAIC
	GG(θ, λ)	-369.200	742.400	742.618	744.005	742.618
	Weibul(λ, k)	-371.839	747.678	747.896	749.283	747.896
	Gumbel(µ, σ)	-381.104	766.207	766.426	767.813	766.426
	EE(α, λ)	-371.153	746.305	746.523	747.910	746.523
	GGu(α, σ, µ)	-410.482	826.963	827.408	829.371	827.408
	Kappa(α, θ, β)	-369.537	745.075	745.519	747.483	745.519
	WG( λ , a , b)	-369.040	744.080	744.524	746.488	744.524
	Distributions	A *	W *	KS	p
	GG(θ, λ)	0.26411	0.0282419	0.0586334	0.988461
	Weibul(λ, k)	0.563524	0.0712539	0.0869865	0.772452
	Gumbel(µ, σ)	3.94994	0.727	0.197178	0.0219974
	EE(α, λ)	0.570699	0.0915234	0.0938397	0.686709
	GGu(α, σ, µ)	25.9378	4.87121	0.419101	2.8×10 -9
	Kappa(α, θ, β)	0.314206	0.0400859	0.0737592	0.910551
	WG( λ , a , b)	0.248297	0.0308173	0.067118	0.956358

Table 5 :

 5 Estimates of the parameters (standard errors in parentheses) for jiwani data

	Distributions	Estimates
	GG(θ, λ)	0.453529	0.0182027
		(0.105628)	(0.00303367)
	Weibul(λ, k)	77.5581	1.16238
		(12.8831)	(0.165544)
	Gumbel(µ, σ)	45.4199	43.7452
		(8.35456)	(6.80433)
	EE(α, λ)	1.34541	0.0163824
		(0.340988)	(0.00365062)
	GGu(α, σ, µ)	10.4962	156.021	226.737
		(1.5249)	(17.0742)	(16.1174)
	Kappa(α, θ, β)	0.881944	1.83423	46.7766
		(1.20898)	(1.74096)	(23.4308)
	WG( λ , a , b)	0.648328	0.00815663	1.40982
		(0.454225)	(0.00451551)	(0.333492)

Table 6 :

 6 Goodness of fit statistics for jiwani data

	Distributions	Log(L)	AIC	AICC	HQIC	CAIC
	GG(θ, λ)	-157.521	319.042	319.487	319.939	319.487
	Weibul(λ, k)	-158.359	320.717	321.162	321.614	321.162
	Gumbel(µ, σ)	-162.542	329.084	329.529	329.981	329.529
	EE(α, λ)	-158.223	320.446	320.891	321.343	320.891
	GGu(α, σ, µ)	-166.937	339.874	340.797	341.219	340.797
	Kappa(α, θ, β)	-159.344	324.688	325.611	326.033	325.611
	WG( λ , a , b)	-158.038	322.077	323.000	323.422	323.000
	Distributions	A *	W *	KS	p
	GG(θ, λ)	0.4193	0.0798668	0.141087	0.589015
	Weibul(λ, k)	0.517542	0.095075	0.165072	0.387045
	Gumbel(µ, σ)	2.10048	0.349666	0.265105	0.0294899
	EE(α, λ)	0.54145	0.102138	0.167958	0.365793
	GGu(α, σ, µ)	5.2192	0.858108	0.321454	0.00405898
	Kappa(α, θ, β)	0.567106	0.101475	0.141822	0.582326
	WG( λ , a , b)	0.503657	0.0945876	0.153904	0.476065