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Applied Mathematical Sciences, Vol. 7, 2013, no. 4, 161 - 166
Confidence Intervals for the Mean of Lognormal
Distribution with Restricted Parameter Space
Sa-aat Niwitpong
Department of Applied Statistics
King Mongkut’s University of Technology North Bangkok
1518 Piboonsongkhram Road, Bangsue, Bangkok, 10800, Thailand
snw@kmutnb.ac.th
Abstract
This paper presents new confidence intervals for the mean of lognormal distribution with restricted parameter. We proved the coverage
probability and expected length of our proposed confidence interval.
Monte Carlo simulation will be used to compare the proposed confidence interval to the existing confidence interval.
Mathematics Subject Classification: 62F25
Keywords: Lognormal distribution, the mean, restricted parameter space
1
Introduction
Let X = (X1 , X2 , . . . , Xn ) be a random variable having a lognormal distribution, and μ and σ 2 , respectively, are denoted by the mean and the variance
of Y where Y = ln(X) ∼ N(μ, σ 2 ). The probability density function of the
lognormal distribution, LN(μ, σ 2 ), is
⎧
⎪
⎨
(ln(x) − μ)2
exp
−
f (x, μ, σ 2 ) =
2σ 2
xσ 2π
⎪
⎩
0
1
√
;
for x >0
;
for x ≤0.
(1)
The mean for the lognormal population is E(X) = exp (μ + σ 2 /2) where E(X)
denotes the expectation of X. We are interesting to construct the confidence
interval of θ∗ = exp (μ + σ 2 /2) when parameter θ∗ is bounded i.e., a < θ∗ < b
where a and b are constant and a < b.
162
Sa-aat Niwitpong
2
Confidence interval for the mean of Lognormal distribution
Krishnamoorthy and Mathew [1] showed that to compare confidence intervals
of θ∗ , we can compare only in terms of θ = μ + σ 2 /2 when a < θ < b.
Zhou and Gao [5] proposed the confidence interval CIcox for θ, where
CIcox
Sy2
− c1
= Ȳ +
2
Sy4
Sy2
Sy2
+
, Ȳ +
+ c1
n
2(n − 1)
2
Sy4
Sy2
+
= [lcox , ucox ]
n
2(n − 1)
(2)
where c1 is the 100(1 − α)% percentile of the standard normal distribution.
Recently, Zhou et al. [6] proposed the new method, called the closed form
method of variance estimation (CFM), to constructed the confidence interval
for the mean of lognormal distribution which is
CIrov = [L, U]
(3)
2 2
ˆ
ˆ
θ1 − l1 + u2 − θ2 ,
where L = θ̂1 − θ̂2 −
2 2
Sy2
u1 − θˆ1 + θˆ2 − l2 , θ̂1 = Ȳ , θ̂2 =
,
U = θ̂1 − θ̂2 +
2
(n − 1)Sy2 (n − 1)Sy2
Sy
Sy
,
,
(l1 , u1 ) = Ȳ − c2 √ , Ȳ + c2 √ , (l2 , u2) =
n
n
2c3
2c4
c2 is the 100(1 − α)% percentile of the t-distribution, c3 and c4 are respectively
the 100(1 − α/2)% and 100(α/2)% percentiles of the Chi-squares distribution
with n − 1 degrees of freedom.
Zou et al. [6] reported that the confidence interval CIrov performs as well as
the generalized confidence interval proposed by Krishnamoorthy and Mathew
[1] in terms of coverage probability. Krishnamoorthy and Mathew [1] also
confirmed that the generalized confidence interval performs better than the
standard confidence interval CIcox . In this paper we modified both confidence
intervals to construct the confidence interval for θ with restricted parameter
space.
We also note that the confidence intervals for θ∗ using the methods of Zhou
and Gao [5] and Zou et al. [6] are respectively, CIlcox = [exp(lcox ), exp(ucox )]
and CIlrov = [exp(L), exp(U)].
163
Confidence intervals for the mean of Lognormal distribution
3
Confidence interval for the mean of Lognormal distribution with restricted parameter
space
Following Wang [4], the confidence interval for θ with restricted parameter,
0 < a < θ < b is
CIr = max a, θ̂ − c1 V ar(θ̂) , min b, θ̂ + c1 V ar(θ̂)
(4)
Similarly to the confidence interval CIr , the confidence interval for θ, using
the confidence intervals CIcox = (lcox , ucox ), CIrov = (L, U) and 0 < a < θ < b,
are respectively
CIrcox = [max (a, lcox ) , min (b, ucox )]
(5)
CIrrov = [max (a, L) , min (b, U)]
(6)
and
We also note that the confidence intervals for θ∗ with restricted parameter
space using the methods of Zhou and Gao [5] and Zou et al. [6] are respectively,
CIlrcox = [exp(max (a, lcox )), exp(min (b, ucox ))] and
CIlrrov = [exp(max (a, L)), exp(min (b, U))]
In the next section, we have proved two Theorems for the coverage probability and the expected length of each interval.
4
Main results
Theorem 1. The coverage probability of CIrcox is
E[Φ(W1 )−Φ(−W2 )] where W1 =
Ȳ +Sy2 /2−min(b,ucox )
σ2
2σ 4
+ 4(n−1)
n
, W2 =
Ȳ +Sy2 /2−max(a,lcox )
σ2
2σ 4
+ 4(n−1)
n
,
E[·] is an expectation operator and Φ(·) is the cumulative distribution function
of N(0, 1) and the expected length of CIrcox are respectively
4 (n+1)
2
1.1. 2c1 σn + σ2(n−1)
2 when max(a, lcox ) = lcox and min(b, ucox ) = ucox ,
4 (n+1)
2
σ2
σ2
1.2. n + 2 − a + c1 σn + σ2(n−1)
2 when max(a, lcox ) = a and min(b, ucox ) =
ucox ,
4 (n+1)
2
σ2
σ2
1.3. b- n − 2 +c1 σn + σ2(n−1)
2 when max(a, lcox ) = lcox and min(b, ucox ) =
b,
164
Sa-aat Niwitpong
1.4. b-a when max(a, lcox ) = a and min(b, ucox ) = b.
Proof. Using the fact that P (θ ∈ CIrcox ) = P (max(a, lcox ) ≤ θ ≤ min(b, ucox ))
and applied Theorems 1 and 2 of Niwitpong and Niwitpong [3] or Niwitpong
[2], Theorem 1 is proved.
Theorem 2. The coverage probability of CIrrov is
E[Φ(W3 )−Φ(−W4 )] where W3 =
Ȳ +Sy2 /2−min(b,urov )
σ2
2σ 4
+ 4(n−1)
n
, W2 =
Ȳ +Sy2 /2−max(a,lrov )
σ2
2σ 4
+ 4(n−1)
n
,
E[·] is an expectation operator and Φ(·) is the cumulative distribution function
of N(0, 1) and the expected length of CIrrov are respectively
2.1.
2
c22 σn +
(n−1)−c4
2c4
2
lrov and min(b, urov ) = urov ,
2.2.
urov ,
2
c22 σn +
(n−1)−c4
2c4
2.3. b-
2
c22 σn +
2
(n−1)−c4
2c4
2
σ4 (n+1)
n−1
-
σ4 (n+1)
n−1
- a when max(a, lrov ) = a and min(b, urov ) =
2
σ4 (n+1)
n−1
2
c22 σn +
c3 −(n−1)
2c3
σ4 (n+1)
n−1
when max(a, lrov ) =
when max(a, lrov ) = lrov and min(b, urov ) =
b,
2.4. b-a when max(a, lrov ) = a and min(b, urov ) = b.
Proof. Similarly to Theorem 1.
5
Simulation Studies
In this section, we compare confidence intervals for θ via Monte Carlo simulation, using functions written in R, in variety of situations to see how coverage
probabilities and expected lengths of confidence intervals CIrcox and CIrrov ,
see Theorems 1-2, may depend on sample sizes and on the constants a and
b. We chose σ 2 = 1, μ = 0, 3, 5 and n = 30, 50, 100, 200. We compare confidence intervals of CIrcox and CIrrov based on their coverage probabilities
and expected lengths, with a nominal value of 0.95 throughout. Comparison of coverage probabilities of above intervals, using Theorems 1-2, based on
M=10,000 simulations, are given in Table 1. The ratio of expected lengths for
each intervals, using Theorems 1-2, are also given in Table 1. From Table 1,
Confidence intervals for the mean of Lognormal distribution
Table 1:
intervals
n μ
30 0
3
5
50 0
3
5
100 0
3
5
200 0
3
5
30 0
3
5
50 0
3
5
100 0
3
5
200 0
3
5
165
The estimated coverage probability and length of a 95% confidence
CIrcox and CIrrov when σ 2 = 1 and M = 10, 000
a b CIrcox CIrrov E(CIrcox )/E(CIrrov )
0 6 0.9007 0.9312
1.6677
0 6 0.9358 0.9855
6.3066
0 6 0.8868 0.9303
0.9303
0 6 0.9307 0.9640
1.7848
0 6 0.9414 0.9846
8.9019
0 6 0.9268 0.9675
0.9856
0 6 0.9453 0.9803
1.8085
0 6 0.9458 0.9823
13.4510
0 6 0.9445 0.9807
2.0220
0 6 0.9478 0.9803
1.8800
0 6 0.9479 0.9803
19.6600
0 6 0.9479 0.9804
14.4650
-6 6 0.9350 0.9852
6.3066
-6 6 0.9358 0.9855
6.3067
-6 6 0.8888 0.9526
0.9249
-6 6 0.9417 0.9846
8.9019
-6 6 0.9413 0.9845
8.9019
-6 6 0.9256 0.9664
0.9957
-6 6 0.9458 0.9823
13.4510
-6 6 0.9454 0.9821
13.4519
-6 6 0.9450 0.9804
19.6600
-6 6 0.9478 0.9806
19.6600
-6 6 0.9479 0.9803
19.6601
-6 6 0.9479 0.9803
16.4370
it is not surprising that the coverage probabilities of the confidence interval,
CIrcox are below the nominal value of 0.95 for all cases. However, the coverage
probabilities of our proposed confidence interval CIrrov are above the nominal
value of 0.95 except for the case of small sample size n = 30 and a = 0, b = 6.
As a result, we conclude that the confidence interval CIrrov performs better
than the confidence interval CIrcox in terms of the coverage probability. In
addition, the ratio of expected lengths of the confidence interval CIrcox compared to CIrrov is greater than 1 for all cases, the confidence interval CIrrov is
definitely shorter than that of the confidence interval CIrcox .
166
Sa-aat Niwitpong
Acknowledgments
The author would like to thank grant number 55451026 from the Faculty of
Applied Sciences, King Mongkut’s University of Technology North Bangkok.
References
[1] K. Krishnamoorthy and T. Mathew, Inference on the means of lognormal
distributions using Generalized p-values and generalized confidence intervals, Journal of Statistical Planning and Inference, 115 (2003), 103–121.
[2] S. Niwitpong, A note on coverage probability of confidence interval for the
difference between two normal variances, Applied Mathematical Sciences,
6 (2012), 3313 - 3320
[3] S. Niwitpong and S. Niwitpong, Confidence interval for the difference of
two normal population means with a known ratio of variances, Applied
Mathematical Sciences, 4 (2010), 347 - 359.
[4] H. Wang, Confidence intervals for the mean of a normal distribution with
restricted parameer spaced, Journal of Statistical Computation and Simulation, 78(9) (2008), 829–841.
[5] X.H. Zhou and S. Gao, Confidence intervals for the lognormal mean,
Statistics in Medecien, 16 (1997), 783–790.
[6] G.Y. Zou, J. Taleban, and C.Y. Huo, Confidence interval estimation
for lognormal data with application to health economics, Computational
Statistics and Data Analysis, 53 (2009), 3755–3764.
Received: September, 2012