MST-Department of Mathematics

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A study on numerical range of operators in a hilbert space
(Kenyatta University, 1987) Oleche, Paul Odhiambo
Application of canonical correlation analysis ;a comparative study on academic perfomance
(1990-06) Makambi, Kepher Henry
Finding the Distribution of a Random Variable from its Moment Function
(2000) Otwombe, N. K.
Consider the problem of r .~/ randomly distributed points in a unit n-ball and the convex hull created by these points. Let ~II be r! times the r-content of an rsimplex whose P vertices are in the interior and r -I- /- p vertices on the boundary of a unit n-ball. Explicit expressions for the exact distribution functions of ~II are given when r I / points are independently, and identically distributed according to the Uniform distribution. The exact distributions are obtained using the technique of Inverse Mellin transforms with the help of the moment functions. The technique is illustrated for the general case p =- r -j J and a particular case p =3, r - 2 . Various representations of the distributions in psi and the generalized zeta functions are given. These representations are also given in the most general case as an H- function distribution. SYMBOLS AND NOTATIONS The following is a list of symbols and notations, with meanings indicated on the right that will frequently occur in this Research. (" ) = 17 1 fII ml(n-m) Binomial Coefficient m-J .(at, =n(a +.j), (a),. = I Pochhammer Symbol j (! r(a) Gamma Function pdf Probability density function The cumulative distribution function The natural logarithm of 10 r-content of the r-simplex generated by r + 1 points 11" = r! 11 r! times the r - content of the simplex R",£" Euclidian n-space Re(.) The real part of (.) arg(.) Argument of (.) GRAPHS AND TABLES Fig. 4.1 Theoretical cdf plot for r=2, n=Z, p=3 Fig. 4.2 Density plot for r=2, n=Z, p=3 Table 1 Table of Theoretical moments and the exact Moments from Equation (4.1.1)
Laminar natural convection in a rectangular cavity
(2012-02-01) Thoya, Patrick Kitsao; Gatheri, F. K.
This project is basically a numerical study of the structure of the flow and heat transfer rates due to collision of opposed laminar natural convection boundary layer in an enclosure. Natural convection plays an important role in the flow and heat transfer in a wide range of technological applications. A fluid motion of a Boussinesq fluid in a three dimension rectangular cavity has been considered. To enable the analysis of flow and heat transfer rates, a complete set of non-dimensional equations governing newtonian fluids and boundary conditions were recast into vorticity/vector potential to eliminate the need for solving the continuity equation. The governing equations with the boundary conditions were discretised using a three-point central difference approximation for a non-uniform mesh. The resulting finite difference equations were then solved using the method of false transients and the Samarskii-Andreyev. A powerful computer code was used to generate the current results. The non-uniform mesh has been implemented in the project to allow for efficient use of computer time and storage. This enables placement of more nodes in regions of high velocity gradients and fewer nodes in other regions so that the total number of nodes used is minimized. This mesh type does not compromise the accuracy of the solution as it is demonstrated that order of (h2) accuracy may be maintained, where h is the spatial step size. The Rayleigh number used to obtain the current results was between 5 x 105 and 5 x 106. The study revealed that at higher Rayleigh number, there is substantial increasing exchange of hot air from the lower region to the upper region and vice versa. Temperature tends to decrease as the flow moves away from the 'active wall'.
Analysis of flow in regular bend pipe
(2012-02-01) Chirchir, Julius Kibet
A fluid is a substance that deforms continuously on application of slight shear stress. The study of fluid mechanics deals with motion of fluids and conditions that support or after motion. Fluid mechanics can be divided into fluid kinematics and fluid dynamics. Fluid kinematics deals with forces involved in motion of fluid, whereas fluid dynamics deals with the state of fluid in motion caused by imbalances of forces acting on it. Kinematics of fluid motion uses vector quantities such as; velocity, acceleration and rate of discharge which is defined in terms of scalar quantities; length and time in some specified coding system. These depend on some boundaries of a particular system under in investigation. Fluid dynamics involves the applications of Newton's second law of motion that states:- rate of change of momentum is directly proportional to the applied force; Transfer of fluids through pipes involves changes in potentials/pressure energy between two ends of the pipes under study. This potential difference can either be caused by: - (a) Gravitational differences (g) (b) Differences in height (h) (Potential energy) (c) Density differences, (p), for example that caused by concentration differences, Beek (1985). These three factors are related by equation (i) on the next page. P=phg .........................................................(i) The flow of fluids in channels can either be turbulent or laminar. Flow is where the streamlines do not intersect such that there is no mixing up of fluids. Laminar flow occurs when fluid has; 1. Low Reynold's number (Re<2000) 2. Low viscosity (internal friction of fluid particals 3.Friction between the channel and the fluid negligible. Turbulent flow occurs where there is continuous mixing up of the fluid in motion. This type of flow occurs when the fluid has; 1. High Reinhold’s number 2. High viscosity. 3. Boundary of channel and fluid have got high friction. Fluids transfer can either occur in open or closed channel depending on the geometry of the channel. Open channels are like surface run-off, rivers and streams, dug-out drains, of which all have got a free surface while confined (closed) flow occurs in porous media, closed cracks and in pipes. Fluid flow in both types of channels have got its uses, but highlighted here are only importance of closed channel flow. That is the used both in domestic and industrial sectors. In domestic side flow in closed channels are important in areas, for example in passage of fluids for example water from one point to other. It is important to have piped water at our vicinity especially at this age of development. On the industrial side, flow in closed channels at factories is important and to mention a few examples, are in areas like; - 1. Sewage collection and disposal. 2. Passage of oil, for example the oil pipeline from Mombasa to Eldoret 3. Passage of raw materials in chemical processes, for example breweries 4. Passage of gases in geothermal plants like in Olkaria In this work, only confined type of flow will be considered in a uniform circular cross sectional bend of a pipe. The fluid here is a liquid, for example water. And since the fluid delivered need not necessarily follow a straight path definitely there will be bends in pipes. Shapes such as this might be met in area like: Sprinkler nozzles. Nozzles of sprayers and spouts of various utensils like kettles. Detailed knowledge on fluid flow and the prevailing suitable conditions for laminar flow must be well defined as given earlier. In this work, however. We seek to give a clear picture on: - 1. Pressure distribution. 2. Velocity distribution and profile where necessary. 3. Forces acting on the pipe at certain parts. We therefore consider in detail how flow factors are affected by the shape of the region of the pipe considered. The proper interaction balance of these factors play a role in flow mechanism. Earlier, various analyses on straight pipe of some geometry Douglas (1995). The writer has analysed flows by application of methods of differential calculus and coupled with a well-defined boundary conditions have given practically near accurate results. On basis of the results obtained by this method it has become evident that complete, continuity, momentum and energy (Bernoulli) equations have to be well understood. In most of flows in pipes, velocity distribution, pressure distribution and force analysis are investigated by use of calculus modified by trigonometric ratios with successful results. Differential models has been powerful in predicting a number of flow problems in diverse fields of flow in the past decade.
Comparison of the performance of estimators in estimation of finite population total
(2012-02-01) Muema, Benjamin K.
In this project we compare the performance of two different estimators of the population total. One estimator is model-based and the other one is model assisted. We look at model-based properties of the two estimators. We observed that under the general model, the biases of the two estimators are different.
A model-based approach to estimation of finite population total using local linear polynomial regression estimator
(2012-02-01) Kasungo, Kithikii
Estimation of finite population totals in the presence of auxiliary information is considered. A class of estimators based on local polynomial regression is proposed. Like generalized regression estimators, these estimators are weighted linear combination of study variables, in which the eights are calibrated to known control totals, but the assumptions on the super population model are considerably weaker. The estimators are shown to be asymptotically model-unbiased and consistent under mild assumptions. Simulation experiments indicate that the local polynomial regression estimator is more efficient than regression estimators when the model regression function is incorrectly specified, while being approximately as efficient when the parametric specification is correct.
An interaction model between Cotesia flavipes and Cotesia sesamiae, parasitoids of the gramineous stem-borers at the Kenya Coast
(2012-02-01) Karuku, Muriithi Simon
One of the greatest challenges facing the people of sub-Saharan Africa is the production of sufficient food to feed a rapidly increasing population in the face of dwindling finances. As the population grows at 3% and food production at 2% per annum, an annual shortage of 250 million tons of food is expected by year 2020. The greatest obstacle to increasing the production of maize and sorghum, the staple food in many African communities south of the Sahara, is damage by phytophagous insects. Larval feeding in the plant Whorl and later through stem tunneling causes plant damage. Infested plants have poor growth and reduced yield and are more susceptible to secondary infection and wind damage. Estimates of yield losses due to stem-borer are in the neighborhood of 20-40% of the potential yield. To realize the potential of the Gramineae family in ensuring food security in the world, the stem-borers have to be effectively controlled. Various methods have been tried in a bid to control these pests. In biological control, one of the approaches is to find an exotic natural enemy that will successfully fit into the community of existent natural enemies. Hampered by a lack of economic and convenient tools, however, advances in biological control have been largely overshadowed by the rush to exploit insecticides and the ready availability and comparative simplicity of cultural methods. But that is changing. Effects on non-target organisms, resistance development and environmental pollution have incapacitated insecticides and other chemical-based methods. In this study, a simple one host-two parasitoids interaction model with a non-linear trend is developed to predict and understand the reasons for the ultimate impact of the exotic parasitoid Cotesia flavipes (Cameron) (Hymenoptera Braconidae) on stem-borer population dynamics in the coastal area of Kenya. Results indicate that the ultimate extent of suppression of the stem-borers is largely determined by three attributes of the parasitoids namely; the net reproductive rate, the degree of aggregation and the searching efficiency. The model predicts coexistence of all the species considered with C. flavipes dominating the interactive system. Implications of the results for introduction scheme of parasitoids to control pest are discussed. We argue that a model of intermediate complexity may offer the pest prospects of predictive biological control in situations where it is not practicable to obtain the information needed to build and parameterize a large tactical simulation model. The conclusions we reach are of relevance to classical biological control practices, and in particular to those programs in which more than one parasitoid species has been introduced to combat a particular pest of a perennial standing crop system.
Solving equilibrium equations for stress and deflection functions in a thin elastic shallow shell
(2012-02-02) Okelo, Jeconia Abonyo
A problem connected with the study of thin elastic shallow shells in the theory of non-linear elasticity is considered. The equilibrium equations are in the form V4 B1 (x, y) + Eh n4 B2 (x,y)=0 DV 4 B2 (x,y) - n 4 B1 (x,y) = S(x,y) The entire domain of the shell n bounded in R3 4 is the biharmonic operator  4 (---)= ( + 2 4 +  4) ( x4  x2  y2  y4) 4 is the pucher 's operator 4 (----)=  2 fd 2 - 2 2 f +  2 + 2 f2  x2  y2  x  y  x  y  x2 ( y2 E: Young's modulus of elasticity of the material  h: the uniform thickness of the shell  D= Eh 3 (I- u2)-1 is the flexural rigidity of the material 12 µ=Poisson's ratio i (x,y) is the stress function 2(x,y) is the deflection function S(x,y) is the external force on the projection of the shell  on xy plane The entire boundary of the shell in the form dn is assumed clamped so that the deflection and slopes are zero boundaries The entire domain of the shell l  is bounded in R3 having middle surface in Monge's form Z= f (x ,y ). Galerkin's orthogonality conditions are applied to solve the equilibrium equations for stress and deflections. Appropriate forms of orhonormal Fourier's double series are formulated for1 (x, y) and 2(x, y) to satisfy the boundary conditions. Finally the existence and the uniqueness of the solutions are established.
The effects of streaming on mathematics achievement among secondary school pupils in Kisumu, Kenya
(2012-02-02) Ukanda, Ferdinand Ingubu
This study investigated the effects of streaming on Mathematics achievement. The study examined whether the Learning setting played a significant role in determining pupil achievement and whether the effect of the Learning setting depended upon the Ability level of the pupils. The subjects of the study consisted of Form Three Secondary School pupils (N=48). Purposive sampling was used to select an urban secondary school that streamed its pupils according to Ability. A stratified random sampling technique was used to select the subjects. The stratifying criteria was pupil Ability as determined by end-of-year examination results. A factorial research design was used. The factors studied were: the Learning setting and the Ability levels. The Learning Setting had two levels namely the Individual Learning Setting and the Group Learning Setting. Ability had three levels namely high-ability, medium-ability and low-ability levels. The dependent measures used were the total error scores on two Mathematics tests namely Algebra and Logarithms and Indices. The instruments used for data collection were; two mathematics achievement, an observation checklist to obtain data on interaction in groups; a pupils Questionnaire to determine the pupils' views on streaming; and Learning materials. The tests were administered under examination conditions. The observation data was collected by trained research assistants, while the pupils' Questionnaire was completed immediately after doing the tests. Two-way factorial Analysis of Variance was used to identify the significant effects of the Learning setting and that of Ability on the total error scores, on both Algebra and Logarithms and Indices. It was also used to identify the interaction effects of the Learning setting and Ability on the total error scores. For the Learning setting, the F-values of 3.749 on Algebra and 0.13 on Logarithms and Indices were not significant (P<. 05, 1, 42 DF). For Ability, the F-values of 3.966 on Algebra and 13.58 on Logarithms and Indices were significant (P<. 05, 2, 42 DF). For the interaction of the Learning setting and Ability, the F-values of 1.562 on Algebra and 0.128 on Logarithms and Indices were not significant (P<. 05, 2, 42 DF). AVONA was also used to test the effects of the Mathematics Tasks and its interaction with the Learning setting. For the Mathematics tasks, the F-value of 10.232 for high-ability pupils was significant (P<. 05, 1, 20 DF), that of 3.158 for medium-ability pupils was not significant (P<. 05, 1, 44 DF) and that 0.011 for low-ability pupils was not significant (P<. 0.5, 1, 20 DF). For interaction of the Learning setting and the Mathematics Tasks, the F-values of 0.605 for high-ability pupils and 0.161 for medium ability pupils, the F-values of 4.825 was significant (P<. 05, 1, 20 DF). The above findings indicated that the Learning setting had no significant effect on the performance of the pupils. However descriptive analysis pointed to a clear-cut effect though it was insignificant. There is therefore need for replicating the study, with a larger sample of pupils and also requiring more time for the pupils to master the Tasks properly. The study can also be replicated using tasks in other subjects than Mathematics. However, Ability had a significant effect on the performance of the pupils as expected. The performance at each Ability level was however seen to depend on the Learning setting though the interaction effect of the Learning setting and Ability was not Learning setting though the interaction effect of the Learning setting and Ability was not significant on both the tasks. The medium-ability and low-ability pupils greatly benefited from the Group Learning setting while the Individual setting seemed to favour the high-ability pupils. Analysis of group interaction indicated that the high-ability pupils in nearly every group directed the group- work and delegated work to another members of the group. They thus engaged in peer tuition. The high-ability pupils were able to locate their own area of difficulty by giving explanations. This was quite necessary for high achievement. The medium-ability pupils rarely explained how to carry out calculations but received explanations about them. The low-ability pupils rarely participated in setting up algorithms but often solicited and received explanations from other members. This must have led to their increased performance. When asked to indicate their preferences, the high-ability pupils showed no preference for any one of the Learning settings. The medium-ability and low-ability pupils showed clear preference for the Group setting. The high-ability pupils indicated they liked the streaming practice used in their school. Half of the medium-ability pupils liked the system while the others did not. All the low-ability pupils did not like the streaming practice and thought it did little to help them learn better. From the findings, it was recommended that mixed-ability group discussions should be encouraged in secondary schools. Peer-to-peer teaching should thus be encouraged especially given our very large classes and teacher's inability to attend meaningfully to individual needs in a classroom. Secondary schools should have classes with pupils of mixed-ability. All the pupils be provided with appropriate learning materials. The practice of streaming should be discontinued. It was also recommended that the study be replicate using a larger sample and requiring more time and also in other subjects. A similar study should be carried out at other levels of education like primary schools, colleges and the University.
The extent to which Kenya certificate of primary education (KCPE) mathematics results predict performance in mathematics at Kenya certificate of secondary education(KCSE): a case study of national schools
(2012-02-02) Kihara, Joseph M.
The study set out to investigate the extent to which Kenya Certificate of Primary Education (KCPE) mathematics examination results predict performance in mathematics at the Kenya Certificate of Secondary Education (KCSE). It focused on the national schools. To achieve this objective, the study examined such factors as: • Skills and abilities tested in KCPE and KCSE mathematics examinations. • Teaching techniques adopted by teachers at KCSE level • Attitudes of students towards mathematics • Gender of the student • KCPE and KCSE mathematics performance. Each of the above factors was examined in the light of their combined impacts on the overall KCSE mathematics performance. The target population included all the 18 national schools in Kenya of which 10 are boys and 8 are girls' schools. A sample of six secondary schools was selected at random, using stratified random sampling to proportionately represent the boys and girls school categories. Data was collected through questionnaires and interviews administered on students and mathematics teachers. Performance data was obtained from the six national schools students' performance records on KCPE examinations and corresponding KCSE examinations. Both quantitative generated. Quantitative data were subjected to statistical analysis using measures of correlation and the textual data were analyzed qualitatively. The main findings developed from the study indicated that: There is a strong relationship between the grade a student obtains in KCPE mathematics and the grade s/he obtains in the KCSE mathematics. The predictive validity of KCPE mathematics examination results on KCSE mathematics performance is high and significant in national schools. There is a progressive linkage between the content skills and cognitive abilities in mathematics at KCPE and those tested at KCSE level. Variations in mathematics performance among students in national schools by gender is minimal. Key recommendations made in this study are that: Students with low performance grades at KCPE level require increased individual attention by mathematics teachers in order to excel in KCSE mathematics. Teaching strategies used by mathematics teachers at KCSE level should be enhanced to strengthen understanding and performance.
Finding the distribution of a random variable from its moment function
(2012-02-24) Otwombe, Naviava Kennedy; Jairu, D. N.
Consider the problem of r + l randomly distributed points in a unit n-ball and the convex hull created by these points. Let “ n" be r! Times the r-content of an r-simplex whose p vertices are in the interior and r + 1-p vertices on the boundary of a unit n-ball. Explicit expressions for the exact distribution functions of " n" are given when r + 1 points are independently, and identically distributed according to the Uniform distribution. The exact distributions are obtained using the technique is illustrated for the general case p = r + 1 and a particular case p = 3, r = 2. Various representations of the distributions in psi and the generalized zeta functions are given. These representations are also given in the most general case as an H-function distribution.
State space models and estimation of missing observations in time series
(2012-02-24) Biwott, Daniel Kiprotich; Odongo, L. O.
In this project we have considered a non-linear time series model, which encompasses several standard non-linear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithm and the other using optimal estimating equation theory. Recursive estimation of missing observations in an Autoregressive Conditionally Heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. For the case of prediction and fixed point smoothing algorithm, we have generalised the formula developed by Abraham and Thavaneswaran (1991) for estimating missing observations to a case when there are more than two missing observations. Simulation studies have been carried out on AR 91) data to illustrate the application of the formula.
Weibull model for dose response data and akaike information criterion for model selection
(2012-02-28) Stephen, P. M.; Otieno, Romanus Odhiambo
Statistical linear models are used to study dose response models in the bioassay. These have given rise to many statistical problems since the dose response data do not follow linear model. This has led to the use of non-linear models such as probit and Logit. The Logit model has been widely used to analyse the data. Several non-linear models have also been proposed which can be treated in a fashion similar to the parametric logistic model. In this project, we review the parametric logistic model and study the analytical method used in its analysis thoroughly. We study in detail the Weibull dose-response model, following the same method of logistic model. We are able to show that their structures are similar. We study the Akaike Information Criterion for model selection and use it to select a better model between logistic and Weibull models.
Some contributions on the use of auxiliary information in sample surveys.
(2012-02-28) Mulati, Omukoba Nyukuri; Wafula, C.
In this project, we assume that we have a population from which a sample is taken and that not all sample units are observed... It is also assumed that auxiliary information is available for all units in the population. In this case to make inference about the population, we encounter two procedural problems, of estimation and imputation. The big question is do we use auxiliary information for imputation or do we use it for estimation? We have tackle this problem by considering the following strategies: use of hot deck imputation technique to fill in missing values and then of ratio imputation method to impute the missing values and then use of an expansion estimator to estimate the population total. We have suggested some recommendation on when to use auxiliary information. In particular, when non-response is random the use of auxiliary information at imputation is preferred. This is evident from both our theoretical and empirical study carried out to support our decisions and further recommend others.
Turbulent natural convection in a rectangular enclosure
(2012-04-03) Sigey, Johana Kibet
Equations governing natural convection have been solved using a fast and stable finite difference approximation, which has been developed and validated. The technique relies on using different false transient factors in different flow regions. The efficiency was illustrated by solving a three-dimensional heated cavity and the flow and thermal fields in a rectangular enclosure in which one wall is heated and cooled with all the remaining walls adiabatic. A second order central difference approximations in space and first order in time is used to discretized the governing equations together with the boundary conditions. The vorticity transport equations and the energy equation were solved to the steady state using the Sarmaski-Andreyev (1963) Alternative-Direct Implicit (ADI). Solutions are presented for the colliding boundary layer problem for air in the Rayleigh number range 5 x 1010 < Ra > 5 x 1011. The turbulent boundary layers form on the hot and cold end wall. A parametric study of the window problem with a variable area and a fixed center is performed for Ra = 5 x 1011 the velocity and temperature distribution was considered when the heater is placed between the window and the floor of the cavity. In conclusion the vorticity vector potential formulation has been successfully used in solving natural convection problems due to non-uniformities in temperature boundary condition.
Analysis of seasonal time series with missing observations
(2012-04-11) Kihoro, John Mwaniki
In the past, Kenya's Tourist data has been analysed by among others, Mutiso (1982) who carried out Spectral Analysis to the monthly data for a series covering 10 years (1971-1980). Mutiso came up with general conclusion on cycles experience by the tourism industry in Kenya. Onyango (1993) fitted non-linear model to the tourist data before testing it for linearity. In an effort to confirm or disapprove their findings we decided to analyse a longer series, that is, to collect and use data for the years 1971 to 1990. It is in this process that we encountered the problem of missing data. Monthly data for the year 1984 could not be located in the Kenya central Bureau of Statistics (CBS) records, but quarterly totals were available. Our immediate problem was to fill in the block of missing values and as such most of the work in this project point to this direction. In this dissertation we have used the two known segments [(1971 - 1983) and (1985 - 1990)] of tourist data to estimate the block of missing values. We have considered two methods under regression (indirect) imputation methods and imputed the missing values from eventual forecasts. We have showed how two forecasts arising from the two regression methods can be combined to come up with more accurate estimates. We have also suggested a method of adjusting the estimates to incorporate the information from the already known quarterly totals. We have proposed simpler and direct method of imputing the missing values and we have showed that the final adjusted estimates from the direct and the indirect methods are similar. Finally we modelled the two data sets arising from the direct and the indirect imputation and generated eventual forecasts, which further showed that the two methods give almost similar estimates.
Incorporation of convariates in the dose response model
(2012-04-11) Waweru, Peter Matheru
In the study of dose response curves, there are two main assumptions that are normally made. First, we assume that the dose response data can be modeled using linear models (linear probability models), and secondly we assume that the response is only determined by the dose and nothing else. In practical situations this is not the case. In this project, we have considered these two assumptions and explored them in details. Particularly, we have looked at alternative models for the dose response data, and incorporated cavariate effects into the dose response model. Cavorts are the other factors influencing the response in addition to the dose. The incorporation of the cavariates is done in two ways, through the parameters (parametric approach) and through use of the logic difference of the success rate (semi-parametric approach). We have also suggested that we can use the non-parametric methods to analyze dose response data. In particular, we have looked at the estimation of the median dose using the smoothed responses (Graphical approach) and using the Spearman-Karber's method.
Non-parametric regression to finite population estimation.
(2012-04-18) Tobias, Mbithi Mwalili; Otieno, Romanus Odhiambo
Nonparametric regression is used here to estimate the finite population mean. The variance of the derived estimate is obtained and procedures for its estimation suggested. The appropriateness of the variance estimators is established by derivation of their mean square error, which is shown to diminish with rise in sample size. Empirical study is performed using real and simulated data, and the outcomes support theoretical findings.
Residuals influence and weighting in estimation of regression parameters
(2012-04-20) Ombui, Thomas Mageto; Kibua, T. K.
We consider analysis and fitting models of the regression data in the fields, which exhibit non-constant variances, which are referred to as regression models. We focus on various approaches and procedures used in estimating the variances in such models as a way of estimating the regression parameters. Chapter 1 comprises the introduction to the subject. In the first part, Chapter 2 and 3 purely discusses the parametric approach, which is commonly used due to its outstanding features of simplicity in computation, compatibility with model assumptions and for its mathematical convenience. The procedures are fully formulated in chapter 2 and the empirical study using the same procedures is covered in Chapter 3. In the second part, Chapter 4 discusses non-parametric method. The central problems of interest are the choice of the smoothing methods, choice of the Kernel and bandwidth. In Chapter 4 we illustrate both parametric and non-parametric methods in a practical situation. A contrast and the conclusion has been done in the same chapter. All the computation has been done in Splus programming language. Table formats and other organization matters are comfortably done in Microsoft Office (Word) while graphics; figure representation and analysis are computer drawn in Microsoft Office (Excel).

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