MCom I Semester Statistical analysis Decision Theory Study Material Notes
Table of Contents
MCom I Semester Statistical analysis Decision Theory Study Material Notes: Meaning and Definition of Statistical Decision Theory Components or Ingredients of the Decision theory Basic Elements of Decision problems Decision tree Diagram some Important Numerical Issusration for Clarifications Basic Concepts Being Used Statistical decision Regret or Loss Table regret or Loss Table Decision Criteria or Decision-Making Environment :
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Statistical Decision Theory
Decision-making is a part of everybody’s life and it is especially so in the case of a manager of business. The decision-makers in an organisation may have to take a decision on whether to introduce a new product or not. In marketing, the decision makers may have to decide whether to adopt a new promotional strategy or continue the old one, In production, they may have to decide whether to make or buy an item. In finance, they may have to decide whether to invest in a new business venture or not.The decision-makers select one course of action (or strategy) over others based on some criteria such as utility, maximum sales, minimum cost or maximum rate of return. Thus, decision-making is a process of choosing a course of action out of several courses of action for the purpose of achieving an objective In earlier days, decisions were made mainly on personal judgement. However, these days there are several statistical techniques which help in this process. Some of which were discussed earlier. For example, the techniques of correlation and regression help in establishing suitable functional relationships between variables, while forecasting helps in scientific prediction. Time series analysis gives us estimation of variable, subject to some condition. The test of hypothesis procedures are designed to test a statistical statement about a population (the null hypothesis) given a level of significance.
Decision Theory Study Material
According to Jerome D. Braverman, there are mainly three defects in hypothesis testing technique :
Firstly, it provides for only two possible actions which correspond to either acceptance or rejection of the null hypothesis. Also it allows only two states of the parameter being tested, that is those parameter values that make the null hypothesis true and those values for which the null hypothesis is false. However, in most practical decision problems, a decision-maker should be able to make a choice from among several different acts or subject to a wide variety of states or conditions over which decision-maker has no control.
Secondly, there may be some serious economic consequences that takes place by making wrong decisions. But such consequences are not considered by the decision-maker in hypothesis testing technique.
Thirdly, classical decision procedure does not recognise the validity of any information pertaining to the decision that does not exist in the form of empirical data result from a process oi sampling.
The statistical decision theory, also called “Bayesian decision theory”, removes the above shortcoming and enables us in taking optimal decisions. In short, the main features of this theory are as follows:
(1) It provides a model for decision-making in situations that involve multiple stages of the parameter.
(2) It incorporates the economic consequences of wrong decisions directly into the decision-making model.
(3) It allows the use of information pertinent to the problem which exists prior to any sampling or experimentation, whether this information is in the form of empirical data or is a subjective assessment of the decision-maker.
The statistical decision theory tries to reveal the logical structure of the problem into alternative action, states of nature, possible outcomes and likely pay-off from each such outcome.
In this chapter, a brief description of the statistical decision theory is done.
Decision Theory Study Material
Meaning and Definition of Statistical Decision Theory
Statistical decision theory consists of a large number of quantitative techniques which help in analysing a decision situation and enables us to arrive at a strategy which is the best under given circumstances of the case. We can say that decision analysis refers to the logical and quantitative analysis of all the factors that influence a decision. The theory tries to uncover the logical structure of the problem through proper analysis and classification of a decision problem into say the alternative courses of action, the possible states of nature, the possible outcomes with each of the above and the likely pay-offs from each such outcome.
Broadly defined ‘Statistical Decision Theory’ is a term used to apply to those methods for solving decision problems in which uncertainty plays a crucial role. Two principal foundations of this theory are probability and statistical inference.
“Statistical Decision Theory is concerned with the selection of an optimal course of action from among several alternatives where the outcome associated with an action is uncertain.”
– Joseph Van Matre and Glenn Gilbreath Conclusion : Decision theory provides a method for rational decision-making when the consequences are not fully known. The probability theory therefore, has to play an important role.
Components or Ingredients of the Decision Theory
or
Basic Elements of Decision Problems
or
Basic Concepts being used in Statistical Decision Theory
One of the purposes of studying decision theory is to put the problem into a suitable logical framework. This includes identifying the problem, generating alternative courses of action, determining the results of each course of action and finally evolving criteria for evaluating the alternatives to arrive at the best choice of action. Before, going over to the various decision strategies and the processes of making optimal choices let us grasp the basic components of a decision situation. The basic components of Statistical Decision Theory’ are as follows:
(1) Acts (Course of Action): The various possible alternative courses of action to solve the particular problem are known as “Acts’. Acts are independent to provide solution to a problem. For example, a man wants to go to Lucknow, he can go
by bus or by air. It means there are three acts out of which one act is to Thus, acts are the several alternatives available to the decision-maker.
The final choice of any one will depend on the pay-off (or money value) of each strategy under circumstances.
Decision-making problems deal with the selection of a single act from a set of alternative acts.
A set of various alternative actions or acts can be indicated in any one of the following ways:
a = (a, a, az,…. a.) or A = (A,,A,,A3,…,4) The totality of all possible acts is known as action place.
In the matrix form of presentation, these acts may either be shown in the rows or columns. But, in practice columner form is preferred.
(2) Events (States of Nature or States of the Parameter) : The term “state of nature” does not mean nature in the ordinary sense of the word. It is a general term which is used to encompass all those factors beyond the control of the decision-maker that affect the outcome of his decision. In every ‘Act’ there are events which are uncertain and beyond the control of a decision-makerBefore applying decision theory, we must develop an exhaustive list of possible future events. Such future events are referred to as states of nature and it is assumed that these are mutually exclusive and collectively exhaustive. When there are many possible outcomes (state of nature) of an event, one cannot predict what will happen-it is only in terms of probability one may forecast.
The various states of nature are outside the control of a firm and are not fully under its control, e.g., consumer demand, competition from rival firms, change of technology, etc. These affect the pay-off and hence the choice of an act. A set of state of nature can be indicated in anyone of the following ways: S = (S,,S,S3,…,.) or E = (E1, E2, E3,…,En or 2 = (@,,,,03,…,0)
Totality of all outcomes is called nature space or state space symbolised by S2. For example, if a product is marketed it may be highly appreciated (outcome ), it may not appeal to the customers (outcome ) or it may be liked by a certain fraction of the customers, say 25% (outcome 03).
In the matrix form these can be stated in either of the two forms given below:
The state of nature can have a numerical description such as demand of some units of a given item or a non-numerical description like an employee’s strike.
(3) Outcomes : There is an outcome or consequence of the combination of each of the likely acts and the possible states of nature. This outcome is also known
(4) Pay-off: Whereas outcome refers to the result of the combination of an act and each of the states of nature, the pay-off deals with the monetary gain or loss from each of the outcomes. This means that the expression pay-off should be in quantitative form. Pay-off may be also in terms of cost saving or time saving. Thus, a pay-off table is prepared and it shows the relation between all possible states of nature, all possible actions and the values associated with the consequences. A specimen of pay-off table is given below:
General Format of Pay-off Table State of Nature
(5) Opportunity Loss : The resultant outcomes of the various combinations of the acts and events (the states of nature) can be expressed in an alternate way. This is in terms of the opportunity loss. Also called regret, the opportunity loss is defined as the amount of pay-off foregone by not adopting the optimal course of action that which would give the highest pay-off, for each possible event. In other words, the difference between the highest possible profit for a state of nature and the actual profit obtained for the particular action taken is known as opportunity loss. Opportunity losses are calculated separately for each state of nature that might occur. For a given state of nature, the opportunity loss of possible course of action is the difference between the pay-off value for that course of action and the pay-off for the best possible course of action that could have been selected.
Decision-Tree Diagram
A decision-tree is a graphic representation of the sequences of action-event combinations available to the decision-maker. It depicts in a systematic manner all possible sequences of decisions and consequences. Each such sequence is shown by a distinct path through the tree. A decision-tree enables the decision maker to see the various elements of his problem in proper perspective and in a systematic manner. It may be mentioned that the criterion on the basis of which the decisions are made in the decision-tree approach is generally the expectation principle. Thus, we may choose the alternative that maximizes the expected profit, or the alternative that minimizes the expected cost and so on.
There are basically two types of decision-trees-deterministic and probabilistic. These can further be divided into single stage and multi-stage trees. A single stage deterministic decision-tree involves making only one decision under conditions of certainty (no chance events). In a multi-stage deterministic tree a sequence or chain of decisions are to be made. A problem involving only one decision to be made under conditions of risk or uncertainty (more than one chance event) can be represented in a single stage probabilistic decision-tree while the same type of
