# Nonparametric Partial Sequential Test for Location Shift at an Unknown Time Point

##### Citations

24 citations

15 citations

### Cites background from "Nonparametric Partial Sequential Te..."

...We assume that, for each m there are positive numbers ri = ri m and positive integers i = i m i = 1 2 s such that, as m → , i → but ri m i m → i ∈ 0 (4.1) Set T imk = 1√ ri ( Fm Yik − 1 2 ) k = 1 2 i i = 1 2 s Then, as in Bandyopadhyay and Mukherjee (2007), we observe the following simple results....

[...]

...Then, as in Bandyopadhyay and Mukherjee (2007), we observe the following simple results....

[...]

...In fact, as in Bandyopadhyay and Mukherjee (2007), it can be seen that ami i = 1 2 s converges to an s-variate normal distribution with mean vector 0 and...

[...]

...…null distribution of ∑ i k=1 T i mk is normal with mean ami and variance v 2 m where ami = im 1√ri ∑m k=1 1 2 − F Xi and v2m = E Fm Yk − 12 2 In fact, as in Bandyopadhyay and Mukherjee (2007), it can be seen that ami i = 1 2 s converges to an s-variate normal distribution with mean vector 0…...

[...]

12 citations

### Cites background or methods from "Nonparametric Partial Sequential Te..."

...Recently, Bandyopadhyay and Mukherjee (2007) updated the partially sequential stopping rule using the concept of sequential ranks. Such sequential ranks were used earlier in connection with a quasi-sequential stopping rule by Bhattacharya and Frierson (1981). However, all such sequential stopping rules terminate with probability one under both the null and alternative hypotheses....

[...]

...Recently, Bandyopadhyay and Mukherjee (2007) updated the partially sequential stopping rule using the concept of sequential ranks. Such sequential ranks were used earlier in connection with a quasi-sequential stopping rule by Bhattacharya and Frierson (1981). However, all such sequential stopping rules terminate with probability one under both the null and alternative hypotheses. Thus, the monitoring stops even when there is no fluctuation in the populations. This is really unwarranted in much econometric as well as environmental monitoring. Here a process needs to be monitored ceaselessly. Particularly, when we assume a negligible cost of sampling under no fluctuation, we do not need to stop at all unless there is a signal. Chu et al. (1996) emphasizes the need for developing partial sequential tests which rarely terminate when no fluctuation is observed....

[...]

...Recently, Bandyopadhyay and Mukherjee (2007) updated the partially sequential stopping rule using the concept of sequential ranks. Such sequential ranks were used earlier in connection with a quasi-sequential stopping rule by Bhattacharya and Frierson (1981). However, all such sequential stopping rules terminate with probability one under both the null and alternative hypotheses. Thus, the monitoring stops even when there is no fluctuation in the populations. This is really unwarranted in much econometric as well as environmental monitoring. Here a process needs to be monitored ceaselessly. Particularly, when we assume a negligible cost of sampling under no fluctuation, we do not need to stop at all unless there is a signal. Chu et al. (1996) emphasizes the need for developing partial sequential tests which rarely terminate when no fluctuation is observed. They also discussed the importance of controlling Type I error in monitoring structural changes. Thus, we require framing a rule in such a way that a termination will signal instability. Hence we certainly need to construct tests with asymptotically or approximately power one and simultaneously to achieve control over Type I error rate. Sequential procedures are effective in several real bio-statistical and econometric problems. Sequential plans often save precious sample sizes and give efficient inference. On the other hand, in most of the real life situations, the normality assumption does not work well. It is difficult to suggest suitable non normal model in many situations. Moreover, inference based on non normal models is not always easy. Thus, nonparametric models have a wide applicability in the present era. At the same time, research related to controlling Type I error rate in statistical inference are getting more and more importance. As the pattern of the alternative is unknown or vague in most cases, deriving optimal tests become complicated. Statisticians are, in these days, avoiding the long practice of using mathematically sound optimal tests. Those tests, mainly in the area of sequential clinical trials, are continually being replaced by some heuristic tests that are more practical and easy to handle. If properly designed, such heuristic tests are likely to have greater appeal in testing some econometric or environmental hypothesis as well. One may see Huang et al. (2005) for an illustration of controlling Type I error in adjusted O’Brien’s test....

[...]

...Bandyopadhyay and Mukherjee (2007) show for linear stopping boundary that sequential rank test can even improve power in such occasion if a shift occurs at a later stage....

[...]

...…by the following stopping variable: M = min { n n∑ k=1 Fm Yk ≥ r/2 } (2.1) where r is a prefixed positive number and Fm · is the empirical df based on Xm. Bandyopadhyay and Mukherjee (2007) suggest a stopping rule by updating the empirical df using the available second stage observations also....

[...]

^{1}

11 citations

### Cites background or methods from "Nonparametric Partial Sequential Te..."

...and sequential rank-based Bandyopadhyay and Mukherjee (2007)-type stopping rule becomes...

[...]

...We may readily see from Bandyopadhyay and Mukherjee (2007) that, under H0, both Tn and Sn have symmetric distributions around the common expected value n/2, although Tn has larger variance than Sn....

[...]

...Among others, to name a few are Page (1955), Bhattacharya and Johnson (1968), and Bandyopadhyay and Mukherjee (2007). As in Bandyopadhyay and Mukherjee (2007), we define Fm · be the empirical d....

[...]

...4 of Bandyopadhyay and Mukherjee (2007) and can easily be extended for bivariate situation employing Cramer–Wold device....

[...]

...Among others, to name a few are Page (1955), Bhattacharya and Johnson (1968), and Bandyopadhyay and Mukherjee (2007). As in Bandyopadhyay and Mukherjee (2007), we define Fm · be the empirical d.f. based on m. Further suppose Hm+k−1 · be the empirical distribution function (d.f.) based on m and 1 2 k−1 , k ≥ 1. Then Tn = ∑n k=1 Fm k gives the usual rank sum statistic, whereas the sequential rank sum statistic is given by Sn = ∑n k=1 Hm+k−1 k . Hence usual rank-based Orban and Wolfe (1980)-type stopping rule becomes...

[...]

11 citations

##### References

1,763 citations

[...]

1,653 citations

758 citations

674 citations

### "Nonparametric Partial Sequential Te..." refers background in this paper

...Testing a problem of this type related to the location shift at an unknown time point was first introduced by Page (1955) for a known initial level....

[...]

180 citations

### "Nonparametric Partial Sequential Te..." refers background in this paper

...Our work is motivated by Wolfe (1977) as well as Orban and Wolfe (1980)....

[...]

...For various inferential problems based on usual ranks, one can also go through the book by Sen (1981)....

[...]