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The spirit of Islam binds Muslims into an Ummah. This bonding of a unique feature of any Muslim Society. As Muslims, we should defy splits or differences within our societies to avoid factions or divisions. There must prevail tolerance peaceful co-existence to promote the universal brotherhood amongst Muslims. Only then, the Muslims may rise to supremacy and lead the nations of the world.

Generalized Designing of Systematic Sampling Schemes

After a detailed review of existing sampling schemes, a new class of systematic sampling design, called a Generalized Linear Systematic Sampling (GLSS) for estimation ofnite population mean is introduced. The proposed design is found to be better than Simple Random Sampling (SRS) and is the generalization of the several existing systematic sampling schemes such as Linear Systematic Sampling (LSS), Diagonal Systematic Sampling (DSS) and Generalized Diagonal Systematic Sampling (GDSS). All of these designs become special cases of the proposed design. In this design an optimum choice of sampling interval under linear trend is also be discussed. Sampath and Varalakshmi (2008) proposed an equal probability scheme called Diagonal Circular Systematic Sampling (DCSS) under the conditions stated by Sudhakar (1978). However, it is observed that DCSS does not ful ll these conditions. Therefore, a necessary and su cient condition has been suggested for DCSS after a slight modi cation in the theorem proposed by Sengupta and Chattophadyay (1987). Under this condition, one can easily decide when and where DCSS is applicable. Some de ciencies in traditional selection procedure of circular version of systematic sampling schemes are also investigated and alternative methods are proposed. Some rules of thumb for coincidence of units in the sample are also introduced. The end corrections proposed by Bellhouse and Rao (1975) and Sampath and Varalakshmi (2008) for circular systematic sampling (CSS) and DCSS respectively are also modi ed. Theoretical selection procedure has also been established for several cyclic CSS regarding the suggestion of Sudhakar (1978). Mean and variance expressions of CSS for perfect linear trend are not available in the literature. Therefore, a new approach is introduced to study the characteristic of circular version of systematic sampling. By using it, mean and variance expressions of CSS for perfect linear trend has been derived. Mean and variance of DCSS can be deduced from these expressions. Average variance expressions of corrected sample vii means for modi ed CSS and DCSS are derived under the super population model. Based on the average variances, numerical e ciency comparison of CSS and DCSS has also been carried out. In the current study a new sampling design called Modi ed Systematic Sampling (MSS) is proposed. In this design each unit has an equal probability of selection. Moreover, it works for both situations: N = nk or N 6= nk. Modi ed Systematic Sampling reduces to LSS, if N = nk and becomes CSS, if N and n are co-prime. The proposed MSS performs better than CSS in every aspect of systematic sampling, speci cally, simplicity, e ciency and even coverage of sample unit over the entire population. E ciency comparison of MSS with CSS is also carried out for natural populations. Furthermore, MSS is also studied for populations having a linear trend. Expressions for mean and variance of sample mean are obtained for the population having perfect linear trend among population values. Average variance of corrected sample mean under super population linear model and average variance of sample mean under super population auto-correlated model are also obtained. Further,numerical e ciency comparisons using these average variances are also obtained for di erent sample sizes. One of the major and long-standing problem of unbiased estimation of population variance is also discussed in the current study. In this case, the concept of multiple random start is extended from linear version (where N = nk) to the general case (where N 6= nk). As a result, a new sampling design called \Universal Systematic Sampling (USS)" is introduced. Linear systematic sampling and Simple Random Sampling (SRS) are the two extreme cases of this design. Mean and variance of mean for LSS and SRS can be extracted from the derived expressions of mean and variance of mean of USS. An explicit expressions of unbiased estimator of population variance and its variance are also derived. Finally, an e ciency comparison with SRS is also carried out for natural populations, simulated populations and population having linear trend.