We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equationsf(ax+by)+f(ax−by)=(b(a+b)/2)f(x+y)+(b(a+b)/2)f(x−y)+(2a2−ab−b2)f(x)+(b2−ab)f(y)wherea, bare nonzero fixed integers withb≠±a,−3a, andf(ax+by)+f(ax−by)=2a2f(x)+2b2f(y)for fixed integersa, bwitha,b≠0anda±b≠0.
Abstract. In this paper, we prove a generalization of Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487].
In this paper, we have proposed an estimator of finite population mean in stratified random sampling. The expressions for the bias and mean square error of the proposed estimator are obtained up to the first order of approximation. It is found that the proposed estimator is more efficient than the traditional mean, ratio, exponential, regression, Shabbir and Gupta (in Commun Stat Theory Method 40:199-212, 2011) and Khan et al. (in Pak J Stat 31:353-362, 2015) estimators. We have utilized four natural and four artificial data sets under stratified random sampling scheme for assessing the performance of all the estimators considered here.
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