Applications: Large-Sample Confidence Intervals for θkl

We would like to construct a confidence interval for % centered at its corresponding estimator H_w. Having fixed 0 < a < 1, with n large, we explore the performances of the following approximate 100(1 - a)% confidence interval for the IIM, 0_kf.

where z_a/₂ is the upper 100(a /2)% point of the standard normal distribution.

In order to compare effectiveness of the proposed IIMs in terms of the attained coverage probabilities and confidence intervals' lengths, we conducted a simulation study with sample sizes n = 20,30,50,75 and confidence coefficients fixed at 90%, 95%, and 99%.

The results shown in Tables 1.10-1.12 and 1.13-1.15 are respectively based on random samples drawn from the gamma distribution (shape = 2.649, rate = 0.84) and the lognormal distribution (location = 2.185, shape = 0.562). In all cases, the number of replications used was fixed as 2000 and these tables show the averages of lengths (L) and coverage probabilities (p) along with the standard errors.

We observe that the average coverage probability associated with the Gini index (= Hu) falls below that associated with CV whereas the average lengths of such confidence intervals tend to go the other way. Indeed, with regard to CV, we observe that the average coverage probability in most cases tends to be substantially better than the target probability 1 - a, but the associated confidence intervals appear wider on average in comparison with those associated with the confidence intervals centered at the Gini index.

Next, we briefly summarize the performance of the confidence intervals corresponding to our proposed IIMs, namely H21, H22, and H31. We note that although the IIM H31 gives on an average the shortest confidence interval (for 03i) in all cases, the corresponding average coverage probability appears rather low compared with the target 1 - a. Performances of confidence intervals for 021 centered at the IIM H21 look similar, but a bit worse, on an average to those centered at H31.

TABLE 1.10

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for 0w under the Gamma Distribution from 2000 Replications: a = 0.10

TABLE 1.11

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for вц under the Gamma Distribution from 2000 Replications: a = 0.05

TABLE 1.12

Average Length (L) and Average Coverage Probability (p) Associated with the Confidence Interval from (1.25) for the IIMs for 0_k! Under the Gamma Distribution from 2000 replications: a = 0.01

Based on empirical evidence from Tables 1.10-1.12 and Tables 1.13-1.15, we look at the ИМ H22 more favorably as an ad hoc inequality measure since it provides on average a shorter confidence interval than H31 with near-target coverage probability. We feel that the confidence intervals for в₂2 centered at our proposed ИМ H22 go nearly head-to-head with the confidence intervals for в centered at Gini index (=Нц). This sentiment is clearly consistent with our interpretation of H22 laid down precisely in Section 1.2. After all, H22 does 'Took like" a Gini index based on the dataset shown in (1.9) constructed from a U-statistic with degree 4.

Interesting and more complex estimation problems for Gini Index from (1.1) have been handled in the papers of Chattopadhyay and De (2016) and De and Chattopadhyay (2017). They constructed estimators of Gini Index with controlled risks under appropriately designed sequential sampling methodologies.