Abstract 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) = râˆ‘i=1 Ç¾ (Âµ,o2/i ) Where Ç¾ (Âµ,o2/1 ) = (2Ð», o2/i)1/2 And râˆ‘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.
