^{1}

^{*}

^{2}

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^{2}

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Conflicting views had greeted the use of systematic sampling for sample selection and estimation in stratified sampling in terms of the precision of the population mean base on the inherent characteristics of the population. These conflicting views were analyzed using Cochran data (1977, p. 211) [1]. When the population units are ordered, variance of systematic sampling for all possible systematic samples provides equal, non-negative and most precise estimates for all the variance functions considered
* i.e.* , unlike when a single systematic sample is used and when variance of simple random sampling is used to estimate selected systematic samples.

Cochran (1977) [^{th} unit thereafter. The selection of the first k^{th} units determined the whole sample. This is called an every k^{th} systematic sample.

Murthy (1967) [^{th} unit starting with the unit corresponding to a number r chosen at random from 1 to k, where k is taken as the integer nearest to

A sample selected by this procedure is termed a systematic sample with a random start r. Therefore, the value of r determines the whole sample. In other words, this procedure amounts to selecting with equal probability one of the k possible groups of units (samples) into which the population can be divided in a systematic manner.

Same view was expressed by Raj and Chandhok (1998) [^{th} unit thereafter. It is therefore called a 1-in-ksystematic sample.

Early studies on the development of theory of systematic sampling was as reported by Murthy (1967, p.134) [

Guatschi (1957) [

gram, single random start

Murthy (1967) [

Thus for a population of Y units

Considering all the k possible samples, the sample mean

Showing that when N = nk,

Random Start (Sample Number) | |||||
---|---|---|---|---|---|

1 | 2 | 3 | i | k | |

Means_{ } | _{ } | _{ } | _{ } | _{ } | _{ } |

Above is the applicable systematic sampling in a situation in which N = nk. In practice, it is common to encounter situations in which

Lahiri (1952) [

for

It implies from CSS, therefore, that the usual procedure of selecting a random start r from 1 to k and including in the sample the units corresponding to

Murthy (1967) [^{ }were the quotient and remainder obtained respectively on dividing

Then, the units’

This approach is suitable in situations in which the sample size n is not fixed or predetermined and the sampler is free to adjust the sample to suit the above application. Therefore, Murthy’s approach to handle

Another approach when

Leu and Tsui (1996) [

Also reviewed in this section is Remainder Linear Systematic Sampling (RLSS) due to Chang and Huang (2000) [

a) Divide the population units into two strata with the first stratum containing the front

b) From stratum II, random start

Sample of size n is the combination of

Therefore, in stratified systematic sampling when

Estimation of the population mean of a systematic sample over all possible samples is as given by relation (1). For the variance of the population mean, Murthy (1967) [

where

Equivalent to

It is simplified as

where

Note that

and

Therefore

where

Other expressions for the estimation of variance of the mean of systematic samples by various authors are reported by Murthy (1967, Section 5.8, pp. 153-155) [

On the efficiency of systematic sampling in relation to other sampling scheme, literature agreed that efficiency of systematic sampling was strongly anchored on the arrangement of the population units. Cochran (1977) [

Same view was expressed by Murthy (1967) [

Cochran (1977) [

1) The variances of the mean of systematic sample given by Cochran are:

where

This can further be expressed as

which is the weighted variance over all possible systematic samples generated by random start

2) The second one is given as

where

where the numerator is averaged overall

The two expressions of

3) The third is expressed in terms of variance of stratified random sample in which the strata are composed of the first k units, the second k units and so on.

The subscript j in

where

This is the variance among units that lie in the same stratum. The divisor

This quantity is the correlation between the deviations from the stratum means of pairs of units that are in the same systematic sample.

It implies therefore from relation (9) above that a systematic sample has the same precision as that of a stratified random sampling sample with one unit per stratum if

Thus, we have examined systematic sampling in terms of procedure and estimation process. But our concern is taking a systematic sample of fixed sample size n from each stratum for estimation purpose.

Much have been said in Section 2 on the significance of the arrangement of the population units on the precision

Notations

Cochran (1977, p. 91) [

The subscript h denotes the stratum and i the unit within the stratum.

The subscript “sy” in this section denotes systematic sample.

is the mean of systematic sample in stratum h, equivalent to relation (1).

is the population mean of the stratified systematic sample.

is the variance of stratified systematic samples in stratum h when

Therefore,

is the variance of the population mean of stratified systematic samples.

is the MSE of the population mean of stratified systematic samples.

The mean and the variance of RLSS are given below (see relation 2.2 and 2.3, p. 251 of Chang and Huang (2000) [

To suite our applications, expression (17) and (18) are modified as follows:

It should be noted that that expressions

Systematic samples are easy to draw and to execute but may not be simple in term of estimation as there are competing estimators. This drew our attention for an empirical investigation to ensure the right choice of estimator in the face of conflicting reports. Murthy (1967, section 5.8, p. 153) [

Raj and Chandhok (1998) [

In view of the above, the question is: should a single systematic sample be used to estimate

Empirical investigation reveals that when we select a single systematic sample, the result is as shown in

Since the efficiency of systematic sampling depends on the arrangement of the population units, an attempt is also made to rearrange Cochran (1977) [

while

This analysis brings to the lime light the caution by Murthy (1967, p. 145) [

Groups | ||||
---|---|---|---|---|

g_{1} | −314.7056 | 30.0944 | 33.0578 | 30.825 |

g_{2} | −309.3056 | 11.5944 | 31.1928 | 36.375 |

g_{3} | −307.0744 | 4.6569 | 30.6481 | 38.4563 |

g_{4} | −305.5244 | 14.6569 | 30.6975 | 35.4563 |

*g_{5} | −295.1744 | 19.6569 | 30.4062 | 33.9563 |

g_{6} | −298.8244 | 19.6569 | 30.3950 | 33.9563 |

g_{7} | −288.3056 | 22.0944 | 30.8478 | 33.225 |

g_{8} | −281.9306 | 2.3444 | 31.0459 | 39.15 |

g_{9} | −278.5744 | −0.8431 | 31.4389 | 40.1063 |

g_{10} | −281.9306 | −7.6556 | 30.7959 | 42.15 |

*In _{5} indicates the center for systematic sample estimates when Madow’s procedure is used while the subscript i = 1, ・・・, k = 10 is the random start in the interval 1 to 10.

Groups | ||||
---|---|---|---|---|

g_{1} | −311.4244 | 14.6569 | 31.7250 | 35.4563 |

g_{2} | −309.3056 | 11.5944 | 31.1928 | 36.375 |

g_{3} | −307.0744 | 4.6569 | 30.6481 | 38.4563 |

g_{4} | −305.5244 | 14.6569 | 30.6975 | 35.4563 |

*g_{5} | −298.8244 | 19.6569 | 30.3950 | 33.9563 |

g_{6} | −295.1744 | 19.6569 | 30.4062 | 33.9563 |

g_{7} | −289.3244 | 15.1569 | 30.5918 | 35.3063 |

g_{8} | −285.1744 | 0.1569 | 30.6031 | 39.8063 |

g_{9} | −281.9306 | −2.6556 | 30.9209 | 40.65 |

g_{10} | −279.7056 | −2.4056 | 31.2328 | 40.575 |

*In _{5} indicates the center for systematic sample estimates when Madow’s procedure is used while the subscript i = 1, ・・・, k = 10 is the random start in the interval 1 to 10.

shown in