شریعت کی حقیقت اور تعبیرات: افکارِ مجدد الف ثانی کا ایک فنی زاویہ نگاہ

When we analyse the preaching of religion by the Holy Prophets (peace an blessing of Allah be upon them) in the background of history of mankind, we come to know that Allah Almighty offered at least a Shari'ah (a code of social, moral and religious conduct) to every Prophet or bound them to a Shari'ah previousl preached by another Prophet before them. Shari'ah is, in fact, based upon the principles and practices to lead life. There had been at least some differences in the Shari'ah different Prophets with regard to the difference of their time and place. It is a proven fact that the followers of a Prophet had always been bound to follow the Shari'ah of their own Prophets. The Shareeya of the Prophet of Islam (peace and blessing of Allah be upon him) is absolutely consummate in every respect and it also fulfills the needs of all times to come. It is because of the fact that Mujtahidin who kept on resorting to Ijtehad. The topic under consideration is an intellectual research analysis of the efforts of Shaykh Ahmad Sirhindi in the field of Ijtehad in the history of Islam.

Use of Adaptive Cluster Sampling under Different Sampling Designs

In this dissertation, a class of Hartley Ross type unbiased estimators is proposed for estimation of finite population mean under adaptive cluster sampling and stratified adaptive cluster sampling. Hartley Ross type unbiased estimator is also proposed utilizing two auxiliary variables. These estimators employ information on known parameters of the auxiliary variable. The variances of proposed class of unbiased estimators are obtained up to first degree of approximation. Computations related to proposed estimators are illustrated via numerical example. Proposed estimators are more efficient than the usual mean estimator, ratio and modified ratio estimators in adaptive cluster sampling and stratified adaptive cluster sampling under certain realistic conditions. Exponential-ratio-type and difference-type estimators are propounded for general parameter in adaptive cluster sampling and stratified adaptive cluster sampling. The proposed estimators coherently utilize information on two auxiliary variables in three different situations i-e. none, partial and full information about population parameters of auxiliary variables. The proposed estimators for general parameter can be used to estimate the population mean, population coefficient of variation, population standard deviation and population variance of the variable of interest. Proposed estimators are also presented to be used with multi auxiliary variables. Difference-type estimators are recommended for estimation of population coefficient of variation under adaptive cluster sampling. Proposed estimators utilize mean, ranks and coefficient of variation of auxiliary variables. Difference-type and difference-cum-exponential-ratio-type estimators are presented utilizing two auxiliary variables for estimation of general parameter under adaptive cluster sampling and stratified adaptive cluster sampling. These estimators utilize auxiliary information in terms of ranks, variances and means of auxiliary variables. Such estimators are generalized for multi auxiliary variables. xviii Generalized ratio-type and ratio-exponential-type estimators are proposed for population mean under adaptive cluster sampling based on modified Horvitz-Thompson estimator. The proposed estimators utilize auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, mid-range) to increase efficiency. Finally, three new sampling schemes are proposed to select initial sample in adaptive cluster sampling. These schemes are proposed adopting, ranked set sampling to increase precision of estimates. Usual Hansen-Hurwitz and Horvitz-Thompson estimators for population mean under adaptive cluster sampling are modified for employment under the proposed schemes. Procedures related to the proposed schemes are also illustrated with the help of examples. Expressions for bias and mean square error of proposed estimators are derived using first order of approximation. Empirical and simulation studies are conducted to evaluate the proposed estimators. Behaviors of existing and proposed estimators are analyzed for several initial sample sizes and at different levels of correlation between study and auxiliary variables. Comparisons of existing and proposed estimators are also illustrated. The results reveal that whenever the efficiency conditions are fulfilled, proposed estimators performed more efficiently than competing estimators for estimation of population mean, population variance and population coefficient of variation. The proposed estimators are found to be more efficient under both adaptive cluster sampling and stratified adaptive cluster sampling. The sampling schemes which are recommended by adopting ranked set sampling are found to be more efficient than adaptive cluster sampling when initial sample is drawn by simple random sampling without replacement.