On this page:

• Frequentist Estimation of \( p \) (Probability)
• Frequentist Estimation of \( \lambda \) (Rate)

Frequentist Estimation of p (Probability)

Suppose that similar components or systems are repeatedly demanded, that the success or failure on any demand is independent of the outcomes on other demands, and that the probability of failure on any demand is \( p \), the same for all demands. Under these assumptions, the number of failures in \( n \) demands, \( X \), is a random variable with a Binomial(\( n \)) distribution. RADS performs some statistical checks on the plausibility of these assumptions.

The maximum likelihood estimator (MLE) of \( p \) is \( \hat{p} = X / n \). Before the data have been observed, the number of failures is random and \( \hat{p} \) is random. After the data have been observed, the value of \( \hat{p} \) is some number, which may or may not be close to the true \( p \).

If many sets of outcomes of \( n \) demands were collected, some would give \( \hat{p} \) close to \( p \) and some would give \( \hat{p} \) far from \( p \). The mean, or long-run average, value of \( \hat{p} \) is \( p \), the true unknown parameter. The variance of \( \hat{p} \) is \( p(1-p)/n \). This shows that as \( n \) becomes large, the variance of \( \hat{p} \) becomes small. As a result, with high probability \( \hat{p} \) is close to the true unknown \( p \) when \( n \) is large.

A confidence interval is another kind of estimator of \( p \), an interval estimator instead of a point estimator. Given any data set with \( x \) failures in \( n \) demands, a rather complicated formula (given below) yields a lower and upper bound, calculated from \(x \) and \( n \). The interval is random because it depends on the data - a different data set would give a different interval. The intervals are constructed to be long enough so that most of the intervals contain the true unknown parameter \( p \). More precisely, suppose that the interval \( I \) consists of a lower end point \( L \) and an upper end point \( U \), both calculated from the data. \( I \) is a 90% confidence interval for \( Pr( L \leq p \leq U) \geq 0.90 \). This means that if many data sets were randomly generated, at least 90% of the resulting intervals would contain the unknown true \( p \). If, instead, 95% confidence intervals were constructed, at least 95% of them would, in the long run, contain the true \( p \), and so forth for other confidence coefficients.

RADS calculates both the MLE and a confidence interval for \( p \). The default confidence coefficient is 90%, but the user may select any confidence coefficient. The higher the desired confidence, the longer will be the resulting intervals.

One version of the complicated formula for the confidence interval is as follows. The lower limit of a \( 100 ( 1 - \alpha ) \)% confidence interval is \[ p_{conf,\> \alpha / 2} = \frac{d_1}{d_1 + d_2 F_{1-\alpha/2}(d_2, d_1)} \] where \[d_1 = 2x \] \[d_2 = 2(n - x + 1) \] and \( F_{\alpha/2} \) is the \( \alpha/2 \) quantile of the \( F \) distribution with \( d_2 \) and \( d_1 \) degrees of freedom.

If \( x = 0 \) then \( d_1 = 0 \) and the \( F \) distribution is not defined; in that case, simply set \( p_{conf,\> \alpha / 2} = 0 \).

Similarly, the upper limit is \[ p_{conf,\> 1-\alpha / 2} = \frac{ d_1 F_{1-\alpha/2} } {d_2 + d_1 F_{1-\alpha/2}(d_1, d_2)} \] where \[d_1 = 2(x+1) \] \[d_2 = 2(n - x) \] and \( F_{1-\alpha/2} (d_1, d_2) \) is the \( 1 - \alpha/2 \) quantile of the \( F \) distribution with \( d_1 \) and \( d_2 \) degrees of freedom.

If \( x = n \), then \( d_2 = 0 \) and the \( F \) distribution is not defined; in that case, set \( p_{conf,\> \alpha / 2} = 1 \).

Frequentist Estimation of λ (Rate)

Suppose that one or more similar components or systems are observed over time, that events, such as failures, occur randomly. Suppose further, that exactly simultaneous events do not occur, and that the probability of an event in a time interval of length \( \Delta t \) is approximately \( \lambda \Delta t \), with \( \lambda \) unchanging, and with \( \lambda \) being the event occurrence rate. Finally, suppose that event occurrences in disjoint time periods are independent. Under these assumptions, the number of failures in time \(t \) is a random variable with a Poisson( \( \lambda t \) ) distribution. RADS performs some statistical checks on the plausibility of these assumptions.

Let \( x \) be the number of events in time \( t \). The maximum likelihood estimator (MLE) of \( \lambda \) is \( \hat{\lambda} = x/t \). Before the data have been observed, the number of events is random and \( \hat{\lambda} \) is random. After the data have been observed, the value of \( \hat{\lambda} \) is some number, which may or may not be close to the true \( \lambda \).

If many sets of event observations \(n \) were collected, some would give \( \hat{\lambda} \) close to \( \lambda \) and some would give \( \hat{\lambda} \) far from \( \lambda \). The mean, or long-run average, value of \( \hat{\lambda} \) is \( \lambda \), the true unknown parameter. The variance of \( \hat{\lambda} \) is \( \lambda / t \). This shows that as \(t \) becomes large, the variance of \( \hat{\lambda} \) becomes small. As a result, with high probability, \( \hat{\lambda} \) is close to the true unknown \( \lambda \) when \( t \) is large.

A confidence interval is another kind of estimator of \( \lambda \), an interval estimator instead of a point estimator. Given any data set with \( x \) failures in time \( t \), a rather complicated formula (given below) yields a lower and upper bound, calculated from \( x \) and \( t \). The interval is random because it depends on the data - a different data set would give a different interval. The intervals are constructed to be long enough so that most of the intervals contain the true unknown parameter \( \lambda \). More precisely, suppose that the interval \( I \) consists of a lower end point \( L \) and an upper end point \( U \), both calculated from the data. \( I \) is a 90% confidence interval for \( \lambda \) if \( Pr( L \leq \lambda \leq U) \geq 0.90 \). This means that if many data sets were randomly generated, at least 90% of the resulting intervals would contain the unknown true \(\lambda \). If, instead, 95% confidence intervals were constructed, at least 95% of them would, in the long run, contain the true \( \lambda \), and so forth for other confidence coefficients.

RADS calculates both the MLE and a confidence interval for \( \lambda \). The default confidence coefficient is 90%, but the user may select any confidence coefficient. The higher the desired confidence, the longer will be the resulting intervals.

The rather complicated formula for the confidence interval is now given. For a \( (1 - \alpha) \) confidence interval, or equivalently a \( 100 (1 - \alpha) \)% confidence interval, the lower limit is \[ \lambda_{conf,\> \alpha / 2} = \frac{\chi_{\alpha/2}^2 (2x)}{2t} \]

If \( x = 0\) this formula is undefined and simply set \( \lambda_{conf,\> \alpha / 2} = 0 \)

Similarly, the upper limit is \[ \lambda_{conf,\> 1 - \alpha / 2} = \frac{\chi_{1-\alpha/2}^2 (2x+2)}{2t} \]

Notice that an upper confidence limit is defined in the case \(x = 0\). It is reasonable that observing no occurrences of the event would provide some information about how large \( \lambda \) might be, but not about how small it might be.