File may be more up-to-date. One of the most confusing concepts to a novice reliability engineer is estimating the precision of an estimate. This is an important concept in the field of reliability engineering, leading to the use of confidence intervals or bounds. In this section, we will try to briefly present the concept in relatively simple terms but based on solid common sense. To illustrate, consider the case where there are millions of perfectly mixed black and white marbles in a rather large swimming pool and our job is to estimate the percentage of black marbles.

The only way to be absolutely certain about the exact percentage of marbles in the pool is to accurately count every last marble and calculate the percentage. However, this is too time- and resource-intensive to be a viable option, so we need to come up with a way of estimating the percentage of black marbles in the pool.

In order to do this, we would take a relatively small sample of marbles from the pool and then count how many black marbles are in the sample. First, pick out a small sample of marbles and count the black ones.

Say you picked out ten marbles and counted four black marbles. Which of the two is correct? Both estimates are correct! As you repeat this experiment over and over again, you might find out that this estimate is usually between andand you can assign a percentage to the number of times your estimate falls between these limits. If you now repeat the experiment and pick out 1, marbles, you might get results for the number of black marbles such as,etc.

The range of the estimates in this case will be much narrower than before. The same principle is true for confidence intervals; the larger the sample size, the more narrow the confidence intervals. We will now look at how this phenomenon relates to reliability. Overall, the reliability engineer's task is to determine the probability of failure, or reliability, of the population of units in question.

However, one will never know the exact reliability value of the population unless one is able to obtain and analyze the failure data for every single unit in the population.

Since this usually is not a realistic situation, the task then is to estimate the reliability based on a sample, much like estimating the number of black marbles in the pool. If we perform ten different reliability tests for our units, and analyze the results, we will obtain slightly different parameters for the distribution each time, and thus slightly different reliability results.

However, by employing confidence bounds, we obtain a range within which these reliability values are likely to occur a certain percentage of the time.

This helps us gauge the utility of the data and the accuracy of the resulting estimates. Plus, it is always useful to remember that each parameter is an estimate of the true parameter, one that is unknown to us.

This range of plausible values is called a confidence interval. When we use two-sided confidence bounds or intervalswe are looking at a closed interval where a certain percentage of the population is likely to lie. That is, we determine the values, or bounds, between which lies a specified percentage of the population.

One-sided confidence bounds are essentially an open-ended version of two-sided bounds. A one-sided bound defines the point where a certain percentage of the population is either higher or lower than the defined point. This means that there are two types of one-sided bounds: An upper one-sided bound defines a point that a certain percentage of the population is less than.

Conversely, a lower one-sided bound defines a point that a specified percentage of the population is greater than. Care must be taken to differentiate between one- and two-sided confidence bounds, as these bounds can take on identical values at different percentage levels.

For example, in the figures above, we see bounds on a hypothetical distribution. If obtaining the confidence bounds on probability of failure we will again identify the lower numeric value for the probability of failure as the lower limit and the higher value as the higher limit. This section presents an overview of the theory on obtaining approximate confidence bounds on suspended multiple censored data.

These bounds are employed in most other commercial statistical applications. In general, these bounds tend to be more optimistic than the non-parametric rank based bounds. This may be a concern, particularly when dealing with small sample sizes.

Some statisticians feel that the Fisher matrix bounds are too optimistic when dealing with small sample sizes and prefer to use other techniques for calculating confidence bounds, such as the likelihood ratio bounds. In utilizing FM bounds for functions, one must first determine the mean and variance of the function in question i.

An example of the methodology and assumptions for an arbitrary function is presented next. For simplicity, consider a one-parameter distribution represented by a general function which is a function of one parameter estimator, say For example, the mean of the exponential distribution is a function of the parameter. Then, in general, the expected value of can be found by:.

The term is a function ofthe sample size, and tends to zero, as fast as as For example, in the case of andthen where. Thus aswhere and are the mean and standard deviation, respectively. Using the same one-parameter distribution, the variance of the function can then be estimated by:. Consider a Weibull distribution with two parameters and.

For a given value of. Repeating the previous method for the case of a two-parameter distribution, it is generally true that for a functionwhich is a function of two parameter estimators, saythat:. Note that the derivatives of the above equation are evaluated at and where E and E. The determination of the variance and covariance of the parameters is accomplished via the use of the Fisher information matrix. For a two-parameter distribution, and using maximum likelihood estimates MLEthe log-likelihood function for censored data is given by:.

In the equation above, the first summation is for complete datathe second summation is for right censored data and the third summation is for interval or left censored data.

The subscript 0 indicates that the quantity is evaluated at and the true values of the parameters.

So for a sample of units where units have failed, have been suspended, and have failed within a time interval, and one could obtain the sample local information matrix by:.

Then the variance of a function can be estimated using equation for the variance. Values for the variance and covariance of the parameters are obtained from Fisher Matrix equation. Once they have been obtained, the approximate confidence bounds on the function are given as:. We address finding next. Thus if is the MLE estimator forin the case of a single parameter distribution estimated from a large sample of units, then:. We now place confidence bounds on at some confidence levelbounded by the two end points and where:.

Now by simplifying the equation for the confidence level, one can obtain the approximate two-sided confidence bounds on the parameter at a confidence level or:. If must be positive, then is treated as normally distributed.

The two-sided approximate confidence bounds on the parameterat confidence levelthen become:. The one-sided approximate confidence bounds on the parameterat confidence level can be found from:. The same procedure can be extended for the case of a two or more parameter distribution. Lloyd and Lipow [24] further elaborate on this procedure. Type 1 confidence bounds are confidence bounds around time for a given reliability. For example, when using the one-parameter exponential distribution, the corresponding county durham core strategy issues and options for a given exponential percentile i.

Bounds on time Type 1 return the confidence bounds around this time value by determining the confidence intervals around and substituting these values into the above equation. The bounds on are determined using the method for the bounds on parameters, with its variance obtained from the Fisher Matrix. Note that the procedure is slightly more complicated for distributions with more than one parameter. Type 2 confidence bounds are confidence bounds around reliability.

For example, when buying cheap stocks online the two-parameter exponential distribution, the reliability function is:.

Reliability bounds Type 2 return the confidence bounds by determining the confidence intervals around and substituting these values into the above equation. Once again, the procedure is more complicated for distributions with more than one parameter. Another less mathematically intensive method of calculating confidence bounds involves a procedure similar to that used in calculating median ranks see Parameter Estimation.

This is a non-parametric approach to confidence interval calculations that involves the use of rank tables and is commonly known as beta-binomial bounds BB. By non-parametric, we mean that no underlying distribution is assumed.

Parametric implies that an underlying distribution, with parameters, is assumed. In other words, this method can be used for any distribution, without having to make adjustments in the underlying equations based on the assumed distribution. The median rank was obtained by solving the following equation for:.

The same methodology can then be repeated by changing for 0.

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For one would formulate the equation as. Keep in mind that one must be careful to select the appropriate values for based on the type of confidence bounds desired.

Using this methodology, the appropriate ranks are obtained and plotted based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound. Full details on this methodology can be found in Kececioglu [20].

These binomial equations can again be transformed using the beta and F distributions, thus the name beta binomial confidence bounds. Another method for calculating confidence bounds is the likelihood forex news trade trading academy review bounds LRB method.

Conceptually, this method is a great deal simpler than that of the Fisher matrix, although that does not mean that the results are of any less value. In fact, the LRB method is often preferred over the FM method in situations where there are smaller sample sizes. If is the confidence level, then for two-sided bounds and for one-sided. Recall from the Brief Statistical Background chapter that if is a continuous random variable with pdf:.

The likelihood function is given by:. The maximum likelihood estimators MLE of are are obtained by maximizing These are represented by the term in tsla option trade denominator of the ratio in the likelihood ratio equation.

Since the values of the chutney makers lancashire points are known, and the values of the parameter estimates have been calculated using MLE money management rules forex trading, the only unknown term in the likelihood ratio equation is the term in the numerator of the ratio.

It remains to find the values of the unknown parameter vector that satisfy the likelihood ratio equation. For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy religare demat brokerage charges likelihood ratio equation.

The values of the parameters that satisfy this equation will change based on the desired confidence level but at a given value of there is only a certain region of values for and for which the likelihood ratio equation holds true. This region can be represented graphically as a contour plot, an example of which is given in the following graphic.

The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of the likelihood ratio equation. Contour plots can be used for comparing data sets. The engineer would like to determine if the two designs are significantly different and at what confidence.

If, for example, there is no overlap i. An example of non-intersecting contours is shown next. The bounds on the parameters are calculated by finding the upper and lower bounds put option values of the contour plot on each axis for a given confidence level.

Since each axis represents the possible values of a given parameter, the boundaries of the contour plot represent the extreme values of the parameters that satisfy the following:. Unfortunately, there is no closed-form solution; therefore, these values must be arrived at numerically.

This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant.

In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists. Five units were put on a reliability test and experienced failures at 10, 20, 30, 40 and 50 hours. We can now rearrange the likelihood ratio equation to the form:. The next step is to find the set of values of and that satisfy this equation, or find the values of and such that.

The solution is dubai stock market emaar iterative process that requires setting the value of and finding the appropriate values ofand vice versa.

The following table gives values of based on given values of. Note that this plot is generated with degrees of freedomas we are only determining bounds on one parameter. As can be determined from the table, the lowest calculated value for is 1. Note that the points where are maximized and minimized do not necessarily correspond with the points where are maximized and minimized.

This is due to the fact that the contour plot is not symmetrical, so that the parameters will have their extremes at different points.

The manner in which the bounds on the time estimate for a given reliability are calculated is much the same as the manner in which the bounds on the parameters are calculated. The difference lies in the form of the likelihood functions that comprise the likelihood ratio. In the preceding section, we used the standard form of the likelihood function, which sm forex locations in terms of the parameters and.

In order to calculate the bounds on a time estimate, the likelihood function needs to be rewritten in terms of one parameter and time, so that the maximum and minimum values of the time can be observed as the parameter is varied.

This process is best illustrated with an example. The ML estimate for the time at which is As was mentioned, we need to rewrite the likelihood ratio equation so that it is in terms of and This is accomplished by using a form of the Weibull reliability equation, This can be rearranged in terms ofwith being considered a known variable or:.

This can then be substituted into the term in the likelihood ratio equation to form a likelihood equation in terms of and or:. Note that the likelihood value for is the same as it was for Example 1. This is because we are dealing with the same data and parameter estimates or, in other words, the maximum value of the likelihood function did not change. It now remains to find the values of and which satisfy this equation.

This is an iterative process that requires setting the value of and finding the appropriate values of. The following table gives the values of based on given values of.

As can be determined from the table, the lowest calculated value for is The likelihood ratio bounds on a reliability estimate for a given time value are calculated in the same manner as were the bounds on time. The only difference is that the likelihood function must now be considered in terms of and. The likelihood function is once again altered in the same way as before, only now is considered to be a parameter instead ofsince the value of must be specified in advance.

Once again, this process is best illustrated with an example. The ML estimate for the reliability at is As was mentioned, we need to rewrite the likelihood ratio equation so that it is in terms of and This is again accomplished by substituting the Weibull reliability equation into the term in the likelihood ratio equation to form a likelihood equation in terms of and:.

It now remains to find the values of and that satisfy this equation. As can be determined from the table, the lowest calculated value for is 2. A fourth method of estimating confidence bounds is based on the Bayes theorem. This type of confidence bounds relies on a different school of thought in statistical analysis, where prior information is combined with sample data in order to make inferences on model parameters and their functions.

An introduction to Bayesian methods is given in the Parameter Estimation chapter. Bayesian confidence bounds are derived from Bayes's rule, which states that:. In other words, the prior knowledge is provided in the form of the prior pdf of the parameters, which in turn is combined with the sample data in order to obtain the posterior pdf. Different forms of prior information exist, such as past data, expert opinion or non-informative refer to Parameter Estimation. It can be seen from the above Bayes's rule formula that we are now dealing with distributions of parameters rather than single value parameters.

For example, consider a one-parameter distribution with a positive parameter. Given a set of sample data, and a prior distribution for the above Bayes's rule formula can be written as:. In other words, we now have the distribution of and we can now make statistical inferences on this parameter, such as calculating probabilities. Specifically, the probability that is less than or equal to a value can be obtained by integrating the posterior probability density function pdfor:.

The above equation is the posterior cdfwhich essentially calculates a confidence bound on the parameter, where is the confidence level and is the confidence bound. Substituting the posterior pdf into the above posterior cdf yields:.

The only question at this point is, what do we use as a prior distribution of? For the confidence bounds calculation application, non-informative prior distributions are utilized.

Non-informative prior distributions are distributions that have no population basis and play a minimal role in the posterior distribution. The idea behind the use of non-informative prior distributions is to make inferences that are not affected by external information, or when external information is not available.

In the general case of calculating confidence bounds using Bayesian methods, the method should be independent of external information and it should only rely on the current data. Therefore, non-informative priors are used.

Specifically, the uniform distribution is used as a prior distribution for the different parameters of the selected fitted distribution. For example, if the Weibull distribution is fitted to the data, the prior distributions for beta and eta are assumed to be uniform. The above equation can be generalized for any distribution having a vector of parameters yielding the general equation for calculating Bayesian confidence bounds:.

If is given, then from the above equation and and for a giventhe bounds on are calculated. If is given, then from the above equation and and for a given the bounds on are calculated.

For a given failure time distribution and a given reliabilityis a function of and the distribution parameters. To illustrate the procedure for obtaining confidence bounds, the two-parameter Weibull distribution is used as an example. For the two-parameter Weibull distribution:. For a given reliability, the Bayesian one-sided upper bound estimate for is:.

The above equation can be rewritten in terms of as:. Applying the Bayes's rule by assuming that the priors of and are independent, we then obtain the following relationship:.

The above equation can be solved forwhere:. The same method can be used to get the one-sided lower bound of from:. The above equation can be solved to get. The Bayesian two-sided bounds estimate for is:.

Using the same method for the one-sided bounds, and can be solved. For a given failure time distribution and a given timeis a function of and the distribution parameters. The bounds in other types of distributions can be obtained in similar fashion. For example, for two parameter Weibull distribution:. The Bayesian one-sided upper bound estimate for is:. Using the same method for one-sided bounds, and can be solved. This utility can assist the analyst to a better understand life data analysis concepts, b experiment with the influences of sample sizes and censoring schemes on analysis methods, c construct simulation-based confidence intervals, d better understand the concepts behind confidence intervals and e design reliability tests.

This section describes how to use simulation for estimating confidence bounds. SimuMatic generates confidence bounds and assists in visualizing and understanding them. In addition, it allows one to determine the adequacy of certain parameter estimation methods such as rank regression on X, rank regression on Y and maximum likelihood estimation and to visualize the effects of different data censoring schemes on the confidence bounds.

Comparing Parameter Estimation Methods Using Simulation Based Bounds.

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The purpose of this example is to determine the best parameter estimation method for a sample of ten units with complete time-to-failure data for each unit i. The data set follows a Weibull distribution with and hours. To obtain the results, use the following settings in SimuMatic. The following plot shows the simulation-based confidence bounds for the RRX parameter estimation method, as well as the expected variation due to sampling error.

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Create another SimuMatic folio and generate a second data using the same settings, but this time, select the RRY analysis method on the Analysis tab. The following plot shows the result. The results clearly demonstrate that the median RRX estimate provides the least deviation from the truth for this sample size and data type.

However, the MLE outputs are grouped more closely together, as evidenced by the bounds. This experiment can be repeated in SimuMatic using multiple censoring schemes including Type I and Type II right censoring as well as random censoring with various distributions.

Multiple experiments can be performed with this utility to evaluate assumptions about the appropriate parameter estimation method to use for data sets.

Confidence Bounds From ReliaWiki.

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