The Probability of a Confidence Interval Based on Minimal Estimates of the Mean and the Standard Deviation

A confidence interval is an interval in which a measurement falls with a given probability [1]. This paper addresses the question of defining the probability of a confidence interval that can be constructed when only a minimal number of measurements are known. The problem has a couple of significant applications. In geophysics, time-lapse 3D seismic monitoring is used to monitor oil and gas reservoirs before and after production. Because of the high cost of 3D seismic monitoring, a minimal-effort time-lapse 3D seismic monitoring approach was proposed by Houston and Kinsland [2]. This approach proposes a minimal measurement interval based on preliminary measurements. It implies successively smaller seismic surveys asmonitoring continues and knowledge of the behavior of the reservoir grows. The success of the minimal-effort method is based on the probability that the reservoir is detected by a seismic survey. The minimal-effort method is based on a minimal number of measurements, which connects it to the topic of this paper. Another significant application of a confidence interval based on a minimal number of measurements is the problem of cancer treatment based on the irradiation of a tumor [3]. The purpose of irradiation is to destroy the tumor, but targeting the tumor can be problematic because tumors often exhibit small random motions within the body. Because of random tumor motion, the result of radiation treatment often includes the destruction of healthy tissue. Clearly, it is beneficial that cancer radiation treatment incorporates minimal intervals. The success of the cancer treatment is based on the probability that the tumor is irradiated while using a minimal irradiation interval. This success hinges on the ability of the radiation treatment to incorporate preliminary measurements that are often minimal. It is upon this basis that the cancer treatment problem connects to the topic of this paper. In this paper, we construct a confidence interval based on minimal estimates of the mean and the standard deviation. Explicitly, we use two measurements to specify an interval that contains a subsequent measurement with a given probability. The mathematical effort includes the derivation of a specific probability for the confidence interval. The probability is computed using the cumulative distribution function, and this probability is averaged over both the estimate of the mean and the sample standard deviation to yield a specific value.The results are compared to a computer simulation that estimates the probability based on frequency.


Introduction
A confidence interval is an interval in which a measurement falls with a given probability [1].This paper addresses the question of defining the probability of a confidence interval that can be constructed when only a minimal number of measurements are known.
The problem has a couple of significant applications.In geophysics, time-lapse 3D seismic monitoring is used to monitor oil and gas reservoirs before and after production.Because of the high cost of 3D seismic monitoring, a minimal-effort time-lapse 3D seismic monitoring approach was proposed by Houston and Kinsland [2].This approach proposes a minimal measurement interval based on preliminary measurements.It implies successively smaller seismic surveys as monitoring continues and knowledge of the behavior of the reservoir grows.The success of the minimal-effort method is based on the probability that the reservoir is detected by a seismic survey.The minimal-effort method is based on a minimal number of measurements, which connects it to the topic of this paper.
Another significant application of a confidence interval based on a minimal number of measurements is the problem of cancer treatment based on the irradiation of a tumor [3].The purpose of irradiation is to destroy the tumor, but targeting the tumor can be problematic because tumors often exhibit small random motions within the body.Because of random tumor motion, the result of radiation treatment often includes the destruction of healthy tissue.Clearly, it is beneficial that cancer radiation treatment incorporates minimal intervals.The success of the cancer treatment is based on the probability that the tumor is irradiated while using a minimal irradiation interval.This success hinges on the ability of the radiation treatment to incorporate preliminary measurements that are often minimal.It is upon this basis that the cancer treatment problem connects to the topic of this paper.
In this paper, we construct a confidence interval based on minimal estimates of the mean and the standard deviation.Explicitly, we use two measurements to specify an interval that contains a subsequent measurement with a given probability.The mathematical effort includes the derivation of a specific probability for the confidence interval.The probability is computed using the cumulative distribution function, and this probability is averaged over both the estimate of the mean and the sample standard deviation to yield a specific value.The results are compared to a computer simulation that estimates the probability based on frequency.

Probability Based on the Cumulative Distribution Function
The cumulative distribution function [4] is given as where  is a measurement,  is the mean, and  is the standard deviation.Let V = ( − )/ √ 2.So, V = / √ 2, and (1) becomes where erf is the error function [5].We know that the cumulative distribution function designates probability as Therefore, we can write where  is the estimate of the mean given as and  is the sample standard deviation given as This allows us to write Let  = 2.This implies that So we have This equation can be simplified.Let  =  1 +  2 and  =  1 −  2 .Then, (9) becomes ) . (10)
Let  = ( − 2)/2, so that  = /2.Equation (11) becomes At this point, consider the following integral: We will find three conditions on  that determine its structure.First, the following limit is clear from (14).
Now, let us determine the following integral: where V = (/√) +  and V = /√.Using the known integral of the error function, we find that Therefore,  must vary as  and √.Now, because the error function is odd, we can write Therefore, (19) can be added to  to obtain we have the following.
Based on Conditions 1, 2, and 3,  must have the following form: where  > 0. Equations ( 14) and ( 23) suggest that we can write Based on (24), we can numerically determine  as For  = 200 and Δ = 0.1, we find and, thus,  = 1, so that or or

The Probability Averaged over the Sample Standard Deviation
The expectation with respect to  can be written as where we have used the fact that the standard deviation of  is √ 2 based on previous information and the fact that var() =  2 var() [6].Equation (33) can be expressed on a semi-infinite interval as Let  = /2.So,  = /2, and (34) becomes or we can write

Computational Simulation
We can estimate ⟨( =2 −  =2 ≤  ≤  =2 +  =2 )⟩ computationally.Simulate the normal, independent random variables  and {  }, for which Let the condition  be  : If () is the number of trials in which the condition  is met and  is the total number of trials, then an estimate of Figure 1 shows a plot of (2/√) ∫ ∞ 0 erf(√2/3) − 2  versus ()/ for  = 2000.

Conclusions
We have computed the probability of a confidence interval based on minimal estimates of the mean and the standard deviation.Specifically, the estimates are based on two measurements.The effort addresses the problem of constructing a confidence interval based on minimal knowledge.The result is in the form of an integral of the error function that we have evaluated numerically and compared to a computer simulation that estimates the probability based on frequency.The comparison shows a high level of agreement that supports the validity of the derivation.
This work is part of a research effort interested in constructing confidence intervals based on various levels of knowledge.Example applications exist in geophysical exploration and cancer treatment.