1. No category

How to compute the confidence intervals for Mean
Time to Failure (MTTF)
By Eduardo Santiago
The following section describes the theory needed to construct confidence intervals for MTTF in
Minitab. This section can be skipped as the math is explained with two examples in sections 2 and 3.
1
Theory
To construct a confidence interval for the Mean Time to Failure (MTTF) of a process, Minitab uses
the Delta Method to estimate the standard error of the MTTF. If the estimate for MTTF is a function
of µ and σ, e.g. g1 = f (µ) + f (σ), then for a location-scale distribution a confidence interval can be
constructed as:
gˆ1 ± zα/2 · SEgˆ1 .
(1)
where gˆ1 = f (ˆ
µ) + f (ˆ
σ ).
To calculate the standard error for the MTTF we use the Delta Method:
s
2
2
∂g1 (µ, σ)
∂g1 (µ, σ) ∂g1 (µ, σ)
∂g1 (µ, σ)
V ar(ˆ
µ) + 2
Cov(ˆ
µ, σ
ˆ) +
V ar(ˆ
σ)
SEgˆ1 =
∂µ
∂µ
∂σ
∂σ
(2)
The three estimates corresponding to V ar(ˆ
µ), Cov(ˆ
µ, σ
ˆ ) and V ar(ˆ
σ ) can be obtained from the
Variance-Covariance matrix which can be stored from the dialog box. Minitab stores these elements in
a matrix as shown below:
"
#
V ar(ˆ
µ)
Cov(ˆ
µ, σ
ˆ)
M=
(3)
Cov(ˆ
µ, σ
ˆ)
V ar(ˆ
σ)
For a log-location-scale distribution – which includes the Weibull and Lognormal distributions –
Minitab approximates a 1 − α confidence interval by exponentiating the confidence interval summarized
in Equation 1, as:
P r egˆ1 −zα/2 ·SEgˆ1 ≤ MTTF ≤ egˆ1 +zα/2 ·SEgˆ1 = 1 − α
(4)
In general, follow the steps indicated next to construct an approximate confidence interval for the
MTTF.
1. Estimate the function gˆ1 (µ, σ) from the parameter estimates in the Session window.
2. Estimate the Standard Error of gˆ1 (µ, σ) using the Delta Method stated in Equation 2.
3. The confidence interval for gˆ1 (µ, σ) can be estimated from Equation 1.
4. After constructing the confidence interval in Equation 1, for log-location-scale distributions we can
estimate an approximate confidence interval for h(g1 (µ, σ)) by applying the same transformation
to the entire confidence interval in Equation 1, as illustrated in Equation 4.
1
2
MTTF for Lognormal data
The mean of a Lognormal distribution can be defined in terms of its location (µ) and scale (σ) parameters. If X represents a random variable that follows a Lognormal distribution then, its expected value
is represented as:
E(X) = eµ+σ
2
/2
.
Therefore, it is straightforward to find an estimate of the mean, using the respective estimates of µ
ˆ
and σ
ˆ . This implies that:
2
b
E(X)
= eµˆ+ˆσ /2 .
To construct the confidence interval for the MTTF see the next steps:
1. Estimate the function gˆ1 (µ, σ). For Lognormal data this estimate is gˆ1 (µ, σ) = µ
ˆ+σ
ˆ 2 /2.
2. Estimate the Standard Error of gˆ1 (µ, σ). Using the Delta Method, the standard error can be
calculated using the equation below:
q
2
2
µ) + 2(1)(σ)Cov(ˆ
µ, σ
ˆ ) + (σ) V ar(ˆ
σ)
(5)
SEgˆ1 = (1) V ar(ˆ
3. A 95% confidence interval for gˆ1 (µ, σ) can be estimated from:
gb1 (µ, σ) − 1.96 · SEgˆ1 ≤ g1 (µ, σ) ≤ gb1 (µ, σ) + 1.96 · SEgˆ1 .
(6)
4. After constructing the confidence interval in Equation 6, exponentiate both sides of the interval
to get the final approximated confidence interval.
2.1
Example
To illustrate this procedure, consider the data worksheet Reliable.mtw. In particular consider the
variable Temp80 and fit a Lognormal distribution to the data.
The information about the estimates needed to construct the confidence interval for the MTTF is
provided in the Session window.
Parameter Estimates
Parameter
Location
Scale
Estimate
4.03430
0.413458
Standard
Error
0.0599960
0.0414962
95.0% Normal CI
Lower
Upper
3.91671
4.15189
0.339626 0.503340
Characteristics of Distribution
Mean(MTTF)
Standard Deviation
Median
Estimate
61.5452
26.5736
56.5033
Standard
Error
3.82954
3.70119
3.38997
95.0% Normal CI
Lower
Upper
54.4791 69.5279
20.2253 34.9145
50.2348 63.5539
Now for this example to replicate the output for MTTF follow the steps below:
2
1. Estimate gˆ1 (µ, σ) as shown below:
gˆ1 (µ, σ) = µ
ˆ+σ
ˆ 2 /2 = 4.03430 + 0.4134582 /2 = 4.1198.
2. Estimate the standard error of gˆ1 (µ, σ).
s
2
2
∂g1 (µ, σ)
∂g1 (µ, σ) ∂g1 (µ, σ)
∂g1 (µ, σ)
SEgˆ1 =
V ar(ˆ
µ) + 2
Cov(ˆ
µ, σ
ˆ) +
V ar(ˆ
σ)
∂µ
∂µ
∂σ
∂σ
p
=
(1)2 (0.0036) + 2(1)(0.413458)(−0.0000268) + (0.413458)2 (0.001722) = 0.0622232.
The three estimates corresponding to V ar(ˆ
µ), Cov(ˆ
µ, σ
ˆ ) and V ar(ˆ
σ ) can be obtained from the
Variance-Covariance matrix which can be stored from the dialog box. For this particular example:
"
#
V ar(ˆ
µ)
Cov(ˆ
µ, σ
ˆ)
M=
Cov(ˆ
µ, σ
ˆ)
V ar(ˆ
σ)
Last, the partial derivatives can be obtained directly as
∂[µ+σ 2 /2]
∂µ
= 1 and
∂[µ+σ 2 /2]
∂σ
= σ.
3. Estimate an initial 95% confidence interval:
σ
ˆ2
− 1.96 · SEg1 ≤
2
4.1198 − 1.96(0.06222) ≤
µ
ˆ+
3.998 ≤
σ
ˆ2
+ 1.96 · SEg1
2
g1 (µ, σ) ≤ 4.1198 + 1.96(0.06222)
g1 (µ, σ) ≤ µ
ˆ+
g1 (µ, σ) ≤ 4.242
4. Then after constructing the confidence interval above, transform this interval appropriately.
σ
ˆ2
σ
ˆ2
ˆ+
ˆ+
− 1.96 · SEg1 ≤ exp{g1 (µ, σ)} ≤ exp µ
+ 1.96 · SEg1
exp µ
2
2
3
e3.998 ≤
MTTF
≤ e4.242
54.479 ≤
MTTF
≤ 69.528
MTTF for Weibull data
The mean of a Weibull random variable is defined by its shape (β) and scale (α) parameters. The Mean
Time To Failure (MTTF), or expected value, for a Weibull process is:
1
MTTF = α · Γ 1 +
,
(7)
β
where Γ(r) is the gamma function evaluated at r.
To use the Delta Method, we use the reparameterization of a Weibull to a Smallest Extreme Value
and use µ = log(α), σ = 1/β. Thus, the MTTF = eµ Γ(1 + σ). So we construct a confidence interval
for log(MTTF) = µ + log[Γ(1 + σ)].
r
2
∂g1
\
V ar(MTTF),
Since g1 = log(MTTF), the standard error is constructed as SEg1 =
∂MTTF
which yields an estimate of:
s
SEg1 =
1
MTTF
2
\ =
V ar(MTTF)
SEMTTF
.
MTTF
(8)
So, an approximate 1 − α confidence interval for the MTTF of a Weibull process is:
elog(MTTF)±zα/2 ·SEg1 .
3
(9)
3.1
Example
To illustrate this procedure, consider the same dataset presented previously. This time consider the
variable Temp80 and fit a Weibull distribution to the data.
The information about the estimates needed to construct the confidence interval for the MTTF is
provided in the Session window.
Parameter Estimates
Parameter
Shape
Scale
Estimate
10.8694
989.827
Standard
Error
0.849608
9.60135
95.0% Normal CI
Lower
Upper
9.32547 12.6689
971.186 1008.83
Characteristics of Distribution
Mean(MTTF)
Standard Deviation
Median
Estimate
944.913
105.063
957.007
Standard
Error
10.5037
7.18749
10.3784
95.0% Normal CI
Lower
Upper
924.549 965.726
91.8792 120.138
936.880 977.566
Now for this example, the steps below replicate the MTTF output:
1. Estimate gˆ1 (µ, σ) = 989.827 · Γ(1 + 1/10.894) = 944.913.
2. Estimate the standard error of gˆ1 (µ, σ).
SEgˆ1 =
10.5037
SEMTTF
=
= 0.011116.
MTTF
944.913
3. Estimate an initial 95% confidence interval:
log MTTF − 1.96 · SEg1 ≤
6.82931 ≤
g1 (µ, σ) ≤ log MTTF + 1.96 · SEg1
g1 (µ, σ) ≤ 6.87288
4. Then after constructing the confidence interval above, transform this interval appropriately.
exp {log MTTF − 1.96 · SEg1 } ≤ exp{g1 (µ, σ)}
4
≤ exp {log MTTF − 1.96 · SEg1 }
≤
MTTF
≤ e6.87288
944.913 ≤
MTTF
≤ 965.726
e
6.82931
References
[1] Meeker and Escobar, Statistical Methods for Reliability Data(1998) page 189
4