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Title of Thesis

Muhammad Siddiq Ansari
Institute/University/Department Details
Institute of Mathematics/ Quaid-i-Azam University, Islamabad
Number of Pages
Keywords (Extracted from title, table of contents and abstract of thesis)
mixture of normal distribution, problem of identification, estimation of the parameters, moment estimators, asymptotic dispersion matrix

Muhammad Siddiq Ansari, Mixture of Normal Distribution. (Under the directions of Dr. Ashiq Hussain Baloch). This thesis is concerned with the mixture of normal distributions, characterized by the density function,

F(x) = rˆ‘i=1 Ǿ (µ,o2/i )


Ǿ (µ,o2/1 ) = (2л, o2/i)-1/2


rˆ‘i=1 pi = 1

Four main problems connected with a mixture of distributions are Problem of identification. Problem of distributions arising from a mixture. Problem of estimation of the parameters of the mixture. Problem of testing of meaningful hypotheses about the parameters of the mixture. And (d) listed above. To our knowledge, these problems have not been adequately analyzed so far. Identified case of the mixture of normal distributions is considered and some properties are stated. Also some this mixture are formulated and proved. The case when some of the mixing proportions pi's are negative is important and is studied. In this case a necessary and sufficient condition for the mixture under study to represent a probability density function is derived. Then the problem of estimation is taken up. The method seems to be suitable. First of all a general mixture of r components is considered and the moment estimators of the unknown parameters are derived as roots to some equations. It is shown that these estimators are consistent and asymptotically normal. Their asymptotic dispersion matrix is found. Attempt is made to drive tests for some pertinent hypotheses on the mixture. It is found that exact tests are not available, so that we use approximate tests. This is followed by a discussion of a particular case of the mixture under consideration. In this case explicit expressions for the estimators and their asymptotic dispersion matrix are derived.

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4341.31 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 1 Introduction 1
599.82 KB
2 2 Properties Of The Mixture
544.65 KB
3 3 Estimation Of Parameters Of General Case
1517.93 KB
  3.1 Summary
  3.2 Construction Of Estimators For The General Case
  3.3 Asymptioc Bhaviour Of The Moment Estimators
  3.4 Estimation When Some Of 2i's And Pi's
  3.5 Estimation When Some Of The Parameters Are Known Of Equal
  3.6 Extension To The Multivariant Situation
  3.7 Inedequacy Of Maximum Likelihood Procedure
4 4 Test Of Hypotheses On The Mixture
421.24 KB
  4.1 Some Difficulties In Testing Hypotheses
  4.2 Tests Of Normality
  4.3 Test About The Parameters
  4.4 Test About The Mixture( 1) To Be A Density
5 5 Estimation For A Mixture Of Two Normal Distributions
1210.91 KB
  5.1 Construction Of The Moment Estimators
  5.2 Asymptotic Dispersion Matrix Of The Estimators
  5.3 Numerical Example
  5.4 Summary And Conclusions